Constant-overhead fault-tolerant quantum computation

Abstract

Constant-overhead fault-tolerant quantum computation is provided. A first plurality of physical qubits is configured to encode a first plurality of logical qubits using a quantum low-density parity-check (qLDPC) code. A second plurality of physical qubits is configured to encode a second plurality of logical qubits using an error-correcting code; and a plurality of ancilla qubits. The quantum computing system is configured to teleport at least one of the first plurality of logical qubits into a corresponding at least one of the second plurality of logical qubits via the plurality of ancilla qubits. The quantum computing system is configured to perform at least one logical operation on the at least one of the second plurality of logical qubits. The quantum computing system is configured to teleport the at least one of the second plurality of logical qubits into the at least one of the first plurality of qubits via the plurality of ancilla qubits.

Description

HQU-01425 HU 9476 CONSTANT-OVERHEAD FAULT-TOLERANT QUANTUM COMPUTATION CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. Provisional Application No.63 / 530,032, filed July 31, 2023, which is hereby incorporated by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under W911NF-23-1-0077 and W911NF-21-1-0325 awarded by U.S. Army Research Office (ARO) and under FA9550-19- 1-0399 and FA9550-21-1-0209 awarded by Air Force Office of Scientific Research (USAF / AFOSR) and under 1936118, 1941583, and 2137642 awarded by National Science Foundation (NSF). The government has certain rights in this invention. BACKGROUND

[0003] Embodiments of the present disclosure relate to quantum computation, and more specifically, to constant-overhead fault-tolerant quantum computation on platforms including neutral atom arrays. BRIEF SUMMARY

[0004] In various embodiments, a quantum computing system is provided, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a quantum low-density parity-check (qLDPC) code; a second plurality of physical qubits, configured to encode a second plurality of logical qubits using an error-correcting code; and a plurality of ancilla qubits. The quantum computing system is configured to teleport at leastHQU-01425 HU 9476 one of the first plurality of logical qubits into a corresponding at least one of the second plurality of logical qubits via the plurality of ancilla qubits. The quantum computing system is configured to perform at least one logical operation on the at least one of the second plurality of logical qubits. The quantum computing system is configured to teleport the at least one of the second plurality of logical qubits into the at least one of the first plurality of qubits via the plurality of ancilla qubits.

[0005] In some embodiments, the plurality of ancilla qubits is configured to encode a hypergraph product of a classical code corresponding to a logical operator of the qLDPC code and a classical code corresponding to a logical operator of the error-correcting code.

[0006] In some embodiments, teleporting the at least one of the first plurality of qubits and / or the at least one of the second plurality of logical qubits comprises performing a measurement- based circuit with the plurality of ancilla qubits. In some embodiments, performing the measurement-based circuit comprises performing lattice surgery.

[0007] In some embodiments, the quantum computing system is further configured to perform a plurality of quantum error correction (QEC) cycles on the first plurality of physical qubits. In some embodiments, the plurality of QEC cycles comprises a plurality of intervals, and wherein performing the plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals. In some embodiments, applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph. In some embodiments, performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered-statistical decoding (OSD). In some embodiments, applying the final BP and OSD comprises constructing a decoding graph of data errors.

[0008] In some embodiments, the qLDPC code is a hypergraph product (HGP) code. In some embodiments, the qLDPC code is a lifted-product (LP) code. In some embodiments,HQU-01425 HU 9476 the error-correcting codes is a topological code. In some embodiments, the topological code is a surface code.

[0009] In some embodiments, the first plurality of physical qubits comprises a first array of neutral atoms and the second plurality of physical qubits comprises a second array of neutral atoms, wherein each neutral atom has a first state and an excited Rydberg state, and each neutral atom is arranged to impose a Rydberg blockade on at least its nearest neighbors in its array when in the excited Rydberg state.

[0010] In some embodiments, the first plurality of physical qubits comprises a plurality of data qubits and a plurality of ancilla qubits, and wherein encoding the first plurality of logical qubits using the qLDPC comprises: providing the plurality of data qubits, each of the plurality of data qubits disposed in a corresponding trap; providing the plurality of ancilla qubits, each of the plurality of ancilla qubits disposed in a corresponding trap; arranging the plurality of data qubits and the plurality of ancilla qubits in a plurality of rows and a plurality of columns, thereby forming a lattice; performing a plurality of permutations of the plurality of rows and the plurality of columns, each of the plurality of permutations placing each of the plurality of data qubits within an interaction radius of one of the plurality of ancilla qubits, thereby forming a plurality of proximate pairs; and subsequent to each of the plurality of permutations, applying a global control pulse to the lattice, thereby applying a gate to each of the plurality of proximate pairs, and thereby encoding a parity check matrix between the plurality of ancilla qubits and the plurality of data qubits.

[0011] In some embodiments, encoding the first plurality of logical qubits using the qLDPC further comprises, prior to forming the lattice: applying a control laser pulse to the plurality of data qubits to thereby prepare them in an initial state.HQU-01425 HU 9476

[0012] In some embodiments, arranging the plurality of data qubits and the plurality of ancilla qubits in the lattice comprises: moving, in parallel, the plurality of ancilla qubits into the lattice.

[0013] In some embodiments, performing the plurality of permutations comprises: moving, in parallel, one or more rows within the lattice and moving, in parallel, one or more columns within the lattice.

[0014] In some embodiments, the plurality of ancilla qubits comprises Z stabilizers and X stabilizers.

[0015] In some embodiments, encoding the first plurality of logical qubits using the qLDPC further comprises: removing a subset of the plurality of ancilla qubits from the lattice and performing a measurement on the subset. In some embodiments, the subset corresponds to Z stabilizers. In some embodiments, the subset corresponds to X stabilizers.

[0016] In some embodiments, performing the plurality of permutations comprises: determining a collision-free path for each of the plurality of data qubits and each of the plurality of ancilla qubits for each of the plurality of permutations. In some embodiments, determining the collision-free path comprises: bipartition and recursive sorting of the plurality of data qubits and the plurality of ancilla qubits.

[0017] In some embodiments, the collision-free path is a cubic spline. In some embodiments, determining the collision-free path comprises: decomposing the parity check matrix into a first product graph and a second product graph; determining a routing of the plurality of rows according to the first product graph; and determining a routing of the plurality of columns according to the second product graph. In some embodiments, the plurality of permutations comprises row-specific and / or column-specific permutations. In some embodiments, performing the plurality of permutations comprises: applying a pinning beam to at least oneHQU-01425 HU 9476 qubit of the plurality of data qubits or the plurality of ancilla qubits, thereby maintaining the position of the at least one qubit according to the routing of the plurality of rows or columns.

[0018] In some embodiments, the gate applied to each of the plurality of proximate pairs is a CZ gate.

[0019] In some embodiments, the global control pulse is a laser pulse.

[0020] In some embodiments, the trap corresponding to each of the plurality of data qubits and to each of the plurality of ancilla qubits is an optical trap. In some embodiments, the optical traps corresponding to the plurality of data qubits and to the plurality of ancilla qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), and moving the one or more rows and moving the one or more columns comprises varying a drive frequency of the at least one AOD. In some embodiments, one or more rotations is applied during said moving. In some embodiments, applying the one or more rotations comprises applying a Raman pulse.

[0021] In various embodiments a method of performing quantum computation is provided. A first plurality of physical qubits is provided. A first plurality of logical qubits is encoded using a quantum low-density parity-check (qLDPC) code on the first plurality of physical qubits. A second plurality of physical qubits is provided. A second plurality of logical qubits is encoded using an error-correcting code on the second plurality of physical qubits. A plurality of ancilla qubits is provided. At least one of the first plurality of logical qubits is teleported into a corresponding at least one of the second plurality of logical qubits via the plurality of ancilla qubits. At least one logical operation is performed on the at least one of the second plurality of logical qubits. The at least one of the second plurality of logical qubits is teleported into the at least one of the first plurality of qubits via the plurality of ancilla qubits.HQU-01425 HU 9476

[0022] In various embodiments, a method of performing quantum error correction is provided. A first plurality of physical qubits is provided. A first plurality of logical qubits is encoded using a quantum low-density parity-check (qLDPC) code on the first plurality of physical qubits. A plurality of quantum error correction (QEC) cycles is performed on the first plurality of physical qubits, the plurality of QEC cycles comprising a plurality of intervals. The plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals.

[0023] In some embodiments, applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph. In some embodiments, performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered- statistical decoding (OSD). In some embodiments, applying the final BP and OSD comprises constructing a decoding graph of data errors.

[0024] In various embodiments, a computing system is provided, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a first homological product code, the first homological product code being a product of a first plurality of classical codes; and a second plurality of physical qubits, configured to encode a second plurality of logical qubits using a second homological product code. The quantum computing system is configured to apply a first homomorphic gate to at least a subset of the first plurality of logical qubits and at least a subset of the second plurality of logical qubits. In some embodiments, the quantum computing system is further configured to measure a subset of the second plurality of physical qubits, thereby measuring the subset of the first plurality of logical qubits.

[0025] In some embodiments, the second homological product code is determined by puncturing some of the first plurality of classical codes. In some embodiments, the second homological product code is determined by adding checks to at least some of the firstHQU-01425 HU 9476 plurality of classical codes. In some embodiments, the first homological product code and / or the second homological product code is a hypergraph product (HGP) code.

[0026] In some embodiments, the quantum computing system is further configured to perform a plurality of quantum error correction (QEC) cycles on the first plurality of physical qubits. In some embodiments, the plurality of QEC cycles comprises a plurality of intervals, and wherein performing the plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals. In some embodiments, applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph. In some embodiments, performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered-statistical decoding (OSD). In some embodiments, applying the final BP and OSD comprises constructing a decoding graph of data errors.

[0027] In some embodiments, the first plurality of physical qubits comprises a first array of neutral atoms and the second plurality of physical qubits comprises a second array of neutral atoms, wherein each neutral atom has a first state and an excited Rydberg state, and each neutral atom is arranged to impose a Rydberg blockade on at least its nearest neighbors in its array when in the excited Rydberg state. In some embodiments, performing the first homomorphic gate comprises placing the first and second pluralities of physical qubits such that each physical qubit of the first plurality of physical qubits is within a blockade radius of exactly one corresponding physical qubit of the second plurality of physical qubits and illuminating the first and second plurality of physical qubits with a first laser.

[0028] In some embodiments, the first plurality of physical qubits comprises a plurality of data qubits and a plurality of ancilla qubits, and encoding the first plurality of logical qubits using the first homological product code comprises: providing the plurality of data qubits, each of the plurality of data qubits disposed in a corresponding trap; providing the plurality of ancilla qubits, each of the plurality of ancilla qubits disposed in a corresponding trap;HQU-01425 HU 9476 arranging the plurality of data qubits and the plurality of ancilla qubits in a plurality of rows and a plurality of columns, thereby forming a lattice; performing a plurality of permutations of the plurality of rows and the plurality of columns, each of the plurality of permutations placing each of the plurality of data qubits within an interaction radius of one of the plurality of ancilla qubits, thereby forming a plurality of proximate pairs; and subsequent to each of the plurality of permutations, applying a global control pulse to the lattice, thereby applying a gate to each of the plurality of proximate pairs, and thereby encoding a parity check matrix between the plurality of ancilla qubits and the plurality of data qubits. In some embodiments, encoding the first plurality of logical qubits using the first homological product code further comprises, prior to forming the lattice: applying a control laser pulse to the plurality of data qubits to thereby prepare them in an initial state.

[0029] In some embodiments, arranging the plurality of data qubits and the plurality of ancilla qubits in the lattice comprises: moving, in parallel, the plurality of ancilla qubits into the lattice. In some embodiments, performing the plurality of permutations comprises: moving, in parallel, one or more rows within the lattice and moving, in parallel, one or more columns within the lattice.

[0030] In some embodiments, the plurality of ancilla qubits comprises Z stabilizers and X stabilizers. In some embodiments, encoding the first plurality of logical qubits using the first homological product code further comprises: removing a subset of the plurality of ancilla qubits from the lattice and performing a measurement on the subset. In some embodiments, the subset corresponds to Z stabilizers. In some embodiments, the subset corresponds to X stabilizers.

[0031] In some embodiments, performing the plurality of permutations comprises: determining a collision-free path for each of the plurality of data qubits and each of theHQU-01425 HU 9476 plurality of ancilla qubits for each of the plurality of permutations. In some embodiments, determining the collision-free path comprises: bipartition and recursive sorting of the plurality of data qubits and the plurality of ancilla qubits.

[0032] In some embodiments, the collision-free path is a cubic spline. In some embodiments, determining the collision-free path comprises: decomposing the parity check matrix into a first product graph and a second product graph; determining a routing of the plurality of rows according to the first product graph; and determining a routing of the plurality of columns according to the second product graph. In some embodiments, the plurality of permutations comprises row-specific and / or column-specific permutations. In some embodiments, performing the plurality of permutations comprises: applying a pinning beam to at least one qubit of the plurality of data qubits or the plurality of ancilla qubits, thereby maintaining a position of the at least one qubit according to the routing of the plurality of rows or columns.

[0033] In some embodiments, the gate applied to each of the plurality of proximate pairs is a CZ gate.

[0034] In some embodiments, the global control pulse is a laser pulse.

[0035] In some embodiments, the trap corresponding to each of the plurality of data qubits and to each of the plurality of ancilla qubits is an optical trap. In some embodiments, the optical traps corresponding to the plurality of data qubits and to the plurality of ancilla qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), and moving the one or more rows and moving the one or more columns comprises varying a drive frequency of the at least one AOD. In some embodiments, one or more rotations is applied during said moving. In some embodiments, applying the one or more rotations comprises applying a Raman pulse.

[0036] In some embodiments, the quantum computing system further comprises a third plurality of physical qubits, configured to encode a third plurality of logical qubits using aHQU-01425 HU 9476 third homological product code, wherein the first homomorphic gate is a CNOT, the quantum computing system is configured to apply a second homomorphic gate to at least a subset of the second plurality of logical qubits and at least a subset of the third plurality of logical qubits, the second homomorphic gate being a CNOT, and the quantum computing system is configured to measure a subset the third plurality of physical qubits, thereby measuring the subset of the second plurality of logical qubits.

[0037] In various embodiments, a quantum computing system is provided, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a first homological product code, wherein the first homological product code has a quasi-cyclic classical base code, the quantum computing system is configured to perform a permutation of the first plurality of physical qubits, the first permutation corresponding to a block-cyclic shift of the classical base code, thereby performing a translation of the first plurality of logical qubits.

[0038] In various embodiments, a method of performing quantum computation is provided, comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits using a first homological product code on the first plurality of physical qubits; providing a second plurality of physical qubits; encoding a second plurality of logical qubits using a second homological product code on the second plurality of physical qubits; and applying a first homomorphic gate to at least a subset of the first plurality of logical qubits and at least a subset of the second plurality of logical qubits.

[0039] In some embodiments, a subset of the second plurality of physical qubits is measured, thereby measuring the subset of the first plurality of logical qubits.

[0040] In some embodiments, the first homomorphic gate is a CNOT, and the method further comprises: providing a third plurality of physical qubits; encoding a third plurality of logical qubits on the third plurality of physical qubits using a third homological product code;HQU-01425 HU 9476 applying a second homomorphic gate to at least a subset of the second plurality of logical qubits and at least a subset of the third plurality of logical qubits; and measuring a subset the third plurality of physical qubits, thereby measuring the subset of the second plurality of logical qubits.

[0041] In various embodiments, a method of performing quantum computation is provided, comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits on the first plurality of physical qubits using a first homological product code, the first homological product code having a quasi-cyclic classical base code; and performing a permutation of the first plurality of physical qubits, the first permutation corresponding to a block-cyclic shift of the classical base code, thereby performing a translation of the first plurality of logical qubits. BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0042] Fig.1 is schematic view of an architecture for qLDPC-based fault-tolerant quantum computation according to embodiments of the present disclosure.

[0043] Fig.2 is a pair of heat plots illustrating the logical failure rate of qLDPC-based architectures according to embodiments of the present disclosure.

[0044] Figs.3A-B are graphs of physical error rate according to embodiments of the present disclosure.

[0045] Fig.4 is a table of physical qubit requirements for various code schemes according to embodiments of the present disclosure.

[0046] Fig.5A is a schematic view of code layout according to embodiments of the present disclosure.

[0047] Fig.5B is a circuit diagram of a measurement-based teleportation circuit according to embodiments of the present disclosure.HQU-01425 HU 9476

[0048] Fig.5C is a graph of physical error rate according to embodiments of the present disclosure.

[0049] Fig.6 is a schematic diagram of the proof of circuit thresholds according to embodiments of the present disclosure.

[0050] Fig.7A is an exemplary space-time decoding graph according to embodiments of the present disclosure.

[0051] Fig.7B is a plot of a subject of a decoding graph according to embodiments of the present disclosure.

[0052] Figs.8A-B are graphs of number of code cycles according to embodiments of the present disclosure.

[0053] Fig.9 is a graph of physical error rate according to embodiments of the present disclosure.

[0054] Figs.10A-B are graphs of gate error rate according to embodiments of the present disclosure.

[0055] Figs.11A-B are graphs of logical failure rates according to embodiments of the present disclosure.

[0056] Fig.12 is a graph of logical failure rates according to embodiments of the present disclosure.

[0057] Fig.13 is a schematic view of a lattice surgery according to embodiments of the present disclosure.

[0058] Fig.14 is a schematic view of a quantum information architecture according to embodiments of the present disclosure.

[0059] Fig.15 is a level diagram showing key87Rb atomic levels according to embodiments of the present disclosure.HQU-01425 HU 9476

[0060] Fig.16 is a schematic view of a quantum processing unit (QPU) according to embodiments of the present disclosure.

[0061] Fig.17 is a schematic view of a lattice of qubits illustrating the product structure of hypergraph product codes according to embodiments of the present disclosure.

[0062] Figs.18A-C are schematic views of lattices of qubits, illustrating column and row permutations according to embodiments of the present disclosure.

[0063] Fig.19 is a schematic view of a 1-dimensional array of qubits over time, showing a syndrome extraction gate sequence according to embodiments of the present disclosure.

[0064] Fig.20 is a schematic view of a 1-dimensional array of qubits over time, showing the steps of a rearrangement according to embodiments of the present disclosure.

[0065] Fig.21 is a schematic view of a 1-dimensional array of qubits over time, showing the steps of another rearrangement according to embodiments of the present disclosure.

[0066] Fig.22 is a schematic view of an exemplary 2-dimensional graph according to embodiments of the present disclosure.

[0067] Fig.23 is a schematic view of a 1-dimensional array of qubits over time, showing the use of pinning beams to prevent movement in some columns according to embodiments of the present disclosure.

[0068] Fig.24 is a schematic view of a lattice of qubits illustrating the product structure of lifted product (LP) codes according to embodiments of the present disclosure.

[0069] Figs.25A-B show pseudocode for an algorithm for arbitrary 1D atom rearrangement in a logarithmic number of steps according to embodiments of the present disclosure.

[0070] Figs.26A-B show pseudocode for a product coloration circuit for HGP syndrome extraction according to embodiments of the present disclosure.

[0071] Figs.27A-B show pseudocode for a pipelined product coloration circuit for multi- round HGP syndrome extraction according to embodiments of the present disclosure.HQU-01425 HU 9476

[0072] Figs.28A-D are schematic diagrams illustrating the efficient implementation of quantum LDPC codes with atom arrays according to embodiments of the present disclosure.

[0073] Figs.29A-B are schematic diagrams illustrating the ordering of operations in pipelined syndrome extraction according to embodiments of the present disclosure.

[0074] Fig.30 is a schematic view of an apparatus for quantum computation according to embodiments of the present disclosure.

[0075] Figs.31A-C are schematic diagrams illustrating homomorphic CNOT and homomorphic measurement for 2D and 3D homological product codes according to embodiments of the present disclosure.

[0076] Figs.32A-C are schematic diagrams illustrating parallel Grid Pauli product measurements for HGP codes according to embodiments of the present disclosure.

[0077] Fig.33 is a table of the space-time cost of Clifford gates on logical qubits using surface codes and HGP codes according to embodiments of the present disclosure.

[0078] Fig.34 is a schematic diagram illustrating a protocol for generating a GHZ state on all logical qubits of a HGP code according to embodiments of the present disclosure.

[0079] Fig.35A-D are schematic diagrams illustrating one round of parallel magic state distillation on diagonal logical qubits of HGP codes according to embodiments of the present disclosure.

[0080] Fig.36A-B are schematic diagrams illustrating parallel inter-block Toffoli gates on HGP codes according to embodiments of the present disclosure.

[0081] Fig.37A-B are schematic diagrams illustrating an efficient parallel implementation of a quantum adder using HGP codes according to embodiments of the present disclosure.

[0082] Fig.38 is a schematic diagram that shows a chain-complex of an HGP code according to embodiments of the present disclosure.HQU-01425 HU 9476

[0083] Fig.39 is pseudocode for an algorithm for Pauli product measurements (PPMs) for HGP codes according to embodiments of the present disclosure.

[0084] Fig.40 is pseudocode for an algorithm for Grid PPMs for HGP codes according to embodiments of the present disclosure.

[0085] Fig.41 is a table of the HGP logical gadgets used for logical computation according to embodiments of the present disclosure.

[0086] Fig.42 is a schematic diagram that illustrates logical translation for HGP codes with quasi-cyclic base codes according to embodiments of the present disclosure.

[0087] Fig.43 is table of the parameters and code matrices of finite-size OGSC classical codes according to embodiments of the present disclosure.

[0088] Fig.44 is pseudocode for an algorithm for selective teleportation between two identical HGP codes according to embodiments of the present disclosure.

[0089] Fig.45A is pseudocode for an algorithm for measuring the diagonal qubits of an HGP code according to embodiments of the present disclosure.

[0090] Fig.45B is a schematic diagram that illustrates measuring all the non-diagonal qubits of an HGP code in log depth according to embodiments of the present disclosure.

[0091] Fig.46 is a schematic diagram that illustrates one sequence of operations of the iterative gadget according to embodiments of the present disclosure.

[0092] Fig.47 is a schematic diagram that illustrates one sequence of operations for swapping logical qubits of a symmetrical HGP code along the diagonal according to embodiments of the present disclosure.

[0093] Figs.48A-C are schematic diagrams that illustrates circuits for CNOT according to embodiments of the present disclosure.HQU-01425 HU 9476

[0094] Figs.49A-D are schematic diagrams illustrating an implementation of a layer of random CNOTs using gadgets shown in Fig.41 according to embodiments of the present disclosure.

[0095] Fig.50 is a schematic diagram that shows a chain complex of a 3D homological product code according to embodiments of the present disclosure. DETAILED DESCRIPTION

[0096] Quantum error correction (QEC) is believed to be essential for the realization of large-scale fault-tolerant quantum information processing. However, traditional schemes for achieving quantum error correction, such as the paradigmatic surface code, are generally very costly in terms of resource overhead, requiring millions of qubits to solve problems of interest. Recently, a new approach based on quantum low-density parity-check (qLDPC) codes has been proposed as a promising route to reduce the resources required. Unlike planar surface codes that encode a single logical qubit per block, qLDPC codes can encode multiple logical qubits per block and achieve a much higher, asymptotically constant encoding rate as well as a better distance scaling. However, in order to realize these appealing features, qLDPC codes require long-range connectivity between qubits, making their physical realization challenging. The required long-range and multi-layer connectivity makes qLDPC codes particularly suitable for neutral atom array based quantum computers. However, hardware capabilities for superconducting qubit architectures may advance to provide these features. In adapting qLDPC codes beyond the quantum memory setting to realize full- fledged quantum computation, further challenges arise.

[0097] qLDPC codes can enable fault-tolerant quantum computing with constant space overhead, but logical gate schemes based on code concatenation are costly and infeasible in practice. It was not previously clear if finite-size qLDPC codes can outperform surface codes in near-term devices with less than 10000 qubits and physical error rates above 10ିଷ. Thus,HQU-01425 HU 9476 there remain various open questions regarding the practicality of qLDPC code based quantum computation.

[0098] The present disclosure addresses these open questions, and provides systems and methods for fault-tolerant quantum computation based on qLDPC codes. In various embodiments, this quantum computation is implemented using reconfigurable atom arrays, such as the hardware architecture described in Bluvstein, D., Levine, H., Semeghini, G. et al. A quantum processor based on coherent transport of entangled atom arrays. Nature 604, 451– 456 (2022). https: / / doi.org / 10.1038 / s41586-022-04592-6, which is hereby incorporated by reference in its entirety.

[0099] A quantum bit (qubit) is the fundamental building block for a quantum computer. By analogy to classical bits which are used to store information in traditional computers (each bit is 0 or 1), qubits can occupy two distinct states labeled |0^ and |1^, or any quantum superposition of the two states. In various applications, multiple qubits are entangled in order to build multi-qubit quantum gates.

[0100] Bits and qubits are each encoded in the state of real physical systems. For example, a classical bit (0 or 1) may be encoded in whether a capacitor is charged or discharged, or whether a switch is ‘on’ or ‘off’.

[0101] The term qudit (quantum digit) denotes the unit of quantum information that can berealized in suitable ^^-level quantum systems. A collection of qubits that can be measured to^^ states can implement an ^^-level qudit.

[0102] Quantum bits are encoded in quantum systems with two (or more) distinct quantum states. There are many physical realizations that may be employed. One example is based on individual particles such as atoms, ions, or molecules which are isolated in vacuum. These isolated atoms, ions, and molecules have many distinct quantum states that correspond toHQU-01425 HU 9476 different orientations of electron spins, nuclear spins, electron orbits, and molecular rotations / vibrations.

[0103] In principle, a qubit may be encoded in any pair of quantum states of the atom / ion / molecule. In practice, a key parameter of qubits is described by their quantum coherence properties. Coherence measures the lifetime of the qubit before its information is lost. It has a close analogy with classical bits: if you prepare a classical bit in the 0 state, then after some time it may randomly be flipped to 1 due to environmental noise. Quantummechanically, the same error may occur:|0^may randomly flip to|1^after somecharacteristic timescale. However, qubits may suffer from additional errors: for example, a superposition state (|0^+|1^) / √2 may randomly flip to (|0^-|1^) / √2. In real quantum computers, the qubits must be encoded in quantum states which have long coherence properties.

[0104] Quantum computers generally can contain many qubits, each encoded in its own atom / molecule / ion / etc. Beyond simply containing the qubits, the quantum computer should be able to (1) initialize the qubits, (2) manipulate the state of the qubits in a controlled way, and (3) read out the final states of the qubits. When it comes to manipulation of the qubits, this is usually broken down into two types: one type of qubit manipulation is a so-called single-qubit gate, which means an operation that is applied individually to a qubit. This may, for example, flip the state of the qubit from |0^ to |1^, or it may take |0^ to a superpositionstate (|0^+|1^) / √2. The second necessary type of qubit manipulation is a multi-qubit gate,which acts collectively on two or more qubits, including those that are entangled. A multi- qubit gate is realized through some form of interaction between the qubits. The various quantum computing platforms (having various physical encodings of qubits) rely on different physical mechanisms both for single-qubit gates as well as multi-qubit gates according to the physical system that is storing the qubit.HQU-01425 HU 9476

[0105] In various embodiments of a quantum computer, a qubit is encoded in two near- ground-state energy levels of an atom, ion, or molecule. An example of this is a hyperfine qubit. Such a qubit is encoded in two electronic ground states that differ by the relative orientation of the nuclear spin with respect to the outer electron spin. Pairs of such states can be chosen so that they are particularly robust / insensitive to environmental perturbations, leading to long coherence times. These states are split in energy by the hyperfine interaction energy of the atom / ion / molecule, which is the interaction energy between the nuclear spin and the electron spin. The robustness of the qubit can be understood as the energy splitting between the two states being particularly stable. For this reason, such states are called clock states because the stable energy splitting can form an excellent frequency-reference and as such forms the basis for atomic clocks. Typical hyperfine splitting between these qubit states is in the 1 – 13 GHz frequency range.

[0106] To perform single-qubit gates on such a hyperfine qubit, it is possible to apply coherent microwave radiation at the exact frequency of the energy splitting between states. However, there are two drawbacks to this approach. First, microwaves cannot be applied to just one qubit without affecting adjacent qubits. This is because qubits are encoded in particles that are typically just a few microns apart from one another, and microwaves cannot be focused to such a small scale due to their large wavelength. Second, the microwave intensity is fairly limited and as such the maximum speed of single-qubit gates is correspondingly limited.

[0107] An alternative approach is based on stimulated Raman transitions. In this case, a laser field, also referred to herein as a Raman pulse, is applied to the atoms / ions / molecules. The laser field is nearly (but not exactly) resonant with an optical transition from one of the ground states to an optically excited state. The laser contains multiple frequency components separated in frequency by exactly the amount equal to the hyperfine splitting of the qubit.HQU-01425 HU 9476 The atom / ion / molecule can absorb a photon from one frequency component and coherently emit into a different frequency component, and in doing so it changes its state. This approach benefits from the capability of focusing the laser field onto individual particles or subsets of particles in the quantum computer. The laser field can also be applied with high intensity, allowing much faster gate operations.

[0108] Neutral atom quantum computers encode qubits in individual neutral atoms. The neutral atoms are trapped in a vacuum chamber and levitated by trapping lasers. Most commonly, the trapping lasers are individual optical tweezers, which are individual tightly focused laser beams that trap an individual atom at the focus. Alternatively, individual atoms may be trapped in an optical lattice, which is formed from standing waves of laser light which produce a periodic structure of nodes / antinodes.

[0109] A typical approach for encoding a qubit in neutral atoms is the hyperfine qubit approach, in which two ground states split by several GHz form the qubit. Multi-qubit gates in neutral atom quantum computers are realized using a third atomic state, which is a highly- excited Rydberg state. When one atom is excited to a Rydberg state, neighboring atoms are prevented from being excited to the Rydberg state. This conditional behavior forms the basis for multi-qubit gates, such as a controlled-NOT gate. The Rydberg state is used temporarily to mediate the multi-qubit gate, and then the atoms are returned back from the Rydberg state to the ground state levels to preserve their coherence.

[0110] Trapped ion quantum computers use atomic species that are ionized, meaning they have a net charge. In most cases, many ions are trapped in one large trapping potential formed by electrodes in a vacuum chamber. The ions are pulled to the minimum of the trapping potential, but inter-ion Coulomb repulsion causes them to form a crystal structure centered in the middle of the trapping potential. Most commonly, the ions arrange into a linear chain. Other ways to trap ions are also possible, such as using optical tweezers, orHQU-01425 HU 9476 trapping ions individually with local electric fields with a more complex on-chip electrode structure.

[0111] Qubits are encoded in trapped ions in multiple ways. One common approach is to use ground-state hyperfine levels, as described for neutral atoms. In trapped ions with hyperfine- qubit encoding, as with neutral atoms, single-qubit gates may use microwave radiation or stimulated Raman transitions.

[0112] Unlike in neutral atoms, trapped ion hyperfine qubits rely heavily on stimulated Raman transitions for performing multi-qubit gates. Stimulated Raman transitions may be used to control both the hyperfine state of the ion but also to change the motional state of the ion (i.e., add momentum). This can be understood as absorbing a photon moving in one direction and emitting a photon in a different direction, such that the difference in photon momentum is absorbed by the ion. Since many ions are often trapped in one collective trapping potential and are mutually repelling one another, changing the motional state of one ion affects other ions in the system, and this mechanism forms the basis for multi-qubit gates.

[0113] According to various embodiments of a quantum computer, individual particles (atoms / ions / molecules) can first be trapped in an array and arranged into particular configurations. Next, one or more particles are prepared in a desired quantum state. Quantum circuits can then be implemented by a sequence of qubit operations acting on individual qubits (single-qubit gates) or on groups of two or more qubits (multi-qubit gates). Finally, the state of the particles can be read out in order to observe the result of the quantum circuit. The readout can be accomplished using an observation system that typically includes an electron-multiplied CCD (EMCCD) camera image to detect particles’ loaded positions, and a second camera image to read out the particles’ final states by, for example, detecting fluorescence emitted by the particles in their final states.HQU-01425 HU 9476

[0114] A quantum circuit, or a set of quantum circuits, providing a particular modular function may be referred to as a quantum gadget. It will be appreciated that, as in classical computing, the separate design of reusable subcomponents enables flexible modular design of larger systems.

[0115] Quantum information platforms rely on interactions between qubits, either for performing quantum gates or for performing analog many-body simulation. Qubits often interact in a local way, however, which limits the connectivity of the circuit or the analog simulation and constrains the possible computations. While some platforms can communicate in a nonlocal way through the use of a shared bus (e.g., trapped ions), these shared-bus approaches are limited to small systems and thus still require a way to dynamically move qubits around in order to truly scale up the platform.

[0116] Neutral atom arrays can be dynamically reconfigured while preserving quantum coherence and entanglement between qubits, by storing quantum information in hyperfine states and shuttling atoms in optical tweezers. This approach offers a scalable way to realize a quantum information system with large numbers of qubits and arbitrary programmability – where any qubit can perform an entangling gate with any other qubit in the array. Using high-fidelity two-qubit Rydberg gates, various quantum information circuits are described herein that leverage the programmability and nonlocal connectivity achievable with these approaches. Examples of high fidelity Rydberg gates are described in Levine, et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett., vol. 123, issue 17, https: / / link.aps.org / doi / 10.1103 / PhysRevLett.123.170503, and Evered, et al., High-Fidelity Parallel Entangling Gates on a Neutral Atom Quantum Computer, arXiv:2304.05420 [quant-ph], https: / / arxiv.org / abs / 2304.05420, which are hereby incorporated by reference.HQU-01425 HU 9476

[0117] As set out in more detail below, the methods provided herein enable a variety of computational scenarios. In some scenarios, a plurality of neutral atom are moved in parallel between multiple regions in space. For example, a source of illumination may be directed to a first region, and atoms are moved in and out of that region between the application of pulses by the source of illumination. Similarly, a camera may be directed to an imaging region, and atoms are moved in and out of that imaging region for imaging. Similarly, atoms may be moved in and out of the blockade radius of other atoms, thereby allowing the application of gates to the different groups of atoms at different stages of an algorithm or layers of a quantum circuit.

[0118] It will be appreciated that various stabilizer codes entail the readout of ancilla qubits, and the present disclosure allows the physical relocation of ancilla qubits to an imaging region separate from the data qubits. In this way, readout of ancilla qubits may be provided without destruction of the data qubits.

[0119] More generally, an array of atoms may be moved between multiple arrangements to facilitate both digital gates between different selections of atoms and analog evolution of the array as a whole. As used herein, an arrangement of an array of atoms or a plurality of atoms refers to the positioning of those atoms relative to each other. It will be appreciated that certain arrangements provide connectivity between qubits that enable particular gates or analog evolution according to a particular Hamiltonian. One advantage of the methods provided herein is that atoms may be moved into proximity of atoms that were not adjacent within an array. A non-adjacent atom is one that is not within a unit cell in a regular lattice or that is not a nearest neighbor in an irregular array. For example, in a rectangular lattice, each atom has eight atoms that are within a unit cell thereof, and thus has eight adjacent atoms (disregarding edges).HQU-01425 HU 9476

[0120] As defined further below, atoms are moved adiabatically in order to preserve entanglement. As used herein, the term adiabatic movement refers to movement that avoids a transition of the subject atom within its trap. For example, where the first time-derivative of the acceleration of the subject atom is not greater than a predetermined value, the movementis considered adiabatic. Typically, adiabatic movement occurs when ^^^^^^^^ <(^^^^^^^^ ^^^^ ^^^^^^^^) × (^^^^^^^^ ^^^^^^^^^^^^^^^^^^)ଷ. In physics, jerk or jolt is the term given to the rate atwhich an object’s acceleration changes with respect to time.

[0121] In addition to adiabatic movement, in some embodiments dynamical decoupling is applied during the movement. As set out further below, a ^^-pulse during movement cancels out dephasing induced by the trap differential light shift. The trap differential light shift changes when the atom is moving (depending on its acceleration) because it will move in the trap, and so sample a different portion of the light intensity and hence have a different differential light shift.

[0122] Generally speaking, the more pulses applied, the more decoupling from fluctuations. For example, fluctuations may come from laser intensity fluctuations at different displacement positions of the atom, or different magnetic fields in space.

[0123] In embodiments where acceleration and deceleration are symmetric, both change the differential light shift in the same way. Accordingly, in such embodiments it is advantageous to apply a ^^-pulse at the midpoint of the motion. In this way, the changes in differential light shift induced by acceleration and deceleration cancel each other out.

[0124] An overview of a qLDPC-based approach to fault-tolerant quantum computation according to the present disclosure is shown in Fig.1. Architecture 100 consists of a qLDPC memory block 101, a processor with computational logical qubits 102, and mediating ancillae 103 between the memory 101 and the processor 102.HQU-01425 HU 9476

[0125] Fig.2 shows a contour plot of the number of physical qubits ^^ required by the architecture of Fig.1, at a 10ିଷphysical error rate, given a target number of logical qubits and a target logical failure rate (LFR), compared to the surface codes. The bottom right part corresponds to the finite-size lifted-product (LP) codes simulated in Fig.3B, which are more qubit-efficient than the hypergraph product (HGP) codes, while the top left part corresponds to the HGP codes using an extrapolation of the numerical data in Fig.3A.

[0126] As noted above, architecture 100 consists of a qLDPC memory block 101 that reliably and efficiently stores the quantum information, a processor 102 with computational logical qubits such as surface codes or color codes that perform logical gates, and mediating ancillae103 that interconnect the memory and processor. Adopting the conventional ^^^, ^^, ^^^notation for a code with ^^ physical qubits, ^^ logical qubits, and distance ^^, the qLDPC blockusing the HGP codes described below can provide a dense ^^^(^^), ^^, ^^^^^^ encoding, wherethe memory distance is lower-bounded ^^^^^This results in a constant encodingrate ^^ / ^^, and LFR exponentially decaying with the code distance. Using LP codes with higher encoding rate and better distance scaling further reduces required resources at small sizes.

[0127] The processor consists of ^^ computational qubits of code parameters^^^൫^^ଶ^^^^ ൯, 1, ^^^^^^^, where the code distance ^^^^^^ = Θ൫^^^^^^^^^^^^^^(^^^^)൯, with ^^ being thedepth of the logical circuit, is chosen to produce a sufficiently low error rate via the exponential suppression with distance.

[0128] The mediating ancillae are formed by taking the hypergraph product between logical operators in the qLDPC memory block and a computational qubit, respectively, resulting incode parameters ^^^൫^^^^^^^^^^^൯, 1, ^^^^^^൫^^^^^^, ^^^^^൯^.HQU-01425 HU 9476

[0129] By performing ancilla-assisted lattice surgery, the stored logical information can be teleported between any given pair of memory and computation qubits. Within this architecture, logical gates can be applied in parallel to a subset ^^ of the stored memoryqubits in each logical circuit step. By choosing ^^ = ^^൫√^^൯, namely that ^^ grows slowerthan√^^, the physical qubit overhead is dominated by the overhead of the memory block, achieving a constant encoding rate for quantum computation.

[0130] The qLDPC-based architecture starts to out-perform surface codes at instance sizes involving around one hundred logical qubits, and achieves an order of magnitude smaller space overhead at instance sizes of a few hundred logical qubits, as shown in Fig.2.

[0131] Referring to Fig.3, logical failure rates as a function of physical error rate for the qLDPC memory using HGP codes (Fig.3A) and LP codes (Fig.3B) are shown.

[0132] Referring to Fig.4, a table is provided showing the total number of physical qubits required to reach target numbers of logical qubits and logical failure rates using HGP codes and LP codes, compared to using surface codes. The physical error rate is set to be 10ିଷ. The estimates for the HGP and LP codes are based on the numerical data in Fig.3 without any extrapolation.

[0133] The following discussion analyzes the fault-tolerant implementation of HGP and LP codes as a robust quantum memory. The discussion here proves the existence of a circuit- level single-shot threshold for qLDPC codes with the linear syndrome confinement property, under a single-ancilla syndrome extraction circuit and a depolarizing error model where error rates do not scale with instance size. The syndrome confinement property, which requires that for all sufficiently small errors, the weight of the syndrome increases with the (reduced) weight of Pauli errors, holds for the HGP and LP codes when using classical codes or protographs with sufficient expansion.HQU-01425 HU 9476

[0134] This theoretical understanding is supplemented with numerical simulations of HGP and LP codes at practically-relevant instance sizes, showing competitive thresholds and LFR for both codes. For the HGP codes, the hypergraph product is taken of classical codes associated with random(3,4)-regular bipartite graphs that have good expanding properties, and have an encoding rate lower bounded by 4%. For the LP codes, a fixed protograph is chosen and a family of codes is obtained with sizes up to 1428 by increasing the shift-lift size and optimizing the entries of the base matrix over a quotient polynomial ring. This LPfamily has parameters ^^ ≈ 0.38^^^.଼ହ, maintaining a higher encoding rate as well as betterdistances than HGP codes of the same sizes. A variation of the coloration syndrome extraction circuit is used, which uses a single ancilla for each stabilizer generator and measures the ^^ and ^^ stabilizers separately. To be compatible with the parallel qubit rearrangement methods used in neutral atom array implementations, some embodiments apply the same coloration in each row or column based on the classical code connectivity.This results in entangling gate depth 4Δ^, where Δ^ = 4 denotes the check weight of theunderlying classical codes.

[0135] A space-time circuit-level decoder is constructed based on belief-propagation and ordered statistics decoding (BP+OSD). Specifically, for a QEC circuit with multiple cycles, a bipartite Tanner graph is constructed over a certain number of cycles, where the check nodes and variable nodes are associated with stabilizer measurement outcomes and circuit faults, respectively, and BP decoding is applied on this Tanner graph to infer the circuit fault locations in all noisy code cycles. In the final round, the BP+OSD decoder is applied to project back into the code space.

[0136] For all memory simulations, space-time fault graphs are used over three cycles, irrespective of the code size. Compared to alternative phenomenological decoders thatHQU-01425 HU 9476 simply assumed phenomenological measurement errors and decoded each code cycle independently, the space-time circuit-level decoder takes the circuit details into account and can perform joint decoding on multiple rounds of QEC cycles, leading to thresholdimprovements for the HGP code under standard depolarizing noise from below 0.23% to0.33%. Moreover, the space-time decoding is also crucial for simulations of logicaloperation, as the merged code no longer retains the single-shot property and repetitions of syndrome measurements are needed to correct measurement errors.

[0137] It is shown that the HGP and the LP codes have a threshold of 0.63% and 0.57%, respectively, under a depolarizing error model where the idling errors are set to 0. In the sub- threshold regime, the logical failure rates (LFR) of the two codes are well approximated by the formulas:Equation 1 where LFR is defined as the probability that any of the logical qubits failed per code cycle, assuming the absence of a decoding error floor.

[0138] The finite-size LP codes have better subthreshold scaling than the HGP codes. When also considering the idling errors ^^^(^^)associated with the atom rearrangement timeoverhead, which grows as ^^for the HGP codes and ^^for the LP codes, there is noasymptotic code threshold. However, it is numerically observed that the effect of adding theidling errors can be approximated by rescaling the gate error ^^^ → ^^^ + 3^^^(^^). Therefore,the idling errors have a negligible contribution when 3^^^(^^) ≪ ^^^, which is the case forHQU-01425 HU 9476 current experimental parameters and practically relevant code sizes. In Figs.3A-B, the simulated logical failure rate versus the bare two-qubit gate error rate ^^^(which is referred to as the physical error rate for simplicity) for the HGP and LP codes is shown, with the idling errors included and rescaled together with the gate error rate.

[0139] Using the subthreshold scaling formulas in Equation 1 and rescaling the gate error ^^^to approximate the presence of the idling errors, one can estimate the total number of data and ancilla qubits ^^ needed to reach a target number of logical qubits ^^ and a target logical failure rate and compare to the corresponding results for the surface code using thesubthreshold scaling formula ^^^^^^(^^^^^^^^^^^^^^) =.

[0140] In Fig.4, the table presents such estimates for finite-size HGP and LP codes that can be directly simulated (see Fig.3) at a realistic physical error rate of 10ିଷ. Both HGP and LP codes outperform surface codes with as few as 25 logical qubits. At a moderate scale of less than 200 logical qubits, LP codes with a few thousand physical qubits already achieve a qubit saving of over an order of magnitude. Fig.2 roughly estimates the space overhead of the HGP codes at a larger scale by extrapolation. One finds the HGP codes can also achieve a qubit saving of over an order of magnitude at a scale of 1000 logical qubits and 10ହphysical qubits.

[0141] Note that the syndrome extraction circuit is not 2^^-local, since coupling two qubits does not incur an error proportional to their Euclidean distance. The effective syndromeextraction circuit depth is ^^ where the constant ^^ is the number of rounds ofCNOT gates. In the regime where 3^^^(^^) ≪ ^^^, the circuit is quasi-nonlocal, with a constantcircuit depth ^^, while, in the regime where 3^^^(^^) ≫ ^^^, the circuit is effectively 4D-localfor the HGP codes, according to the bound on the circuit depth in finite dimensions. So,HQU-01425 HU 9476 although this scheme does not have a threshold, and the logical failure rate cannot be arbitrarily suppressed asymptotically, constant overhead and fault tolerance can be achieved at physically relevant sizes by utilizing the quasi-nonlocal connectivity in atom arrays.

[0142] The following discussion presents a scheme for performing universal fault-tolerant operations on the logical qubits. Numerical simulation of gate performance on qLDPC codes are provided using the space-time decoder developed above. Once finds that the threshold and logical performance remain almost unchanged when performing gates, indicating that the high threshold and low overhead can be maintained for fault-tolerant quantum computation.

[0143] Referring to Figs.5A-C, fault-tolerant teleportation from surface to qLDPC code is illustrated in the architecture previously discussed with reference to Fig.1. Fig.5A shows the layout of the codes. The logical X of the surface code 502 and a logical Z of the qLDPC code 501 are identified and associated with classical codes ^^^and ^^ଶ, respectively. An ancilla patch 503 is constructed as the hypergraph product 504 of ^^^and ^^ଶ. Direct lattice surgery between this ancilla patch and each of the surface and qLDPC codes is conducted by matching similar boundaries. In between similar boundaries, the transpose 505, 506 of the classical code is inserted to mediate the surgery.

[0144] Fig.5B is a diagram of a measurement-based teleportation circuit. Logical state ห^ത^^ is teleported from the surface code to one of the qLDPC’s logical qubits. The joint Clifford measurements are conducted through lattice surgery as illustrated Fig.5A.

[0145] Fig 5C shows the logical failure rates (per code cycle) of the teleportation. Noise is added during the merge and split steps of the ^^^^ lattice surgery. Decode is performed with the same space-time circuit-level BP+OSD decoder used in the memory simulations. The corresponding surface codes paired with the three HGP codes have distances 3, 5, and 7. A logical failure is recorded if there is an error in any of the logical qubits of the qLDPC codeHQU-01425 HU 9476after the teleportation scheme is complete. Denoting the total logical failure probability as^^^, the logical failure rate (per code cycle) is given as ^^^^^^ = 1 − (1 − ^^^)2^^, where thereare 2^^ cycles during the noisy ^^^^ lattice surgery, and ^^ denotes the minimal code distance. The plotted physical error rates are rescaled to account for idling errors.

[0146] Logical information is teleported between the qLDPC memory and topological codes (such as surface code) using a measurement-based circuit (Fig.5B), where the prescribed logical measurements are implemented using lattice surgery. Universal logical operations can then be performed in the topological codes. Since each topological code patch (e.g., 502) is much smaller than the LDPC patch 501, as long as the number of such patches used issmaller than ^^൫√^^൯, the space overhead from ancilla patches (e.g., 503) will be sub-leading.

[0147] Referring again to Fig.5A, as an example, consider the teleportation from a surface code patch 502 to a qLDPC patch 501 (although it will be appreciated that the present scheme allows for teleportation between any two quantum codes). The logical ^^ operator of thesurface code, ^̅^ଶ 507, and the logical ^^ operator of one of the qLDPC code's logical qubits,^ത^^508, are identified. These two logical operators are associated with classical codes ^^^and^^ଶ, respectively, by mapping the qubits supporting the logical operators to bits and thecorresponding incident stabilizer generators to classical checks.

[0148] An ancilla patch 503 is constructed as the hypergraph product 504 of ^^^and ^^ଶ, where the columns resemble ^^^and the rows resemble ^^ଶ. Direct lattice surgery between this ancilla patch and each of the surface and HGP codes is conducted by matching similar boundaries associated with the chosen logical operators. In between similar boundaries, an extra array 509, 510 of ancillary qubits and checks associated with the transpose of the classical code is inserted to mediate the surgery. All checks in the merged code commute with each other, as required. The product of the stabilizers associated with the checks of theHQU-01425 HU 9476 transposed code gives the required joint logical measurement, as in the case of standard surface code lattice surgery.

[0149] The fault tolerance of the above scheme is provided by showing that the minimum weight of any undetectable error during the lattice surgery is lower-bounded by the minimum distance of the two involved codes.

[0150] Numerical simulations of the above teleportation are shown, enabled by the space- time decoder described in the previous section. In the simulations, the teleportation circuitdepicted in Fig. 5B is used with an initial state of |^^^ = |0^. A logical failure is recorded ifany of the logical qubits in the qLDPC block fail.

[0151] As shown in Fig.5C, similar logical failure rates are observed for the teleportation as for the memory, demonstrating that the high threshold and low resource overhead of the qLDPC code can be maintained in the computation level. Since the logical failure rate of the entire code block is being evaluated, the system is far below the break-even rate at, e.g., 10ିଷphysical error rate, even for the smallest code of size 225.

[0152] By demonstrating large space overhead savings in practical regimes, and good performance of logical gate operations, the present disclosure brings the use of qLDPC codes for fault-tolerant quantum computation into the practical regime.

[0153] Circuit-level fault tolerance

[0154] In this section, the fault-tolerance of qLDPC codes is analyzed under circuit-level noise, the details of a space-time decoder are provided, the numerical circuit-level thresholds of the HGP and the LP codes are shown, and the effect of idling errors associated with the atom rearrangement scheme is considered.HQU-01425 HU 9476

[0155] Threshold theorem

[0156] Here, a theorem is established showing that single-ancilla, single-shot circuit thresholds exist for qLDPC code families with a property called linear syndrome confinement. Before introducing the threshold theorem, some notation is introduced for analyzing a faulty syndrome extraction circuit. For simplicity, only the CNOT gates areconsidered to be faulty. Consider a (Δ^, Δ^)-limited ^^^, ^^, ^^^ Calderbank-Shor-Steane (CSS)code, where each check checks at most Δ^qubits and each qubit is checked by at most Δ^checks. Consider a single-ancilla syndrome extraction circuit ^^ for one code cycle, in which each stabilizer ancilla interacts with all the data qubits it checks and then gets measured. The correction for ^^-type errors is analyzed using the ^^ checks, and the correction for the ^^-typeerrors follows. Let≃ ^^ଶ^ଶ be the data error space, ^^ ≃ ^^^ೞଶ be the syndrome space for the^^ checks, ℱ = ({^ ^^, ^^, ^^}be the fault space (a set) associated with asyndrome extraction circuit ^^, where ^^ denotes the number of CNOT gates. A qubit graph(^^^, ∼^) is defined whose nodes represent the data qubits, and a syndrome graph (^^^, ∼^) isdefined, whose nodes represent the X checks. The adjacency relations are defined as:Equation 2

[0157] To analyze circuit-level noise, a gate graph ൫^^^, ∼^൯ is defined where ^^^ ={^^^, ^^ = 1,2, ... , ^^} denote the nodes associated with the gates. For any fault ^^ ∈ ℱ, let ^^: =^^^^^^^^(^^) ⊂ ^^^ be the support of ^^ (capital letters are used to denote the corresponding set ofgraph nodes). The adjacency relation of ^^^is defined after introducing the following circuit maps.HQU-01425 HU 9476

[0158] Given a circuit ^^, one can obtain the circuit-dependent maps: ^^^: ℱ → ^^^ thatdescribes how faults propagate to data errors and ^^^: ℱ → ^^ that describes how faults triggersyndromes. ^^^, ^^^ is used to describe essentially the same maps, with the images replaced byassociated nodes in ^^^, ^^^. Combining with a decoder ^^: ^^ → ^^^ one obtains a map

[0159] With the above maps, the connectivity of the gate graph is defined as: ^^^ ∼^ ^^^ ⇔∃^^^, ^^^ ∈ ℱ with support on ^^^, ^^^ respectively, and ^^^(^^^) is adjacent to ^^^൫^^^൯ or ^^^(^^^) isadjacent to(note that ^^^(^^^), ^^^(^^^) are each connected). Note that ൫^^^, ∼^൯ hasvertex degrees bounded by (Δ^ + Δ^)(Δ^Δ^)ଶ.

[0160] Local stochastic circuit noise. A noise model is a local stochastic noise with noise parameter ^^ if there exists a probability ^^ such that the probability a circuit fault ^^ has asupport ^^ ∈ ^^^ is upper bounded

[0161] For example, the depolarizing noise is a local stochastic circuit-level noise with parameter ^^^, where ^^^is the gate error rate.

[0162] The reduced weight of a data Pauli operator ^^ ∈ ^^^ is defined as the minimum weightof all the Pauli operators that produce the same syndrome as ^^, i.e., |^^|^^ௗ: =^^^^^^{|^^|: ^^(^^) = ^^(^^), ^^ ∈ ^^^}. Based on the definition of reduced weight, the linearsyndrome confinement property of a code is defined as:

[0163] Linear syndrome confinement. A code has a ^^-linear syndrome confinement if for anydata Pauli ^^ such that |^^|^^ௗ ≤ ^^, |^^|^^ௗ ≤ ^^|^^(^^)| for some constant ^^. A code family haslinear confinement if each code of size ^^ has a ^^(^^)-linear confinement where ^^(^^)growswith ^^.

[0164] Referring to Fig.6, an illustration is provided of the proof of the following theorem on the circuit thresholds. Zooming in on one of the successive QEC cycles, it is illustratedHQU-01425 HU 9476 how the circuit faults ^^ cause the data errors ^^^(^^), the syndrome errors ^^^(^^), and finally, the residual data errors ^^.

[0165] For a (Δ^, Δ^)-limited qLDPC code family with growing distances and linearsyndrome confinement, there exists a single-ancilla syndrome extraction circuit for this code family such that a single-shot threshold exists under a circuit-level local stochastic noise.

[0166] First the graph-theoretical proof is sketched at a high level. As shown in Fig.6 one of the QEC cycles is used to examine the relationship between the circuit fault ^^ during thesyndrome extraction, the data error ^^^(^^)and the syndrome error ^^^(^^)that ^^ produces, andultimately the residual error ^^ after the correction. By associating the circuit faults, data errors, and syndrome errors to nodes on the gate graph ^^^, the qubit graph ^^^, and the syndrome graph ^^^described above, respectively, one can see that only errors associated with large connected clusters on their corresponding graph can cause logical failure. Ultimately, it is shown that the sizes of the error clusters corresponding to ^^^(^^), ^^^(^^), and ultimately, ^^, are all upper bounded by the size of the fault cluster ^^. Therefore, the probability of a logical failure (per code cycle) is upper bounded by the probability that a connected fault cluster is over a certain size, which exhibits a threshold behavior according to the percolation theorem under a local stochastic circuit noise.

[0167] To quantify the size of an error cluster using, roughly speaking, the maximum size of their connected components, one uses closeness:

[0168] ^^ closeness. Let G be a connected graph, i.e., a graph in which there exists a path between any two nodes. Given a subset E of nodes and a positive integer ^^, its ^^ closeness is defined as the quantity: Equation 3HQU-01425 HU 9476

[0169] Any connected subset of ^^ nodes is a ^^ patch and any ^^ patch ^^ such that |^^ ∩ ^^| =‖^^‖ఉ is a maximal patch for ^^.

[0170] In the spirit of reduced weight, one can define a reduced ^^ closeness function for thedata errors on the qubit graph ^^^:^^(^^) = ^^(^^), ^^^^^^^^(^^) =^^ ^^^^^^ ^^^^^^^^ ^^ ∈ ^^^}.

[0171] Using the closeness to quantify the size of a cluster, the probability of having a large fault cluster in the fault graph ^^^has a threshold behavior based on the percolation theorem:

[0172] Probability of a large fault cluster is small. Suppose ^^^has vertex degree upperbounded by ^^. Let ^^ be a positive integer and 0 < ^^ ≤ 1. Then there exists ^^௧^ > 0 suchthat, for local stochastic circuit faults ^^ of parameter ^^ < ^^௧^, giving:Equation 4 is the binary entropy function. ^^ =

[0173] Then one can show that the logical failure rate of a syndrome extraction circuit can be upper bounded by the probability of forming large clusters on the gate graph ^^^, which is then upper bounded by Equation 4. To do that, one needs the syndrome extraction circuit to satisfy the following linear circuit confinement:

[0174] Linear circuit confinement. A circuit ^^ has linear circuit confinement if:|^^^(^^)|^^ௗ ≤ ^^^|^^| and |^^^(^^)| ≤ ^^ଶ|^^| for some constants ^^^ and ^^ଶ.HQU-01425 HU 9476

[0175] Any constant-depth syndrome extraction circuit can satisfy the linear circuit confinement. One can show that for a code with linear syndrome confinement using a syndrome extraction circuit with linear circuit confinement, the size of the residual errors after each code cycle can be upper-bounded by the size of the circuit faults during that cycle (in terms of the closeness).

[0176] Residual error is under control. Consider a code that has linear syndrome confinement and a circuit that has linear circuit confinement. Provided that the input error pattern ^^ has ‖^^‖^^ௗ௧ ≤ ௧ସ, on circuit faults ^^, the residual error can be upper-bounded byEquation 5 for some constant ^^.

[0177] Given Equation 4 and Equation 5:Equation 6 where ^^ = (Δ௩ + 1)(2Δ^Δ௩ + 1). For a code family that satisfies the linear syndromeconfinement, ^^ grows with the code size ^^. Therefore, a threshold ^^௧^is proved.

[0178] Now it is shown that the closeness function preserves linear syndrome confinement and linear circuit confinement:

[0179] The closeness function preserves linear syndrome confinement and linear circuit confinement. For a code with linear syndrome confinement and a circuit ^^ with linear circuit confinement, the following relations between the data errors and the syndromes are satisfied:HQU-01425 HU 9476For the syndrome confinement, if௧. For thecircuit error propagation,

[0180] A generalized proof for syndrome confinement is given to show that circuitconfinement is also preserved. First, it is proved thatAbipartite graphis defined with edges ^^^ ∼^^ ^^^ ⇔ there exists ^^^ supportedon ^^^ such that ^^^ ∈ ^^^൫^^^൯. The graph has a few useful properties: (1) ^^^^^^(^^^) ≤ Δ^Δ^. (2).The neighbors of disjoint sets in ^^^are disjoint. (3). The neighbors of a connected component of ^^^are themselves connected.

[0181] Canonical ^^ patch. For any error ^^ on a qubit graph ^^ there exists a maximal ^^ patch^^ such that, for all but one connected component ^^^ of ^^, the following holds:^^^^^^ℎ^^^^ ^^^ ⊆ ^^ ^^^^ ^^^ ∩ ^^ = ∅Equation 7

[0182] In other words, if ^^^, … , ^^^ are the connected components of ^^, reordering ifnecessary, there exists an index ^^ such thatEquation 8

[0183] Any such ^^ is called a canonical ^^ patch for the set ^^.

[0184] For ease of notation, let ^^ = ^^^(^^). Let ^^ be the canonical patch of ^^, whichintersects ^^ with its connected components {^^^, ^^ = 1,2, … , ^^ − 1} and a subset of ^^ఔ. Let ^^ =HQU-01425 HU 9476otherwise,^^^ is a minimal path inwhose neighbor contains a path ^^^ in ^^^ that connects all theconnected components {^^^, ^^ = 1,2, ... ,Note that= ^^. And |^^| ≤∑ఔି^^ୀ^ |Γ(^^^)| + |Γ(^^ᇱఔ )| + ^^ ≤ Δ^Δ^(∑ఔି^^ୀ^ |^^^| + |^^ᇱఔ |) + ^^ ≤ (2Δ^Δ^ + 1)^^. Therefore, ^^ is a(2Δ^Δ^ + 1)-patch. Now, one needs to show that |^^ ∩ ^^| ≥ 1 / ^^^‖^^‖௧.Equation 9 where the first inequality comes from the fact the Γ(^^^)are disjoint. The second inequalityholds because for each Γ(^^^), there exists a ^^^ with ^^^^^^^^(^^^) ⊂ Γ(^^^) ∩ ^^ s.t. ^^^(^^^) = ^^^.Then, using the linear circuit confinement property, |Γ(^^^) ∩ ^^| ≥ 1 / ^^^|^^^|. The proof formirrored.

[0185] Stochastic shadow decoder. The stochastic shadow decoder has variable parameters0 < ^^ ≤ 1, and 0 < ^^, ^^ ∈ ℤ. Given an observed syndrome ^^ =+ ^^^ where ^^^ ⊆ ^^^ isthe syndrome error, the stochastic shadow decoder of parameters (^^, ^^, ^^) performs thefollowing two steps: 1. Syndrome repair: find ^^^ of minimum ^^ closenesssuch that ^^ + ^^^ belongsto the (^^, ^^) shadow of the code, where (^^, ^^) shadow = {^^(^^) such that ‖^^‖ఉ ≤^^^^}.2. Qubit decode: find ^^^ of minimum ^^ closeness ‖^^^‖ఉ such that ^^(^^^) = ^^ + ^^^.^^ = ^^ + ^^^ is called the residual error.

[0186] The residual error ^^ after each QEC cycle has three parts: the residual data error ^^ from the last QEC cycle, the data error ^^^(^^) that the current circuit fault ^^ propagates to,HQU-01425 HU 9476and a data correction returned from the decoder ^^, i.e. ^^ = ^^^(^^)^^[^^^(^^) + ^^(^^)]^^. Onecan upper-bound the closeness of ^^ byEquation 10where the second inequality uses the property of a stochastic shadow decoder of parameters^^^ସ^ , ^^, + 1)^^^ that the output has closeness upper-bounded by ^ସ ^^, i.e. ‖^^[^^^(^^) +^^(^^)]‖ ≤ ௧. Combining this with the fact that ‖^^‖ ≤ ௧ if on ‖ ‖ ௧௧ ସ ௧ ସ e can show ^^ ≤ ସ, one canupper-bound the closeness of ^^[^^^(^^) + ^^(^^)]^^ using its syndrome closeness via the linearsyndrome confinement property. The last inequality uses the fact that ^^(^^[^^^(^^)]) =^^^(^^) + ^^ᇱ with ‖^^ᇱ‖(^ೡା^)௧ ≤ ‖^^^(^^)‖(^ೡା^)௧. Using the linear circuit confinement, ‖^^‖௧ ≤(^^^ + 2^^^^ଶ)‖^^‖(^ೡା^)(ଶ^^^ೡା^)௧.

[0187] The threshold proof above uses the stochastic shadow decoder, which may not be efficient in practice. Additionally, proofs based on the percolation theorem can only demonstrate the existence of a threshold, which is often far below what can be achieved in practice. Therefore, numerical simulations are provided using practical decoders to determine achievable thresholds. These results will be presented in the next section.

[0188] At the end of this section, qLDPC codes are listed that satisfy the linear syndrome confinement. First, HGP codes with classical codes that have enough vertex expansion (also called quantum expander codes) have been proved to satisfy the linear syndrome confinement. The key proof is based on the proposition that given a small error, one can simultaneously reduce the weight of the syndrome and the reduced weight of the error byHQU-01425 HU 9476 flipping a subset of the support of some stabilizer generator. Moreover, the decreased syndrome weight is at least proportional to the number of flipped bits. In other words, the quantum expander codes can be decoded by flip-type decoders, e.g., the small-set-flip decoder. Similar proofs can be applied to codes decodable by flip-type decoders, such as the PK code and various quantum Tanner codes. One can similarly prove the linear syndrome confinement for quasi-cyclic LP codes with expanding protographs and constant lift size by showing that this proposition still holds for these codes.

[0189] Circuit-level space-time decoder

[0190] In this section, details are provided on the circuit-level space-time decoder used herein. The decoder is based on a belief-propagation and ordered-statistical decoding (BP+OSD). When applying consecutive QEC cycles, various embodiments use the BP decoder for each noisy cycle to control error accumulation. For the final round, which is typically noiseless (e.g., during transversal readout of qubits), the BP+OSD decoder is applied to eliminate residual syndromes and project the system into the codespace. To decode noisy syndromes using the BP decoder, a space-time circuit-level fault graph is constructed. This allows the BP decoder to decode over multiple QEC cycles while taking circuit-level details into account. Note that a circuit-level BP decoder may be used to improve the matching graph for decoding the surface codes using the minimum-weight perfect matching decoder, and similar circuit-level fault graphs were used for correcting a generic stabilizer circuit. For the final BP+OSD decoder, a simple graph is used with only data errors, as it decodes noiseless syndromes.

[0191] Now, details are provided on the construction of the space-time circuit-level fault graph. Given a code, a QEC circuit is considered consisting of multiple code cycles. Each code cycle includes the initialization of ancillae, a sequence of parallel entangling gatesHQU-01425 HU 9476 between ancillae and data qubits, and measurements on the ancillae. Each ancilla measurement in the ^^-th cycle produces a stabilizer measurement outcome, denoted as(for the ^^-th ancilla). Detectors are defined as the comparison between consecutive stabilizermeasurement outcomes: ^^(௧) (௧ି^)^ : =+ ^^^ (^^^^^^ 2). A bipartite decoding graph isconstructed over ^^ code cycles. Check nodes are associated with detectors, and variablenodes are associated with circuit faults (e.g., two-qubit Pauli errors for a CNOT gate) for ^^ =1,2, ⋯ , ^^. The connectivity between check nodes and variable nodes is naturally determinedby how circuit faults trigger detectors. The prior error probabilities (used by the BP decoder) for variable nodes are given by the circuit-level noise model used in the simulation. To account for residual data errors after the previous QEC cycle, a layer of depolarizing data errors is added before the first code cycle to the decoding graph. The error prior for these data errors is currently set phenomenologically as a hyperparameter.

[0192] Referring to Figs.7A-B, an example of the space-time decoding graph for a length-3 repetition code is provided. In Fig.7A, the syndrome extraction circuit during a code cycle is shown. In Fig.7B, a subset of the decoding graph is shown consisting of the faults associated with the highlighted CNOT gate in Fig.7A in the first code cycle (circles 701) and the phenomenological data faults (circles 702).

[0193] As a simple example, consider a length-3 repetition code, whose syndrome extraction circuit during one code cycle is illustrated in Fig.7A. In Fig.7B a subset of its decoding graph is plotted, which only includes the faults associated with the CNOT gates between the second check ancilla and the middle qubit in the first code cycle (see CNOT gate 703 in Fig. 7A and faults nodes 701 in Fig.7B). In addition, the phenomenologically added data faults before the first code cycle are also shown (see circles 702). These faults will trigger theHQU-01425 HU 9476 detectors associated with the two check ancillae (represented by squares 704) in the first and second code cycles.

[0194] For the simulations in this work, a min-sum variant of the BP decoder and an OSD-0 decoder with an order 10 are used. For the min-sum BP decoder, the maximum iteration number is set to be ^^ / ^^, where ^^ is the size of the code that the decoder decodes on and ^^ is numerically optimized. The scaling factor ^^ of the min-sum decoder and the phenomenological error prior ^^^(normalized by the gate error rate ^^^) are numerically optimized for the virtual nodes associated with the residual data errors in the decoding graph(see nodes 702 in Fig. 7B). ^^ = 5, ^^ = 0.7, ^^^ = 3 are chosen for the HGP codes and ^^ =20, ^^ = 1, ^^^ = 1 are chosen for the LP codes. For all the memory simulations, fault graphsover ^^ = 3 code cycles are used for the BP decoder, irrespective of the code sizes.

[0195] Numerical thresholds

[0196] Memory simulations are conducted by initializing logical qubits, simulating ^^ ≫ 1code cycles using the coloration syndrome extraction circuit, and performing a transversal qubit readout at the end. Stim is used to sample the above circuit. A logical failure is recorded if any of the logical qubits have a flipped logical observable after the decoding. Denoting the above total logical failure probability as ^^^, the logical failure rate per codecycle is obtained as 1 − (1 − ^^ ^ / ே^) .

[0197] One can verify that the HGP codes and the LP codes can indeed be single-shot decoded using the space-time BP+OSD decoder numerically. The space-time BP decoder is applied over every three code cycles (regardless of the code sizes) and the BP-OSD decoder to the final cycle. In Figs.8A-B, it is shown that the logical failure rate saturates to a stable value as one increases the number of code cycles, thus verifying the single-shot property of the HGP and the LP codes using the space-time decoder described herein.HQU-01425 HU 9476

[0198] Referring to Figs.8A-B, verification of the single-shot property of the HGP codes and the LP codes is illustrated. The logical failure rate (per code cycle) stabilizes as the number of code cycles increases for both the HGP codes and the LP codes. The physical error rate is fixed at 0.15% for the simulations.

[0199] The numerical thresholds of the HGP codes and the LP codes are given below. The space-time decoder can improve upon the alternative phenomenological decoder used in the literature considering the same HGP code family with the same coloration syndrome extraction circuit under a standard depolarizing noise model. As shown in Fig.9, a thresholdof 0.33% is given, which is higher than that (< 0.23%) reported for phenomenologicaldecoders. The subthreshold scaling is also greatly improved by using the space-time circuit- level decoder.

[0200] Referring to Fig.9, the logical failure rates of the HGP codes using the space-time decoder under the standard depolarizing noise model is illustrated. The threshold is 0.33%,which is higher than that (< 0.23%) using a phenomenological decoder. The number of codecycles for the simulation is 12.

[0201] Next, the logical failure rates of the HGP codes and the LP codes are evaluated when only considering the two-qubit gate errors (the state preparation and measurement errors have a negligible contribution). The results are shown by the solid markers in Fig.10. The threshold for both the HGP codes and the LP codes is around 0.55%. For gate error ratesbelow 4 × 10ିଷ, the subthreshold scaling is numerically fit using the logarithm of the ansatz^^^^^^^^^ = ^^ ^^^ మ^బ^ . For the HGP codes, ^^ = 0.07 ± 0.04, ^^^ = 0.60% ± 0.08%, ^^ = 0.94 ±0.21, and ^^ = 0.27 ± 0.03; For the LP codes, ^^ = 2.3 ± 1.2, ^^^ = 0.66% ± 0.06%, ^^ =0.22 ± 0.07, and ^^ = 0.60 ± 0.05. The logical failure rates are plotted using thephenomenological decoder (the dashed lines with empty markers) as a comparison. AsHQU-01425 HU 9476 shown, the space-time decoder improves not only the thresholds but also the subthreshold scaling.

[0202] Using the standard derivations of the subthreshold scaling parameters, one can estimate the numerical uncertainty of the boundary lines for the HGP codes and the LP codes in the space overhead estimation. Let ^^ denote the minimal number of physical qubits required, one calculates the standard derivation of ^^ using the error propagation formula:Equation 11

[0203] Referring to Figs.10A-B, the logical failures of the HGP codes and the LP codes are illustrated when only adding two-qubit gate errors. The solid lines represent the fittedsubthreshold scaling: ^^^^^^ = 0.07for the HGP codes and ^^^^^^ =^ ^^ ^.^^^బ.ల2.3 ^.^^^^^ for the LP codes. For the HGP codes, 42 code cycles are simulated for^^^ ≤ 4 × 10ିଷ and 12 cycles for ^^^ > 4 × 10ିଷ; For the LP codes, 60 cycles are simulatedfor ^^^ ≤ 4 × 10ିଷ and 12 cycles for ^^^ > 4 × 10ିଷ. The empty markers along with thedashed lines represent the results using a simpler phenomenological decoder that uses a decoding graph with phenomenological data and measurement errors.

[0204] Effect of idling errors

[0205] The effect of idling errors between the sequence of parallel CNOT gates connecting check qubits and data qubits is now considered. In Figs.11A-B, the logical failure rates are compared when both gate errors (with rate ^^^) and idling errors (with rate ^^^) are added to thesimulation (solid lines) and when only gate errors (with rate ^^^ + 2^^^) are added (dashedHQU-01425 HU 9476 lines). The logical failure rates in both cases are similar, confirming that the effect of idlingerrors can be well approximated by rescaling the gate error rate to ^^^ → ^^^ + 2^^^.

[0206] Referring to Fig.11A, logical failure rates are plotted versus the ratio between theidling error ^^^ and the gate error ^^^ for the HGP codes. The gate error rate ^^^ is set to be10ିଷ. The solid lines represent the results when both the gate errors and the idling errors areadded to the simulation, while the dashed lines represent the results when only gate errors areadded with an error rate ^^^ + 3^^^.

[0207] Referring to Fig.12, achievable logical failure rates with different idling error strengths are illustrated. The LFRs are first exponentially suppressed by the code size ^^when ^^ is small and 3^^^(^^) ≪ ^^^, and gradually saturate and then increase as 3^^^(^^) > ^^^and finally approaches the gate error threshold. Using the relevant experimental parameters,the idling errors are negligible for ^^ up to ∼ 10^ and the LFRs can go below 10ିଶସ (see thesolid curve), which already suffices for implementing practical quantum algorithms. For even larger sizes, concatenation with another code can be employed to extend the effectivecoherence time and further suppress ^^^(^^).

[0208] Teleportation

[0209] Proof of fault tolerance

[0210] Referring to Fig.13, a lattice surgery between two quantum codes ^^^and ^^^ᇱformeasuring the joint logical operator ^ത^^^ത^^ᇱis illustrated.

[0211] Here, the fault tolerance of the lattice surgery protocol is proved using the subsystem formalism. Fault tolerance means that the minimum weight of any undetectable data Pauli error is no smaller than the minimum distance of the two involved codes. The lattice surgery between two codes ^^^and ^^^ᇱcan be viewed as fixing the gauges of a subsystem code ^^HQU-01425 HU 9476 constructed from ^^^and ^^^ᇱ, and the relevant logical errors are given by the dressed logical operators of ^^.

[0212] Consider the measurement of theoperator, where ^ത^^ (^ത^^ᇱ) is a ^^ logical operatorof ^^^(^^^ᇱ), as shown in Fig.13. Let ^^^and ^^^ᇱbe the stabilizer group for ^^^and ^^^ᇱ, respectively, and ^^^and ^^^ᇱbe the corresponding generator set. For ease of notation, all the phase factors^^^^are ignored. The gauge group of the subsystem code ^^ is given byEquation 12 denotes the anti-commuting conjugate partner of ^ത^^, ^^^^ the Pauli ^^ operator on the^^-th middle qubit (see the middle 1301 circles inthe ^^-th middle ^^ check(see the middle 1301 squares in Fig.13). Note that ^^ includes the Pauli ^^ operators of the middle qubits since they are all initiated to the|0^states. The stabilizer group of ^^, which isthe center of ^^, is given by the stabilizers of ^^ ᇱ ᇱ^ and ^^^ , except that the ^^ checks ^^^௭,^ , ^^௭,^ ^incident to ^ത^^ or ^ത^^ᇱare regrouped with the ^^ operators of the middle qubits in order tocommute with the middle ^^ checks:Equation 13

[0213] The stabilizers in ^^ are essentially what are used for correcting the errors during the merging and the splitting step. The nontrivial gauge operators are a subset of ^^ that act nontrivially on the gauge qubits:HQU-01425 HU 9476 Equation 14 where each tuple gives an anti-commuting conjugate logical operator pair for each gaugequbit, and ^^(^^) is a ^^-dependent subset of the indices {^^} such thatonly anti-commutes with ^^௫^,^.

[0214] One can classify the logical operators of ^^ into two types. The bare logical operators^^^^^^: = ^^(^^)\^^ are those that act nontrivially only on the logical qubits, which in this caseare a subset of the logical operators of ^^ ᇱ^ and ^^^ excluding ^ത^^^ത^ᇱ^ , i.e. ^^^^^^ ⊂^^(^^∪ ^^ᇱ^ ∪ ^ത^^^ത^ᇱ^). The dressed logical operators are those that cannontrivially not only on the logical qubits but also on the gauge qubits:Equation 15

[0215] The minimum weight of the dressed logical operators determines the distance of thesubsystem code ^^, i.e. ^^: = ^^^^^^^∈^^^^ೞೞ^^^^^^(^^). To show that ^^ ≥ ^^^: = ^^^^^^{^^ ᇱ^, ^^^ }, itsuffices to show that the weight of all the logical operators of ^^^and ^^^ᇱ, excluding the gaugeoperatorsand ^̅^^, cannot be reduced below ^^^ by applying the gauge operators in ^^.this case, where ^^^ᇱis a HGP code with one of the basis codes being associated with ^ത^^thatdoes not contain any sublogical, one only needs to consider applying the middle ^^ checkson a logical operator of the form ^ത^ଶ^ത^ᇱ^ , where ^^ଶ is another ^^ logical operator of ^^that potentially has a large overlap with ^ത^^. This is proved by contradiction. Let ^^^^(^ത^^) =^^^^(^ത^ᇱ^) = ^^^, ^^^^(^ത^ଶ) = ^^ଶ, and ^ത^ଶ has a overlap ^^ with ^ത^^ in support. Suppose there exists agauge operator ^^௫ that is the product of some middle ^^ checks such that ^^^^(^ത^ଶ^ത^ᇱ^^^௫) < ^^^.HQU-01425 HU 9476Since ^^^^(^ത^ଶ^ത^ᇱ^^^௫)+ ^^ଶ − 2^^, one has ^^^ + ^^ଶ − 2^^ < ^^^. This leads to ^^^^(^ത^^^ത^ଶ) =^^^ + ^^ଶ − 2^^ < ^^^. Since ^ത^^^ത^ଶ is a logical operator of ^^^, one obtains a contradiction.The above proof does not necessarily hold if ^ത^^ contains any sublogical, i.e., there existsanother logical operator ^ത^ଶ ∈ ^^^ such that ^^^^^^^^(^ത^ଶ) ⊂ ^^^^^^^^(^ത^^).

[0216] Simulation details

[0217] Here, numerical simulations are provided of the teleportation scheme between the HGP code and the surface code. A logical ^ത^^operator is selected for the qLDPC code of minimum weight. This ensures it contains no sublogicals so that the scheme is provenlyfault-tolerant. Logical ^ത^^ of the qLDPC code and logical ^̅^ଶ of the surface code are mappedto classical codes ^^^and ^^ଶ, respectively. An ancilla code patch is constructed as an HGPcode, ^^^^^^(^^^, ^^ଶ). The teleportation circuit is then programmed using Stim. For the ^^^^and ^^^^ blocks, lattice surgery circuits are constructed as explained above. For the logical^^^ measurements, a noiseless destructive measurement is simulated.

[0218] The same space-time decoder described above is used to decode. The decoderparameters chosen were ^^ = 10, ^^ = 0.7, ^^^ = 3 where, recall, ^^ / ^^ is the maximumiteration of the min-sum BP decoder, ^^ is the scaling factor, and ^^^is the error prior of the virtual nodes in the decoding graph. Noisy gates are simulated on the joint logical ^^^^ measurement part of the teleportation circuit. Errors are simulated during both the splitting and merging steps of the ^^^^ lattice surgery. The logical failure rate of the ^^^^ logical measurement would be nearly the same as that of the qLDPC memory if using a computation surface code with a distance slightly larger than that of the qLDPC code. Merging the ancilla code patch with the qLDPC code results in a code that is no longer single-shot. The same is true of merging the ancilla patch with the surface code. Then, to tolerate measurement errors, the syndrome extraction circuit is repeated, and then space-time decoding is performed. ^^HQU-01425 HU 9476 rounds of QEC are performed after each of the merge and split steps of the ^^^^ surgery,where ^^ = ^^^^^^(^^^, ^^ଶ) is the distance of the teleportation scheme, and ^^^ and ^^ଶ are thedistances of the qLDPC code and surface code, respectively.

[0219] A gate error threshold (without idling errors) of ∼ 0.7% is found by looking at thecrossing of ^^ = 3,5,7 schemes, which were constructed with (^^^, ^^ଶ) = (4,3), (6,5), (8,7),respectively. A logical failure is recorded if there is an error in any of the logical qubits of the qLDPC code after the teleportation scheme is complete. Denoting the total logical failureprobability as ^^^, the logical failure rate (per code cycle) is calculated as ^^^^^^ = 1 −(1 − ^^^)^ / ଶௗ, where there are 2^^ cycles during the noisy XX lattice surgery. The slightincrease in the threshold compared to the memory is attributed to the increase in the number of code cycles used by the space-time decoder (the memory simulations are decoded using only three cycles).

[0220] The following discussion provides a scheme to efficiently implement constant-rate qLDPC codes with reconfigurable atom arrays. This scheme utilizes the product structure inherent in many qLDPC codes to provide streamlined circuit implementations of them. Efficient parallel rearrangement methods are provided for one-dimensional atom arrays, which, when repeated in parallel across different rows or columns, enable stabilizer measurements of qLDPC codes with a number of rearrangement steps that grows only logarithmically with the system size. This work enables low resource overhead quantum computing with qLDPC codes.

[0221] Referring to Fig.14, a quantum information architecture enabled by coherent transport of neutral atoms is illustrated. Qubits are transported to perform entangling gates with distant qubits, enabling programmable and nonlocal connectivity. Atom shuttling is performed using optical tweezers, with high parallelism in two dimensions and betweenHQU-01425 HU 9476multiple zones allowing selective manipulations. The inset shows the atomic levels used: the|0^, |1^ qubit states refer to the ^^ 87ி = 0 clock states of Rb, and |^^^ is a Rydberg state usedfor generating entanglement between qubits, which are further described with regard to Fig. 15.

[0222] Fig.15 is a level diagram showing key87Rb atomic levels used. The Rydberg excitation scheme from |1^ to |^^^ is composed of a two-photon transition driven by a 420-nmlaser and a 1013-nm laser. A DC magnetic field of ^^ = 8.5^^ is applied throughout this work.

[0223] As noted above, quantum information systems derive their power from controllable interactions that generate quantum entanglement. However, the natural, local character of interactions limits the connectivity of quantum circuits and simulations. Nonlocal connectivity can be engineered via a global shared quantum data bus, but these approaches are limited in either control or size.

[0224] According to various embodiments of the present disclosure, this long-standing challenge is addressed through dynamically reconfigurable arrays of entangled neutral atoms, shuttled by optical tweezers in two spatial dimensions. Hyperfine states are used for storing and transporting quantum information in between quantum operations, and excitation into Rydberg states is used for generating entanglement. Highly parallel operations are enabled via selective qubit operations in distinct zones that qubits are dynamically shuttled between. Taken together, these ingredients enable a powerful quantum information architecture, which is employed to realize applications including entangled state generation, creation of topological surface and toric code states, and hybrid analog-digital quantum simulations.

[0225] Within this architecture, programming a specific quantum circuit entails control over only a few optical degrees of freedom. Arbitrary tweezer positions in space are controlled by a computer-generated hologram, hundreds of atoms are dynamically reconfigured in parallel by two waveforms in a 2D acousto-optic deflector (AOD), and qubit operations are realizedHQU-01425 HU 9476 by pulsing optical beams. This flexible optical control enables sophisticated quantum circuits with only a few classical controls. This architecture enables an inherently scalable approach: larger codes require no increase in the number of classical controls.

[0226] Various quantum circuits are realizable with this approach, including quantum error correction (QEC) codes such as the surface and Steane codes, with fidelities in this disclosure already comparable to state-of-the-art experiments in other platforms. Moreover, the parallelized, nonlocal connectivity is used to create the toric code state on a torus.

[0227] Referring to Fig.16, a quantum processing unit (QPU) according to the present disclosure is illustrated. This design is centered around efficient classical control over many logical qubits in parallel using optical beams. Single-qubit logical gates can be realized transversally, for example, by illuminating all physical qubits within the same logical qubit block by an optical beam. Two-qubit logical gates can also be realized transversally, by interlacing two logical arrays of qubits and applying a global optical pulse for entangling each twin of the pair. For such a gate to be transversal, it must interact only corresponding qubits from the different logical arrays, such that the first qubit of the first logical array interacts with the first qubit of the second logical array, and so on.

[0228] Neutral atom systems have the potential for utility scale computing: for example, millions of identical neutral atom qubits may be trapped in mm-scale regions of space. The key challenge is the classical control required to assemble these qubits into a large-scale quantum processor. Full programmability of single physical qubits generally requires highly complicated classical control techniques in order to operate on millions of qubits. In contrast, the architectures provided herein allow for full programmability of single logical qubits while only requiring a few classical controls per logical qubit. This enables reaching utility-scale by encoding logical qubits into blocks that can be efficiently controlled in parallel. Using advanced optical microscopy systems (such as those utilized for modern industrial-scaleHQU-01425 HU 9476 lithography) with high numerical aperture and large field of view exceeding several millimeters, and appropriately scaled trapping laser power, direct trapping and manipulation of over a million qubits is possible. Further scaling is possible by creating 10-100 such processing units, each under its own microscope objective, and then connecting these units together utilizing photonic links and / or optical lattice transport. This allows for sufficient space, resolution, and power density for enacting high-fidelity control over 10M qubits and beyond.

[0229] QPU 1600 is segmented into several key zones: a storage zone 1611, entangling zone 1612, readout zone 1613, atom loading zone 1604, and remote entangling zone 1605. Storage zone 1611, entangling zone 1612, and readout zone 1613 form processor core 1601, which in some embodiments contains 104to 106qubits in a footprint of 0.5-5mm. Fresh atoms are continuously reloaded from distant atom loading zone 1604, and a distant remote entangling zone 1605 (using optical interconnects and / or lattice transport) delivers remote Bell pair entanglement resources.

[0230] In storage zone 1611, idle logical qubits are stored for long times, utilizing the long qubit coherence times and high fidelity single-qubit gates, such that an error-correction cycle is only required before a logical two-qubit gate. For coherence times of 10-100 second, and assuming performance 10x below threshold, then roughly 1% single-qubit dephasing errors can be tolerated before a round of ^^ cycles of error correction. This corresponds to approximately 0.1-1 second of allowed storage time before the requirement for correction. Due to the all-to-all connectivity provided by the presently described architectures, idled logical qubits can simply be kept in the storage zone, safe from additional errors. Logical qubits are thus stored in dense blocks, shuttled out when they are needed in the algorithm, and only error-corrected before a two-qubit gate, greatly reducing the error correction overhead. In various exemplary devices, atoms are stored at densities of approximatelyHQU-01425 HU 9476 1 / (2^^^^)ଶin the dense storage zone, and densities of approximately 1 / (10^^^^)ଶin the active zone.

[0231] The active logical qubits are manipulated in active zone 1612. By utilizing qubit transport, all combinations of two-qubit gates can be performed in a fixed region of space. This significantly reduces the classical control complexity. For example, all two-qubit gates can be performed using a single, global optical beam, which is dramatically simpler than addressing each individual qubit. This exceptional degree of parallelism for logical qubit control is a significant advantage of the present architecture relative to alternatives such as those involving individual control of atomic qubits.

[0232] Readout zone 1613 allows selectively reading out a subset of qubits mid-circuit without disturbing the other qubits. This readout happens in parallel with a global beam and a camera, again requiring only one set of classical controls.

[0233] Outside of the core processor 1601, atoms are constantly reloaded from loading zone 1604 and transported into the core processor for running arbitrarily long circuits. Remote Bell pairs with other processing units are generated using optical links and / or optical lattice transport 1605, and are shuttled into the core processor 1601 for creating remote logical entanglement. This allows interconnection of 10-100 single processing units into one error- corrected, utility-scale quantum computer.

[0234] The architecture provided above allows for mid-circuit readout. In particular, this architecture may be paired with fast imaging in the readout zone and a classical control loop. In addition, various methods may be used to suppress crosstalk errors and detect / correct for loss. Arbitrarily long circuit depths may be achieved with continuous reloading of atoms and further crosstalk suppression.

[0235] To connect multiple units, many high-fidelity, long-distance Bell pairs may be generated in parallel, using lattice transport and / or photonic links.HQU-01425 HU 9476

[0236] It will be appreciated that the present architecture is suitable for logical state preservation by repetitive mid-circuit measurement and correction. In addition, a surface code logical qubit may be implemented, for example by moving ancillas from a storage zone reservoir, entangling with data qubits for syndrome extraction, and moving to the readout zone. This allows fast mid-circuit readout and feedback while preserving coherence on data qubits. In various embodiments, the data qubits are protected by placing the imaging zone ~50 microns away, thereby suppressing crosstalk from the readout beam and scattered light from the ancilla atoms.

[0237] In various embodiments, a fast classical control loop uses ancilla measurements to determine errors on the data qubits, and to detect and correct qubit loss. Lost qubits may then be replaced with reservoir atoms. In order to reach surface code distances several times larger than the largest codes created in alternative systems, local detuning patterns may be utilized for space-efficient use of the entangling zone.

[0238] The presently described architectures may also be used to perform algorithms with logical qubits. The zoned approach combined with efficient optical control over many logical qubits in parallel allows construction of large-scale processors. In an exemplary use case, ~10 logical qubits are encoded in the active zone and moved to the storage zone. After encoding all logical qubits, the algorithm is run with appropriate logical single-qubit and logical two-qubit gates. The flexible, local single-qubit control required for logical single- qubit gates is implemented with Raman light from a 2D AOD illuminating the grid of a single code block. Logical two-qubit gates are realized transversally in the entangling zone. Mid- circuit readout is used for the non-Clifford gate-teleportation sequence, followed by fast feedback for logical single-qubit rotation.

[0239] It will be appreciated that while certain operating parameter are provided below by way of example, increased fidelity in two-qubit gate errors may be achieved through variousHQU-01425 HU 9476 further optimizations. For example, increasing Rydberg laser power and detuning will reduce laser scattering errors and also suppress other errors by increasing gate speed. Cooling atoms to the motional ground state (thereby suppressing Doppler dephasing errors), and utilizing 10x higher laser power, theoretically results in >99.8% gate fidelities. Further improvements can be made with continued increases in laser power, but alternative routes such as single- photon excitation to Rydberg P states or alkaline-earth-based systems, are also available. Processor speed can be increased to a ~10 microsecond logical qubit cycle time by increasing collection efficiency or utilizing cavity-based or ensemble-based readout schemes, or by increasing movement speed with deeper optical tweezers.

[0240] To reach arbitrarily deep circuits, atoms may be continuously reloaded. Accordingly, some embodiments employ loading into a distant magneto-optical trap (MOT) and transporting atoms in an optical lattice conveyor belt.

[0241] In various embodiments, cross-talk during readout is suppressed by moving the ancilla atoms away from the data qubits.

[0242] Further scaling of the quantum processors can be achieved by connecting more than one microscope objective, either through atom transport or optical communication links. In various embodiments, the first approach utilizes the novel capabilities of atom rearrangement, combined with the use of optical lattice conveyor belts to coherently transport qubits between multiple active optical control regions and distribute entanglement. In various embodiments, the second approach utilizes photon-mediated entanglement between distinct atom array nodes with >104qubits. High entanglement rates can be achieved through parallel nanophotonic or bulk optical cavities, and the large sizes of atom arrays can provide further parallelism. This approach also enables modular construction of quantum processor units, flexibly rewired and linked together.HQU-01425 HU 9476

[0243] Quantum error correction (QEC) lies at the heart of fault-tolerant quantum computation (FTQC). A crucial component of FTQC is the error-correcting code, which describes how to encode quantum information in a redundant way with the goal of lowering the error rate of computation. Quantum error correction is typically implemented by measuring Pauli operators (called stabilizer generators) of a QEC code (referred to as a stabilizer code) to detect faults. An important subclass of stabilizer codes is Calderbank- Shor-Steane (CSS) codes, in which all non-identity components of stabilizer generators are either all Pauli ^^ or all Pauli ^^ operators. Quantum error correction in CSS codes works by measuring all the stabilizer generators and applying a correction based on the outcomes observed.

[0244] Low-density parity-check (LDPC) codes are a natural class of CSS codes to consider for implementation. They are families of stabilizer codes in which every stabilizer generator acts on a constant number of qubits and every qubit is involved in a constant number of generators. The code words of a parity check code are formed by combining a block of binary-information digits with a block of check digits. Each check digit is the modulo 2 sum (i.e., a sum that equals 1 if the ordinary sum is odd and 0 if the ordinary sum is even) of a pre-specified set of information digits. The formation rules for the check digits are represented by a parity-check matrix H, wherein the columns of H represent the binary- information digits, and the rows of H represent the check digits. Low-density parity-check codes are codes specified by a parity check matrix containing mostly 0’s and relatively few 1’s, such that the columns and rows have a relatively small weight (i.e., sum across columns or rows). The parity check matrix is thus a sparse matrix.

[0245] LDPC codes have been very successful in the classical setting as they approach upper bounds on the amount of information that can be reliably transferred through a noisy channel. Many modern technologies such as WiFi, DVB-T, and 5G are error corrected by LDPCHQU-01425 HU 9476 codes. Their quantum generalization requires additional conditions to be satisfied, namely that the ^^ and ^^ checks commute. Families of such codes have been constructed. However, traditional schemes for achieving quantum error correction, such as the surface code, are typically very costly in terms of resource overhead, requiring millions of qubits to solve problems of interest.

[0246] Approaches based on quantum low-density parity-check (qLDPC) codes provide a promising route to reduce the resources required, potentially enabling quantum computation with constant space overhead. However, long-range connectivity between qubits is necessary to have qLDPC codes with better code parameters (number of encoded qubits, code distance), making their physical realization rather challenging. The long-range and multi-layer connectivity required has not previously been demonstrated.

[0247] The present disclosure demonstrated the implementation of qLDPC codes with reconfigurable atom arrays (RAAs), the hardware architecture for quantum computation described above. The product structure present in many qLDPC codes naturally matches the parallelism afforded by acousto-optic deflectors, a core technology of the RAA platform. Combined with a new algorithm to perform arbitrary 1D qubit rearrangement in log(^^) time (^^ is the linear dimension of the system), while respecting the hardware constraints of current atom shuttling technologies, this results in efficient implementations of qLDPC codes that are within reach of demonstrated experimental capabilities.

[0248] Referring to Fig.17, the product structure of hypergraph product codes is illustrated. The hypergraph product code is constructed from two classical LDPC codes. The classical codes are illustrated on the left and top, where circles indicate data bits and squares indicate check bits. A data qubit is placed at each intersection of two classical data bits (type DD, filled circles with crosses) and of two classical check bits (type CC, filled circles without crosses). ^^ stabilizer checks are placed at the intersection of horizontal data bits and verticalHQU-01425 HU 9476 check bits, while ^^ stabilizer checks are placed at the intersection of horizontal check bits and vertical data bits. Each stabilizer is connected to data bits along the same row and column, with the same connectivity as the classical codes, as illustrated for the top left ^^ check. Other connections have been omitted for ease of visualization.

[0249] Quantum error correcting codes seek to encode a number of ^^ logical qubits into a larger number of ^^ physical qubits. One particularly convenient method to achieve this is with the stabilizer formalism, in which one applies a number of stabilizer checks to the physical qubits, monitoring the eigenvalue of certain products of Pauli operators on them. For simplicity, the following discussion focuses on CSS codes, in which each stabilizer generator is either a product of ^^ operators, or a product of ^^ operators, as described above. Logical operators are operators that commute with all stabilizers, but are not contained in the span of the stabilizers. The minimum weight logical operator (i.e., the logical operator with the fewest number of non-identity elements) defines the code distance ^^, which provides a rough characterization of the number of errors that a given code can handle. Together, theparameters [[^^, ^^, ^^]] provide a useful characterization of a QEC code.

[0250] For quantum codes, all operations are performed with the use of imperfect quantum gates. This is in contrast to the classical communication setting, where the encoding and decoding steps are almost perfect, and errors only occur during the communication itself. Thus, in order for the syndrome extraction to be fault-tolerant, it is likely necessary for all qubits to be involved in a bounded number of operations; in other words, the QEC code should be a low-density parity-check code, where each stabilizer has a constant weight that does not grow with the code size, and each data qubit is involved in a constant number of stabilizers.HQU-01425 HU 9476

[0251] Surface code is an example of a qLDPC code. However, unlike many other such codes, a single surface code patch encodes only a single logical qubit, thus requiring many patches—and hence a large overhead—to encode many logical qubits. In contrast, additional families of qLDPC codes are able to achieve a constant encoding rate, meaning that the ratio of logical qubits to physical qubits stays constant as the code size grows. In addition, “asymptotically good” families of such codes are available, in which both the number of encoded qubits and the code distance scale linearly with the number of physical qubits, thereby enabling low-overhead quantum computing, where the resource costs are much reduced compared to conventional schemes.

[0252] Another example of a qLDPC code, which also forms the basis of subsequent ones, is the hypergraph product code (HGP). A hypergraph is a graph when every edge connects to exactly two vertices and thus each edge has a cardinality of two. Here, one starts from two classical LDPC codes, and constructs a quantum code from the product of the two classical codes that inherits many of the properties of the classical codes. As illustrated in Fig.17, one can construct a hypergraph product starting from two classical LDPC codes, placed horizontally (1701) and vertically (1702), respectively. On the associated 2D grid (1703), a data qubit is placed at every intersection (filled circles with crosses) of a data bit and data bit (e.g., data qubit 1704 at the intersection of data bit 1705 and 1706), and at every intersection (filled circles without crosses) of a check bit and check bit (e.g., data qubit 1707 at the intersection of check bit 1708 and 1709). At every intersection of a horizontal data bit and a vertical check bit, a ^^ stabilizer check is placed (e.g., stabilizer qubit 1710 at the intersection of data bit 1705 and check bit 1708), while at every intersection of a horizontal check bit and a vertical data bit, an ^^ stabilizer check is placed (e.g., stabilizer qubit 1711 at the intersection of check bit 1709 and data bit 1706). Along each row and column of theHQU-01425 HU 9476 quantum code, qubits are connected in the same way as their corresponding classical codes, as illustrated for the top left ^^ check 1712.

[0253] Logical operators are inherited from the underlying classical code, and one can choose a basis such that each logical qubit has support in a single row or column. Mathematically, if the parity check matrix (where rows describe bits that should sum to aneven number in the absence of errors) of the two underlying classical codes is denoted asthe ^^ and ^^ stabilizer check matrices for the HGP code can bewritten asEquation 16

[0254] For classical [^^^, ^^^, ^^^] codes with ^^^ linearly-independent checks (^^ = 1,2), theresulting quantum code has parameters ^^^^^^ଶ + ^^^^^ଶ, ^^^^^ଶ, min{^^^, ^^ଶ}^. The surface codeis a special case of hypergraph product codes, with the classical codes being 1D repetition codes. However, by instead choosing classical expander codes as the underlying classicalcodes, where ^^^ = ^^(^^^), ^^^ = ^^(^^^), the resulting quantum code (known as a quantumexpander code) encodes a linear number of logical qubits ^^ = ^^(^^) and has a distance ^^scaling as ^^The distance ^^ is the minimum number of qubits that musttouched in order to change one logical codeword to a different logical codeword. The distance ^^ addresses the ability of the QEC code to correct errors, and a code with distance ^^can correct arbitrary errors affecting up to (^^ − 1) / 2 physical qubits. This is because anerror ^^ that anticommutes with an element ^^ of the stabilizer changes the eigenvalue of the codeword from +1 for ^^ to −1. Thus, measuring the eigenvalues of the generators of thestabilizer yields a binary vector of length ^^ − ^^ called the error syndrome, which can be usedHQU-01425 HU 9476 to identify which error occurred. Due to the expansion properties of the underlying graphs, quantum expander codes support single-shot quantum error correction, meaning that repeated rounds of syndrome extraction are not required to achieve fault-tolerance, unlike the surface code.

[0255] Although HGPs are not “asymptotically good,” in the sense that their distance does not scale linearly with the number of qubits, they form the basis of subsequent constructions that do achieve linear distance, where additional symmetry reductions are used to lower the number of qubits required to achieve a given distance. Thus, understanding the properties of HGP and being able to realize them also forms the foundation for implementing more complex code families.

[0256] Numerical simulations of HGP codes produce promising performance. Using a hypergraph product between classical expander codes constructed from (3,4)-biregular graphs, several studies have examined the thresholds and logical error rates, finding promising circuit- level thresholds of 0.28% and strong evidence that such codes can outperform surface codes as logical memories at moderate system sizes.

[0257] In order to achieve desirable code parameters, a certain number of long-range connections are required. In the present disclosure, reconfigurable atom arrays (RAAs) are employed as a platform to provide that connectivity. In this approach, qubits are encoded in long-lived hyperfine or nuclear degrees of freedom of the atom, with coherence times exceeding 1 second. Raman transitions are used for single-qubit manipulation, and strongly- interacting Rydberg states are used for two-qubit entangling gates. Due to the blockaded nature of Rydberg interactions, the gate action is insensitive to the precise location of the atoms, and operates whenever two atoms are next to each other under global gate laser illumination.HQU-01425 HU 9476

[0258] By coherently shuttling the atoms around in optical tweezers, one can reconfigure the processor connectivity on the fly and realize parallel two-qubit gate operations across the whole system. Crucially, optical tools such as acousto-optic deflectors (AODs) allow rapid parallel movement of entire grids of atoms, with only a few classical controls per logical qubit, as opposed to common approaches that require a few classical controls per physical qubit. Together with static spatial light modulator (SLM) optical tweezers that can produce arbitrary trap patterns, and the ability to transfer between AOD and SLM traps, one can realize arbitrary connectivity.

[0259] The RAA platform features efficient, parallel control and rearrangement of large numbers of qubits, enabling the implementation of long-range connected quantum processors. In addition, as explained above, qLDPC codes with improved code parameters (number of encoded qubits, code distance) rely on randomized expander graphs, for which the connectivity graph is much more complex. The present disclosure shows that despite these constraints, it is possible to efficiently implement the operations needed for HGP codes, in a rearrangement depth that scales only logarithmically with the number of qubits involved. Key to this construction is the realization that the product structure of HGPs is well-matched to the product structure of current AOD hardware. Combined with a new 1D parallel rearrangement scheme that achieves arbitrary permutations in log-depth without atom crossings, these techniques enable the near-term implementation of qLDPC codes.

[0260] To fault-tolerantly prepare an HGP code state, such as one where all logical qubits are initialized in |0^^, all physical qubits are first prepared in |0^, then all stabilizers are measured to project into the code space. Repeated measurements of all stabilizers and performing Pauli frame tracking allows one to preserve logical quantum information. A transversal readout of all physical qubits in one basis allows fault-tolerant measurement ofHQU-01425 HU 9476 the logical state. Thus, most key operations are straightforward, and the nontrivial part is the stabilizer measurement.

[0261] Referring to Figs.18A-C, syndrome extraction via permutations is illustrated. Fig. 18A shows an initial configuration of the qubits (maintaining the legend from Fig.17). Fig. 18B shows the result of column permutation, and Fig.18C shows the result of subsequent row permutation. Due to the product structure of hypergraph product codes, column (row) permutations are sufficient to implement all stabilizer measurements. The same permutation is applied in parallel across all columns (rows).

[0262] Referring to Fig.19, a syndrome extraction gate sequence is illustrated, with each row of qubits showing a configuration at a given point in time with the grey lines indicating movement between configurations. As discussed in connection with Figs.18A-C, each of the two dimensions of the HGP originates from a classical code and has the same connectivity (top). By interleaving parallel qubit rearrangements with global two-qubit gate laser pulses, the desired syndrome extraction circuit is implemented. Time runs down the page and the gray lines show the path of the data and ancilla qubits from one time step to the next.

[0263] As explained above, the stabilizers of the HGP code are inherited from the corresponding checks in the classical code. As illustrated in Fig.17, each stabilizer check is only connected to qubits in the same row or column. In the horizontal direction (similar for vertical), the parity check connectivities of the classical code are copied along all rows. Thus, by bringing together two columns that were connected in the original classical code, the required gates between all pairs of qubits in those two columns are implemented. Syndrome extraction thus involves first performing parallel column permutations to rearrange atoms into pairs, where each pair involves a single data bit and check bit that is connected in the horizontal classical code. By using the coloration circuits described, the number ofHQU-01425 HU 9476 permutation layers required is equal to the largest stabilizer weight in the classical code. As illustrated in Fig.19, each layer of rearrangement is interleaved with a global Rydberg laser pulse, which implements a CZ gate between each pair of neighboring atoms. Repeating the same procedure along the vertical direction, row permutations are performed to implement all required connections in the vertical direction, thus completing one round of syndrome extraction.

[0264] To ensure that the correct stabilizers are extracted, the relative ordering of the gates involved in an ^^ check and ^^ check that involve two shared data qubits must satisfy the following condition: the ^^ check should interact with both qubits before the ^^ check, or with both qubits after the ^^ check. When the ^^ and ^^ syndromes of the same cycle are extracted simultaneously, this condition cannot be satisfied, implying that in this case, full syndrome extraction requires two full cycles of row and column rearrangements. However, the syndrome extractions can be staggered, such that one round of ^^ syndromes is extracted, then the ^^ syndrome of the current round is extracted simultaneously with the ^^ syndrome of the following round. This ensures that the relative ordering is satisfied and thus that the syndrome extraction schedule is valid.

[0265] Referring to Fig.20, efficient non-intersecting rearrangement in log-depth is illustrated. By using a divide and conquer algorithm, an arbitrary 1D rearrangement is performed in depth logarithm in the number of qubits. Repeating this across the array yields an efficient implementation of the desired rearrangements, without requiring intersecting atom trajectories that may lead to additional loss and decoherence. In order to obtain a set of rearrangement steps to move from initial arrangement 2001 to final configuration 2002, the initial arrangement is prepared 2003 in the workspace. The qubits are bipartitioned 2004 intoHQU-01425 HU 9476 left and right subsets. Recursive sorting is performed 2005 on the subsets. The qubits are then moved 2006 to their final positions.

[0266] Arbitrary row or column permutations can be efficiently performed while respecting the hardware constraints of AOD-based atom shuttling. One constraint is the fact that different tones in an AOD, which generate different optical tweezer beams, are not allowed to cross while moving. This is due to the frequency beating that occurs when two tones approach each other, which can heat the atoms, as well as possible atom-atom collisions. It is thus necessary to develop efficient non-intersecting atom rearrangement schemes, in order to realize the permutations required for syndrome extraction.

[0267] A divide-and-conquer algorithm is provided herein, which decomposes an arbitrary one-dimensional permutation into a logarithm number of layers, where each layer consists of non- intersecting atom moves that can be performed in parallel. As shown in Fig.20, the aim is to place all ^^ atoms with final positions in the right half of the system into the correct side. This is achieved by first moving all atoms to the left-most available static SLM trap locations in the workspace (first layer in Fig.19), then moving all atoms that will end up in the right half to the right-most ^^ / 2 static traps in the system. The same procedure is recursively applied to the left half and right half of the system, as illustrated in the middle of Fig.20, until the desired ordering of atoms is reached. A final parallel AOD move transports the atoms to their desired location for gate operations.

[0268] Since every two layers reduces the system size by half, the total number of layersrequired to achieve the desired rearrangement is ^^(log ^^). Thus, arbitrary rearrangements invery large systems can be achieved in a small number of layers. Although this method shares some similarities to techniques such as bitonic sorting, the different constraints (comparators vs. parallel moves) lead to differences in the algorithm itself.HQU-01425 HU 9476

[0269] For the schemes described above, one can estimate the amount of time required to implement one round of stabilizer measurements using the technology that has been demonstrated. The following assumptions are employed: a transfer time ^^௧between a static SLM trap and dynamic AOD trap, a peak atom moving acceleration rate of ^^^with a cubic spline trajectory, and a uniform grid spacing ^^. For simplicity, it is further assumed that thenumber of atoms on a line to be rearranged is a power of 2, ^^ = 2^. In order to provideenough workspace for shuttling, the total number of traps is 3^^ / 2.

[0270] The compactification step at scale s requires a move of distance at most ^^^^ / 2. Moving all target atoms to the right requires a move of distance at most ^^^^. The two steps can be combined such that atoms for the next move are picked up and atoms from the previous move are dropped off at the same time; thus, each layer requires on average one trap transfer between static and dynamic traps. Using a cubic spline movement trajectory, a moveof distance ^^ requires time^6^^ / ^^^.

[0271] The total time required for one layer of full rearrangement is thusEquation 17

[0272] Recent experiments have demonstrated parameters on the order of ^^௧ = 50^^^^, ^^^ =^.^ଶఓ^ఓ^మ , ^^ =For a moderately sized code consisting of 10,000 qubits (including bothdata and ancilla qubits), ^^ ≈ 100. The total trap transfer time is 0.7 ms and the atommovement time is 2.3 ms, for each gate layer. Assuming a [3,4]-biregular graph for the underlying classical expander code, 8 rounds of rearrangement are required to measure one full round of stabilizers, resulting in a total time overhead of 24 ms, a small fraction of the coherence time of 2 s that has been demonstrated in neutral atom arrays. Although thisHQU-01425 HU 9476 timescale is somewhat longer than the typical readout timescales, the HGP code is single shot and consequently, only a single round of stabilizer measurement is required to perform error correction.

[0273] The present disclosure provides techniques for the efficient implementation of quantum low-density parity-check codes in reconfigurable atom arrays. These methods exploit the inherent parallelism of existing optical tools and product structure of many code constructions, enabling their efficient implementation on existing hardware.

[0274] These techniques are applicable to other code families that introduce additional ingredients on top of the hypergraph product structure. Moreover, these schemes can be readily extended to the implementation of lattice surgery gates on qLDPC codes, where the ancilla code patches can be viewed as a hypergraph product code between a subsection of the original code and a repetition code. The present disclosure enables near-term implementations of qLDPC codes and significantly reduces the resources required for large- scale fault-tolerant quantum computation.

[0275] Referring to Figs.21-23, an additional algorithm is illustrated for arbitrary atom rearrangement in two dimensions. In Fig.21, the parallel implementation of swap operationsbetween atoms separated by 2^ is shown. In Fig. 22, a graph ^^ × ^^ generated by the productof two sub-graphs ^^ and ^^ is illustrated. In Fig.23, the use of pinning beams (squares) to prevent movement in some columns is illustrated. By shining strong, local light spots, some of the atoms are prevented from moving under the parallel movement beam, allowing different rows to perform different permutations.

[0276] The algorithm involves three key components. First, it involves a technique for swapping all pairs of qubits separated by a distance 2^, where ^^ is an integer, using a number of extra workspace traps that is a constant factor of the total number of traps. Second, it usesHQU-01425 HU 9476 the idea of extra fast switchable optical traps as pinning beams to allow different operations on different rows or columns. Third, it uses a routing algorithm that decomposes routing problems on a graph with a product structure into three steps on the individual components.

[0277] Pairs of qubits that are separated by a distance 2^are swapped. By shining tweezer light on half of the atoms, corresponding to one atom of each pair of qubits separated by 2^, one sublattice is picked up and moved to a temporary buffer location. The other half of the atoms is then picked up and moved to the original position of the first sublattice. Finally, moving the first sublattice from the temporary buffer location to the original position of the second sublattice completes the swap operation.

[0278] This process is illustrated in Fig.21, where atoms separated by 2^positions are swapped in steps 2101. Atoms separated by 2ଶpositions are then swapped in steps 2102.

[0279] As noted above, a routing algorithm is used that decomposes routing problems on a product graph into routing problems on subgraphs. One example of such an algorithm known in the art is described in Baumslag, M., Annexstein, F. A unified framework for off-line permutation routing in parallel networks. Math. Systems Theory 24, 233–251 (1991). https: / / doi.org / 10.1007 / BF02090401, which is hereby incorporated by reference. However, it will be appreciated that a variety of alternative algorithms may be employed provided that they meet the criteria set out above.

[0280] In order to perform routing on a product graph ^^ × ^^ (Fig. 22), routing is performedin each row based on the routing algorithm on graph ^^, followed by routing in each column based on the routing algorithm on graph ^^, and finally another round of row routing based on graph ^^. A key distinction from the more constrained moves for a hypergraph product code is that here, different rows or columns have to execute different movements.HQU-01425 HU 9476

[0281] Accordingly, local pinning beams that can be rapidly switched on and off are used to realize different movements in different columns or rows. As illustrated in Fig.23, by shining pinning beams (shown as squares, e.g., 2301) on select atoms, certain atoms can be prevented from moving while still applying the same overall motion to all rows or columns. A pinning beam may be implemented by, e.g., a digital mirror device (DMD) that can be switched on a very fast timescale.

[0282] Comparing Fig.21 and Fig.23, one sees that pinning allows the implementation of swaps on only a selective set of qubit sites.

[0283] Putting all of these ingredients together provides a rearrangement algorithm that achieves an arbitrary 2D atom rearrangement in ^^(^^^^^^^^)depth, where ^^ is the total number of qubits. First, the 2D atom rearrangement is decomposed into row, column, row permutations using the product routing algorithm above. Then, the permutations within eachrow or column are implemented by combining swaps at distance 2^, ^^ = 1,2, ... , ^^^^^^^^ withlocal pinning beams.

[0284] An arbitrary 1D permutation can be implemented by ^^(^^^^^^^^) swaps by viewing it as a routing problem on a hypercube, and applying recursively the product routing algorithm. More specifically, routing on 2^qubits can be achieved by viewing the 2^qubits as a product of two graphs with 2^ି^qubits and 2 qubits, then routing on the 2-qubit graph, followed by the 2^ି^qubit graph, and then again on the 2-qubit graph. Local pinning beams allow performing different permutations in the different rows and / or columns. Accordingly, in some embodiments, the permutations are row- or column- specific rather than universal across all rows or columns.

[0285] While the above description uses a routing algorithm that decomposes routing problems on a product graph into routing problems on subgraphs after Baumslag, alternativeHQU-01425 HU 9476 routing algorithms may be employed. For example, algorithms based on bitonic sorting, such as that described in Litinski, D., Nickerson, N. Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections. arXiv:2211.15465 [quant-ph] (2022). https: / / doi.org / 10.48550 / arXiv.2211.15465, which is hereby incorporated by reference. This bitonic based sorting algorithm has complexity ^^(^^^^^^ଶ^^).

[0286] The algorithms described herein, improve this complexity to ^^(^^^^^^^^) by exploiting the fact that unlike bitonic sorting networks, which require bit-wise comparison to determine where each element should go, the use cases herein have information about the target locations in advance.

[0287] In addition to the HGP codes described in other examples, the present disclosure may be used to implement lifted product (LP) codes.

[0288] LP codes are a modified version of the HGP codes. A quasi-cyclic LP code is obtained from two base protographs associated with two base matrices ^^^and ^^ଶover thequotient polynomial ring ℝ[^^] / (^^^− 1). Suppose the two base matrices are of size^^^భ × ^^^భ and ^^^మ × ^^^మ, respectively. One can obtain two matrices (over the samepolynomial ring) ^^௫ and ^^௭ by taking the hypergraph product between ^^^ and ^^ଶ:Equation 18

[0289] Then, the ^^ (^^) check matrix ^^௫(^^௭) is obtained by replacing each entry of ^^௫(^^௭) with its matrix representation with ^^ by ^^ circulant matrices, a process known as lift. Thecode size is ^^ = ^^൫^^^భ^^^మ +and the number of ^^ and ^^ checks is ^^௫ = ^^^^^భ^^^మand ^^௭ = ^^^^^భ^^^మ, respectively. Assuming that ^^^^ and ^^௭ are full rank, then this yields a^^^, ^^ − ^^, ^^^-LP code.HQU-01425 HU 9476

[0290] The above construction can also be described using graphs. As an example, Fig.24 shows a LP code using a 3 by 5 protograph and a lift size 2. The LP code is constructed by taking a lift over the hypergraph product of two classical protographs. The protographs and their hypergraph product are indicated by the dashed nodes (e.g., 2401) and the lift is illustrated by the multiple inner nodes within each dashed node (e.g., 2402, 2403). The inner connectivity between two dashed nodes is given by the matrix representation of the ring elements in Equation 18. When flattening the inner nodes vertically (horizontally), the vertical (horizontal) connectivity between the qubits and the checks for each column (row) is the same as the left 2404 (top 2405) lifted classical code.

[0291] An ^^-th dashed check node is connected to the ^^-th dashed variable node if ^^^^ (^^: =^^^ = ^^ଶ) is non-zero. A lift of the protograph is done by replacing each dashed node with itstwo inner solid nodes, and setting up the connectivity between the inner nodes according to the matrix representation of each ring element ^^^^. Equation 18 corresponds to taking the hypergraph product between the protograph and itself, obtaining a grid of dashed nodes. Similar to the hypergraph product code, the connectivity between the dashed nodes (the entries of ^^௫and ^^௭) is inherited from ^^^and ^^ଶ. Then the qubits and the quantum checks are given by the inner nodes after the lift, and their connectivity is given by the matrix representation of ^^௫and ^^௭. An important feature of the LP codes is that they still have some remaining product structure even after the lift. As shown in Fig.24, when flattening the inner nodes vertically (horizontally), the vertical (horizontal) connectivity between the qubits and the checks for each column (row) is the same as the left (top) lifted classical code.

[0292] For a matrix entry over the polynomial ring, its weight is denoted as its number of terms. For the LP codes constructed in this disclosure, a base matrix is chosen with all weight-one entries of dimension 3 by 5 and a family of codes is obtained with sizes up toHQU-01425 HU 9476 1428 by increasing the shift-lift size ^^. The classical parity checks constructed are optimized by choosing the base matrix entries over the quotient polynomial ring to obtain the best minimum distance for the particular shift-lift size ^^. The choice of the base matrix entries is also such that the girth (length of the shortest cycle in the Tanner graph) is at least 8, and the minimum distances of the lifted product qLDPC codes are the same as the designed classical minimum distances. Note that the choice of the base weight matrix determines the bounds on the best possible minimum distance for such code construction. Allowing multiple weights and more general protographs give more flexibility in qLDPC code design and improve their minimum distances.

[0293] Referring back to Equation 17, t overheads for LP codes may be estimated as follows.

[0294] For LP codes, one needs to first flatten the code layout before implementing the parallel rearrangement scheme. For a fixed 3 by 5 protograph, the flattened rectangle array has dimensions of 2^^ / 8 by 8. This can be achieved in log depth using a divide-and-conquer algorithm that flattens the code by half each time. For example, as shown in Fig.24, the codes are flattened vertically before establishing vertical connections between atoms via row permutations. The vertical connectivity is then the same as an HGP code, and one can use the efficient 1D rearrangement scheme described earlier. Therefore, the rearrangement time for LP codes is estimated by setting L to n / 8 in Equation 17.

[0295] The rearrangement time in Equation 17 determines the idling errors between sequences of entangling gates in a syndrome extraction circuit. In general, ^^^^^^^^^^^(^^) is a function of the code size ^^, as ^^ is a function of ^^. Assuming the idling errors ^^^(^^) can also improve as the gate error ^^^improves: ^^^0.005Equation 19HQU-01425 HU 9476 where the coherence time ^^^and other constant parameters in ^^^^^^^^^^^(^^)listed above.

[0296] Exemplary Methods

[0297] Code constructions

[0298] The following section focuses on two families of qLDPC codes, although it will be appreciated that these results may be extended to asymptotically good codes.

[0299] The first family of codes are hypergraph product (HGP) codes, formed from the product of two classical LDPC codes. A geometric sketch of the code properties is provided above, and the following discussion focuses on an alternative algebraic description of the codes and provides more details of their code properties. Algebraically, if the parity check matrices (where rows describe bits that should sum to an even number in the absence oferrors) of the two underlying classical codes are given as ^భ×^భ ^మ×^మ∈ ^^ଶ, ^^ଶ∈ ^^ଶ, then the^^ and ^^ stabilizer check matrices for the HGP code can be written asEquation 21

[0300] For classical [^^^, ^^^, ^^^] linear codes defined by ^^^ = ^^^ − ^^^ linearly-independentchecks (^^ = 1,2), the resulting quantum code has parameters ^^^^^^ଶ +^^^^^ଶ, ^^^^^ଶ, ^^^^^^{^^^, ^^ଶ}^. The surface code is a special case of hypergraph product codes,with the classical codes being 1D repetition codes. However, by instead choosing classicalHQU-01425 HU 9476codes with good vertex expansion as the underlying classical codes, where ^^^ = Θ(^^^), ^^^ =Θ(^^^), the resulting quantum code (known as a quantum expander code) encodes a linearnumber of logical qubits ^^ = Θ(^^) and has distanceSuch classical expandercodes can be obtained asymptotically, for example, from random biregular Tanner graphs, and will have sufficient vertex expansion with high probability. Logical operators are inherited from the underlying classical code, and one can choose a basis such that each logical qubit has support in only a single row or column.

[0301] In this discussion, HGP codes are constructed by taking the hypergraph product of classical LDPC codes defined by (3,4)-regular Tanner graphs, i.e., bipartite graphs with degree-3 bit nodes and degree-4 check nodes. By increasing the size of the graph, a familyof HGP codes is obtained with a constant encoding rate ^^ ≥ 0.04. For each code size, theclassical code is selected having the largest distance, Tanner graph girth larger than 6 (length of the shortest cycle in the Tanner graph, obtained through rejection sampling without performing edge swaps), and the largest spectral gap (the gap between the largest two singular values of the check matrices) from randomly generated instances. The hypergraph product of vertex-expanding classical codes yields HGP codes that satisfy the syndrome confinement property, and support single-shot QEC.

[0302] The second family of codes considered are quasi-cyclic lifted product (LP) codes, which can be viewed as a hypergraph product code followed by a symmetry reduction to reduce the number of required qubits. Algebraically, a quasi-cyclic LP code is obtained from two base protographs (analogs of the classical codes in the HGP construction) associated with two base matrices ^^^and ^^ଶover the quotient polynomial ringSuppose the two basematrices are of size ^^^భ × ^^^భ and ^^^మ × ^^^మ, respectively. Two matrices are obtained (overthe same polynomial ring) ^^௫and ^^௭by taking the hypergraph product:HQU-01425 HU 9476Equation 22

[0303] The ^^ (^^) check matrix ^^௫(^^௭) is then obtained by replacing each entry of ^^௫(^^௭) with its matrix representation as ^^ by ^^ circulant matrices, a process known as a lift. The codesize is ^^ = ^^൫^^^భ^^^మ +the number of ^^ and ^^ checks are ^^௫ = ^^^^^భ^^^మ and^^ = ^^^^ ^^ , respectively. The encoding rate is lower bounde௭^భ ^మ d by ே =

[0304] One can also describe the above construction using graphs. As an example, Fig.24shows a LP code using a 3 by 5 protograph associated with a base matrix ^^The checks and bits of the protograph are illustrated by the big dashed nodes. The ^^-th dashed check node is connected to the ^^-th dashed bit node if ^^^^is non-zero. A lift of the protograph is done by replacing each dashed node with its two inner solid nodes, and setting up the connectivity between the inner nodes according to the matrix representation of each ring element ^^^^. Equation 22 corresponds to taking the hypergraph product between the protograph and itself, obtaining a grid of dashed nodes. Similar to the hypergraph productcode, the connectivity between the dashed nodes (the entries of ^^௫ and ^^௭) is inherited from^^. Then the qubits and the quantum checks are given by the inner nodes after the lift, andtheir connectivity is given by the matrix representation of ^^௫and ^^௭. An important feature of the LP codes is that they still have some remaining product structure even after the lift. AsHQU-01425 HU 9476 shown in Fig.24, when flattening the inner nodes vertically (horizontally), the vertical (horizontal) connectivity between the qubits and the checks for each column (row) is the same as the left (top) lifted classical code.

[0305] For the LP codes constructed in this disclosure, a base matrix of dimension 3 by 5 is chosen, where all entries have a single polynomial term, and a family of codes is obtained with sizes up to 1428 by increasing the lift size ^^ from 16 to 42. The classical parity checks are optimized by choosing the base matrix entries over the quotient polynomial ring to obtain the best classical distance for the particular lift size ^^. The choice of the base matrix entries is also such that the girth is at least 8, and the distances of the lifted qLDPC codes match the designed classical distances with a high probability. Allowing multiple polynomial terms for each base matrix entry and more protographs of different sizes gives more flexibility in qLDPC code design and improves their distances. The classical base matrices used to construct the four LP codes used in this example are provided below. Denoting ^^ௗ^as a base matrix with a lift size ^^ and a classical code distance ^^ after the lift, the base matrices areEquation 24HQU-01425 HU 9476

[0306] The quantum code distances are upper bounded by the classical code distances of the above (lifted) base matrices. After extensive search for minimum-weight logical operators using a GAP package, these upper bounds appear to be tight.

[0307] Atom rearrangement algorithm

[0308] The reconfigurable atom array platform features efficient, parallel control and rearrangement of large numbers of qubits, enabling the implementation of long-range connected quantum processors. As discussed herein, optical tools such as crossed acousto- optic deflectors (AODs) can generate a rectangular grid of optical tweezers that can be reconfigured on the fly, allowing the control of large code blocks consisting of thousands of physical qubits with only a handful of classical controls.

[0309] However, the use of AODs for dynamic rearrangement comes with two key constraints. First, as the X and Y direction optical spots are controlled by separate AODs, the same operation needs to be applied across multiple rows and / or columns. Second, different rows of atoms cannot cross each other due to beating between RF tones and atom collisions, although they can be temporarily transferred and stored in static traps, such as those based on spatial light modulators (SLMs). Thus, the implementation of qLDPC codes with improved code parameters (number of encoded qubits, code distance), which often relies on pseudorandom expander graphs with complex connectivity graphs, requires the development of efficient atom rearrangement algorithms.

[0310] Figs.25-27 provide a detailed description of an atom rearrangement algorithm in the form of Algorithms 1-3.

[0311] The first component, arbitrary 1D atom rearrangements with a number of steps that scales logarithmically, is described in detail in Algorithm 1, illustrated in Figs.28A-D, and explicitly worked out for a small example in Fig.20. Since successive layers each reduce the system size by half, the total number of layers required to achieve the desired rearrangementHQU-01425 HU 9476 is ⌈^^^^^^ଶ^^⌉. Thus, arbitrary rearrangements in very large systems can be achieved in a small number of layers. Although this method shares some similarities to techniques such as bitonic sorting, the different constraints (comparators vs. parallel moves) lead to differences in the algorithm itself. The algorithm can also be applied to use parallel qubit swaps at increasing distances as the basic primitive, with the same ^^(^^^^^^^^) scaling with system size.

[0312] Figs.28A-D illustrate the efficient implementation of quantum LDPC codes with atom arrays. Fig.28A is an illustration of an algorithm to perform an arbitrary log-depth rearrangement. First, all atoms that need to end in the right half of the system are moved to the right side, then each half is compacted into adjacent sites, so that there is sufficient workspace for subsequent steps. The same procedure can then be repeated on each half of the system recursively for depth ^^^^^^(^^), where ^^ is the length of the atom array to be rearranged. Fig.28B is an illustration of the HGP code, obtained as a product of two classical codes. Lines indicate that the parity check at the syndrome node involves the corresponding data node. Figs.28C-D show the required connectivity implemented via parallel row permutations, followed by parallel column permutations.

[0313] The second component is the observation that the product structure of crossed AODs matches well with the product structure present in many qLDPC codes. Details of a syndrome extraction circuit for HGP codes are provided, based on this observation in Algorithm 2, which is referred to as the product coloration circuit, as it makes use of coloration circuits for each of the component classical codes. The use of the product coloration circuit, as opposed to alternative coloration or cardinal circuits, is necessary to fully exploit the parallel rearrangement capabilities across rows and columns. Here, the native entangling gate set of current atom array systems is diagonal, so CZ gates and appropriate Hadamard rotations are used to perform syndrome extraction. Under global laser excitation and phase advances, any pair of qubits that are within a certain radius (known asHQU-01425 HU 9476 the blockade radius) of each other will execute a CZ gate, while any individual qubits will undergo an identity gate. In order to analyze these results, CNOT gates are used as the entangling gates in the simulations. This can be physically justified if the CZ gates are much noisier than the Hadamard gates.

[0314] The product coloration circuit separately extracts the ^^ and ^^ syndromes, each requiring both a horizontal and vertical step. Thus, if the coloration of each of the classical codes involves Δ^colors (for the codes constructed from (3,4)-biregular graphs that areconsidered herein), Δ^ = 4, the product coloration circuit will have 4Δ^ entangling layers.

[0315] The product coloration circuit can also be applied to the LP codes used herein. As shown in Fig.24, a LP code has the same product vertical (horizontal) connectivity as a HGP code when flattening the inner nodes vertically (horizontally). Thus, the same product coloration circuit can be applied to the LP codes with an extra step of flattening the inner codes in between establishing the horizontal / vertical connections. As 3 by 5 base matrices are used with all weight-one entries, the product coloration circuit for the LP codes has anentangling gate depth of 4 × 5 = 20.

[0316] To further reduce the depth of the syndrome extraction circuit, a modification of the above circuit is provided in Algorithm 3 and Fig.29, which is referred to as the pipelined product coloration circuit. Here, the main challenge is to choose a gate ordering such that the desired ^^ and ^^ syndromes are correctly extracted. By performing pipelining and extracting the ^^ syndrome of the second round simultaneously with the ^^ syndrome of the first round, one can ensure that the gate ordering is always valid, while reducing the number ofentangling layers required to perform ^^ rounds of syndrome measurement to (2^^ + 2)Δ^.This could be particularly relevant in further suppressing the effect of idling errors as well asHQU-01425 HU 9476 improving the performance of logical gates, which in this scheme require ^^ rounds of repetition.

[0317] Referring to Figs.29A-B, ordering of operations in pipelined syndrome extraction is illustrated. In Fig.29A, successive steps of entangling gates for the pipelined productcoloration circuit described in Algorithm 3 are shown, with ^^ = 3 rounds of syndromeextraction. Numbers at the corners of the ^^ and ^^ ancilla qubits denote the round of syndrome extraction they correspond to. Fig.29B is an illustration of a local circuit that data qubits and ancilla qubits of the same round see, with dashed lines indicating different circuit moments. As the ^^ stabilizer interacts with both qubits before the ^^ stabilizer, the syndrome extraction order is valid. Similar analysis can be performed for the commutation relations with the next round of ancilla qubits.

[0318] Details of teleportation

[0319] The following discussion describes a teleportation scheme between the qLDPC codeand the surface code. A logical ^ത^^ operator is selected for the qLDPC code of minimumweight. This ensures it contains no sublogicals, i.e., inequivalent logical operators contained in its support, so that the scheme is fault-tolerant under data errors. The qubit support of ^ത^^of the qLDPC code (^̅^ଶof the surface code) is associated to the bits of a classical code ^^^(^^ଶ), and associated to the ^^ (^^) stabilizers of the qLDPC (surface) code with support on ^ത^^(^̅^ ) to the checks of ^^ (^^ ). Denoting ^^^ ଶ ^ ଶଶ ^ ଶ (^^ ) as the check matrix for ^^^ (^^ଶ), ^^^^ ൫^^^^ ൯ =1 if the ^^-th ^^ (^^) stabilizer checks the ^^-th qubit of ^ത^^ (^̅^ଶ). An ancilla code patch isconstructed as an HGP code by taking the hypergraph product of ^^^and ^^ଶ, which encodes asingle logical qubit with a logical ^^ and ^^ representative associated with the bits of ^^^ and^^ଶ, respectively.HQU-01425 HU 9476

[0320] Lattice surgery between the ancilla patch and the qLDPC (surface) code is realized by merging and splitting along ^^^(^^ଶ), assisted by an extra array of ancillary qubits (see Fig. 5A). For a classical linear code ^^ with check matrix ^^, denote by ^^்its transposed code defined by the check matrix ^^். Taking the qLDPC-ancilla surgery as an example, an extra array of ^^ stabilizers and qubits (initialized in|+^) is inserted in the middle, associated with the checks and the bits of ^^^், respectively. During the code merging, the Z stabilizers of the qLDPC and the ancilla patch associated with the ^^-th check of ^^^are each modified to include the middle qubit associated with the ^^-th bit of ^^^். The middle ^^ stabilizer associated with the ^^-th check of ^^^்checks the two qubits of the qLDPC and surface codes associated with the ^^-th bit of ^^^as well as some middle qubits given by the incident relation of ^^^். It is easy to verify that all the new stabilizers commute as the added / modified qubits and checks across the merged boundary form an HGP code with ^^^and a length-2 repetition code locally. The product of the middle ^^ stabilizers gives the joint logical operator to measure. The detailed merging and splitting protocol is the same as that for the surface codes. The presented scheme allows for teleportation between any two quantum codes with CSS logical operators, that is, logical operators comprising of all Pauli ^^s or all Pauli ^^s.

[0321] Numerical simulations were performed using Stim, using the same space-time decoder described in the previous section for decoding. For ease of simulation, circuit depolarizing noise is added with no idling errors on the joint logical ^^^^ measurement part of the teleportation circuit, in both the merging and splitting steps, and the gate error rate isrescaled ^^^ → ^^^ + 3^^^ to obtain Fig. 5C. As the merged code no longer supports single-shot error correction,+ 1 rounds of QEC are performed after each of the merge andsplit steps of the ^^^^ surgery, where ^^ = ^^^^^^(^^^, ^^ଶ) is the distance of the teleportationscheme and ^^^and ^^ଶare the distances of the qLDPC code and surface code, respectively.HQU-01425 HU 9476 The logical failure rate of the ^^^^ logical measurement would be nearly the same as that of the qLDPC memory when using a computation surface code with a distance larger than that of the qLDPC code. For the logical ^^^measurement, a noiseless destructive measurement is simulated.

[0322] Dynamic reconfiguration in 2D tweezer arrays

[0323] Exemplary experiments utilize the apparatus described below. Inside the vacuum cell,87Rb atoms are loaded from a magneto-optical trap into a backbone array of programmable optical tweezers generated by a spatial light modulator (SLM). Atoms are rearranged in parallel into defect-free target positions in this SLM backbone by additional optical tweezers generated from a crossed 2D acousto-optic deflector (AOD). Following the rearrangement procedure, selected atoms are transferred from the static SLM traps back into the mobile AOD traps, and then these mobile atoms are moved to their starting positions in the quantum circuit. During this entire process, the atoms are cooled with polarization gradient cooling. Before running the quantum circuit, a camera image of the atoms in their initial starting positions is taken. Following the circuit, a final camera image is taken to detect qubit states |0^ (atom presence) and |1^ (atom loss, following resonant pushout). All data are postselected on finding perfect rearrangement of the AOD and SLM atoms before running the circuit. In some embodiments, each atom remains in a single static or single mobile trap throughout the duration of the quantum circuit.

[0324] The crossed AOD system is composed of two independently controlled AODs (AA Opto Electronic DTSX-400) for ^^ and ^^ control of the beam positions. Both AODs are driven by independent arbitrary waveforms which are generated by a dual-channel arbitrary waveform generator (AWG) (M4i.6631-x8 by Spectrum Instrumentation) and then amplified through independent MW amplifiers (Minicircuits ZHL-5W-1). The time-domain arbitraryHQU-01425 HU 9476 waveforms are composed of multiple frequency tones corresponding to the ^^ and ^^ positions of columns and rows, which are independently changed as a function of time for steering around the AOD-trapped atoms dynamically; the full ^^ and ^^ waveforms are calculated by adding together the time-domain profile of all frequency components with a given amplitude and phase for each component. For running quantum circuits, the positions of the AOD atoms at each gate location are programmed and then the AOD frequencies are smoothly interpolated (with a cubic profile) as a function of time between gate positions. The cubicprofile enacts a constant jerk onto the atoms, which allows movement of roughly 5 − 10 ×faster (without heating and loss) than if moving at a constant velocity (linear profile). In the movement protocol, stretches, compressions, and translations of the AOD trap array are applied: i.e., the AOD rows and columns never cross each other in order to avoid atom loss and heating associated with two frequency components crossing each other.

[0325] The AOD tweezer intensity is homogenized throughout the whole atom trajectory in order to minimize dephasing induced by a time-varying magnitude of differential light shifts. To this end, a reference camera is used in the image plane to gauge the intensity of each AOD tweezer at each gate location and to homogenize by varying the amplitude of each frequency component; during motion between two locations the amplitude of each individual frequency component is interpolated.

[0326] The SLM tweezer light (830 nm) and the AOD tweezer light (828 nm) are generated by two separate, free-running Ti:sapphire lasers (M Squared, 18-W pump). Projected througha 0.5 NA objective, the SLM tweezers have a waist of roughly ∼ 900^^^^ (∼ 1000^^^^ forAODs). When loading the atoms, the trap depths are ∼ 2^^ × 16^^^^^^, with radial trapfrequencies of ∼ 2^^ × 80^^^^^^, and, when running quantum circuits, the trap depths are ∼2^^ × 4^^^^^^, with radial trap frequencies of ∼ 2^^ × 40^^^^^^.HQU-01425 HU 9476

[0327] Raman laser system

[0328] Fast, high-fidelity single-qubit manipulations are critical ingredients of the quantum circuits demonstrated in this work. To this end, a high-power 795-nm Raman laser system isused for driving global single-qubit rotations between ^^ி = 0 clock states. This Raman lasersystem is based on dispersive optics. 795-nm light (Toptica TA pro, 1.8W) is phase- modulated by an electro-optic modulator (Qubig), which is driven by microwaves at 3.4 GHz (Stanford Research Systems SRS SG384) that are doubled to 6.8 GHz and amplified. The laser phase modulation is converted to amplitude modulation for driving Raman transitions through the use of a Chirped Bragg Grating (Optigrate). IQ control of the SG384 is used for frequency and phase control of the microwaves, which are imprinted onto the laser amplitude modulation and thus provide direct frequency and phase control over the hyperfine qubit drive.

[0329] The Raman laser illuminates the atom plane from the side in a circularly polarized elliptical beam with waists of 40^^^^ and 560^^^^ on the thin axis and the tall axis, respectively, with a total average optical power of 150^^^^ on the atoms. The large verticalextent ensures < 1% inhomogeneity across the atoms, and shot-to-shot fluctuations in thelaser intensity are also < 1%. The Raman laser is operated at a blue-detuned intermediate-state detuning of 180 GHz, resulting in two-photon Rabi frequencies of 1 MHz and anestimated scattering error per ^^ pulse of 7 × 10ିହ (i.e. 1 scattering event per 15000 ^^ pulses).

[0330] Qubit coherence and dynamical decoupling

[0331] In the 830-nm traps, hyperfine qubit coherence is characterized by ^^∗ଶ = 4^^^^, ^^ଶ =1.5^^ (XY16 with 128 total ^^ pulses), and ^^^ = 4 s (including atom loss). The experimentsdescribed herein are performed in a DC magnetic field of 8.5 Gauss. Coherence can be further improved by using further-detuned optical tweezers (with trap depth held constant, theHQU-01425 HU 9476 tweezer differential lightshifts decrease as 1 / Δ and 1 / ^^^decreases as 1 / Δଷ) and shielding against magnetic field fluctuations. For practical QEC operation, atom loss can be detected in a hardware-efficient manner and the atom can then be replaced from a reservoir, which could in principle be continuously reloaded by a MOT for reaching arbitrarily deep circuits.

[0332] The transport sequences are accompanied with dynamical decoupling sequences. The number of pulses used is a tradeoff between preserving qubit coherence while minimizing pulse errors. In various embodiments, there is an interchange between two types of dynamical decoupling sequences: XY8 / XY16 sequences, composed of phase-alternated individual ^^-pulses which are self-correcting for amplitude and detuning errors, and CPMG- type dynamical decoupling sequences composed of robust BB1 pulses. The CPMG-BB1 sequence is more robust to amplitude errors but incurs more scattering error. The sequence may be empirically optimized for any given experiment by choosing between these different sequences and a variable number of decoupling ^^ pulses, optimizing on either single-qubit coherence (including the movement) or the final signal. Typically, decoupling sequences are composed of a total 12-18 ^^ pulses.

[0333] Movement effects on atom heating and loss

[0334] The following discusses the effects of movement on atom loss and heating in the harmonic oscillator potential given by the tweezer trap. Motion of the trap potential is equivalent to the non-inertial frame of reference where the harmonic oscillator potential isstationary, but the atom experiences a fictitious force given by ^^(^^) =where ^^ isthe mass of the particle and ^^(^^) is the acceleration of the trap as a function of time. The average vibrational quantum number increase Δ^^ is given byEquation 25HQU-01425 HU 9476 where ^^^(^^^) is the Fourier transform of ^^(^^) evaluated at the trap frequency ^^^, and the zeropoint size of the particleΔ^^ is the same for all initial levels of theoscillator. Experimentally, an acceleration profile ^^(^^) = ^^^^ is applied to the atom, from time−^^ / 2 to +^^ / 2 to move a distance ^^ with constant jerk ^^. Calculating |^^^(^^)|ଶ, simplifyusing ^^^^^ ≫ 1, and assume a small range of trap frequencies to average the oscillatoryterms, results inEquation 26

[0335] Several relevant insights can be gleaned from this formula. First, this expression indicates the ability to move large distances ^^ with comparably small increases in time ^^.Furthermore, to maintain a constant Δ^^, the movement time ^^ ∝ ^^ ଷ / ସ^ି . Moreover, toperform a large number of moves ^^ for a deep circuit, Δ^^ ∝ ^^ / ^^ସ can be estimated,suggesting that the number of moves can be increased from, e.g., 5 to 80 by slowing each move from 200^^^^ to 400^^^^. Move speed could be further improved with different ^^(^^) profiles, but inevitably with finite resources such as trap depth, quantum speed limits will eventually prevent arbitrarily fast motion of qubits across the array.

[0336] Equation 26 is now compared to experimental observations. Atom loss is observed with movement of 55 ^^m in 200^^^^ under a constant negative jerk. This speed limit isconsistent with the above estimates: using ^^^ = 2^^ × 40^^^^^^ and ^^௭^^ = 38^^^^, it ispredicted that Δ^^ ≈ 6 for this move, corresponding to the onset of tangible heating at thismove speed. More quantitatively, a Poisson distribution is assumed with mean ^^ andHQU-01425 HU 9476 variance ^^ and the population is integrated above some critical ^^^^௫upon which the atomwill leave the trap. From this analysis, atom retention is given by

[0337] Additional heating and loss during the circuit can also be caused by repeated short drops for performing two-qubit gates, where the tweezers are briefly turned off to avoid anti- trapping of the Rydberg state and light shifts of the ground-Rydberg transition. However, drop-recapture measurements suggest the 500-ns drops used experimentally have a negligible effect until hundreds of drops per atom (corresponding to hundreds of CZ gates). Atom loss and heating as a function of number of drops are well-described by a diffusion model, whichwould then predict that reducing atom temperature by a factor of 2 × (reducing thermalvelocity by √2 ×) and reducing drop time ^^ௗ^^^ by 2 ×, together would increase the numberof possible CZ gates per atom to thousands.

[0338] Two-qubit CZ gates implementation

[0339] Two-qubit gates and calibrations may be implemented using the techniques provided herein. Specifically, the two-qubit CZ gate is implemented by two global Rydberg pulses, with each pulse at detuning Δ and length ^^, and with a phase jump ^^ between the two pulses. The pulse parameters are chosen such that qubit pairs, adjacent and under the Rydberg blockade constraint, will return from the Rydberg state back to the hyperfine qubit manifold with a phase depending on the state of the other qubit. The numerical values for these pulseparameters are:Δ = −0.377371Ω^^ = −0.621089 × (2^^)^^ = 0.683201 / [Ω / (2^^)]Equation 27HQU-01425 HU 9476

[0340] Exemplary experiments are operated with a two-photon Rydberg Rabi frequency ofΩ / 2^^ = 3.6^^^^^^, giving a theoretical ^^ = 190^^^^ and a theoretical Δ / (2^^) = −1.36^^^^^^.The negative detuning sign is chosen to help minimize excitation into the ^^^ = +1 / 2Rydberg state which is detuned by about 24 MHz under the field of 8.5 G (and experiences a3 × lower coupling to the Rydberg laser than the desired ^^^ = −1 / 2 state due to reducedClebsch-Gordan coefficients). In this work, strong blockade between adjacent qubits is provided, with Rydberg-Rydberg interactions ^^^ / 2^^ ranging from 200 MHz to 1 GHz.

[0341] Managing spurious phases during CZ gates

[0342] The two-qubit gate induces both an intrinsic single-qubit phase, as well as spurious phases which are primarily induced by the differential light shift from the 420-nm laser. Under certain configurations, the 420-nm-induced differential light shift on the hyperfinequbit can be exceedingly large (> 8^^^^^^), yielding phase accumulations on the hyperfinequbit of ≈ 6^^. Small, percent-level variations of the 420-nm intensity can thus lead tosignificant qubit dephasing.

[0343] This 420-induced-phase issue may be addressed by performing an echo sequence: after the CZ gate, the 1013-nm Rydberg laser is turned off, a Raman ^^ pulse is applied, and then the 420-nm laser is pulsed again to cancel the phase induced by the 420 light during the CZ gate. This method echoes out the 420-induced phase, but comes at a cost of a factor of two increase in the 420-induced scattering error, which is the dominant source of error in two-qubit CZ gates.

[0344] Echo between CZ gates. To address these various issues, a Raman ^^ pulse is performed between each CZ gate to echo out spurious gate-induced phases on the hyperfine qubit. This approach has several advantages. The 420-induced phase is now cancelled by pairs of CZ gates, without explicitly applying additional 420-nm pulses to echo eachHQU-01425 HU 9476 individual CZ gate, thereby reducing the scattering error of the CZ gate in this work by a factor of approximately two. This echo technique, having reduced the scattering error incurred during each gate, roughly compensates for the increased scattering rate incurred by spreading optical power over more space in 2D, thereby giving comparable gate fidelities tothe two-qubit CZ gate fidelities of ≥ 97.4(2)%. Further, the echo between CZ gates alsocancels the intrinsic single-qubit phase of the CZ gate, removing errors in the calibration of this parameter, as well as canceling any other gate-induced spurious single-qubit phases suchas a ≈ 0.01 rad phase induced by pulsing the traps off for 500 ns for the two-qubit gate. Ininstances where the number of CZ gates is odd, the echo is performed for the final CZ gate.

[0345] Sign of intermediate-state detuning. To further suppress the effect of the spurious 420-induced phase, the 420-nm laser is operated to be red-detuned (by 2 GHz) from the 6^^ଷ / ଶtransition. For red detunings, the light shift on the|0^state and the|1^state are of the samesign, minimizing the differential light shift, while for blue detunings < 6.8^^^^^^, the light shifton the |0^ state and the |1^ state have opposite signs and amplify the differential light shift.

[0346] Sensitivity to axial trap oscillations

[0347] In typical Rydberg excitation timescales with optical tweezers, the axial trap oscillation frequencies of several kHz are inconsequential. Here, with circuits running as long as 1.2 ms, with Rydberg pulses throughout, the axial trap oscillations can have important effects. In particular, the axial oscillations cause the atoms to make oscillations in / out of theRydberg beams: at an estimated axial temperature of ∼ 25^^^^ and an axial oscillationfrequency of 6^^^^^^, an axial spread ^〈^^ଶ〉 ≈ 1.3^^^^ is estimated. For 20-micron-waistbeams, the effect of this positional spread is relatively small on the pulse parameters of the CZ gate, but can be significant on the sensitive 420-induced phase that should be canceled byechoing out the phase induced by CZ gates separated by ∼ 200^^^^. When using 20-micron-HQU-01425 HU 9476 waist beams, and a 2.5-GHz blue detuning of the 420-nm laser, the dephasing due to the axial trap oscillations is significant. To remedy this deleterious effect, the beam waist of the 420- nm laser is increased to 35 microns (while maintaining constant intensity) and the laser frequency is changed to be 2-GHz red-detuned, together resulting in a significant reduction in the dephasing associated with improper echoing of the 420-nm pulse.

[0348] Rydberg beam shaping and homogeneity

[0349] The Rydberg beams are shaped into tophats of variable size through wavefront control using the phase profile on a spatial light modulator (SLM). This ability allows matching the height of the beam profile to the experiment zone size of any given experiment, thereby maximizing the 1013-nm light intensity and CZ gate fidelities. The Rydberg beam homogeneity is optimized until peak-to-peak inhomogenities are below <1%. To this end, all aberrations are corrected up to the window of the vacuum chamber, which yields an inhomogeneity on the atoms of several percent that is attributed to imperfections of the final window. To further optimize the homogeneity, aberration corrections are tuned on the tophat through Zernike polynomial corrections to the phase profile in the SLM plane (Fourier plane). With this procedure, peak-to-peak inhomogeneities are reduced to <1% over a range of 40-50 ^^m in the atom plane.

[0350] Coherent mapping protocol

[0351] A coherent mapping protocol is provided to transfer a generic many-body state in the{|1^, |^^^} basis to the long-lived and non-interacting {|0^, |1^} basis. To achieve thismapping, immediately following the Rydberg dynamics, a Raman ^^ pulse is applied to map|1^ → |0^, and then a subsequent Rydberg ^^ -pulse to map |^^^ → |1^.

[0352] Even for perfect Raman and Rydberg ^^ pulses (on isolated atoms), there are three key sources of infidelity associated with this mapping process:HQU-01425 HU 9476 (1) Any population in blockade-violating states (i.e., two adjacent atoms both in |^^^) will be strongly shifted off-resonance for the final Rydberg ^^ pulse. As such, this atomic population will be left in the Rydberg state and lost. (2) Long-range interactions, e.g., from next-nearest-neighbors, will detune the final Rydberg ^^ pulse from resonance and thus reduce pulse fidelity. Since the long-range interactions are not the same for all many-body microstates, this effect cannot be mitigated by a simple shift of the detuning. (3) Dephasing of the state occurs throughout the duration of the Raman ^^ pulse, predominantly from Doppler shifts between the ground states|0^,|1^and theRydberg state |^^^. Although these random on-site detunings are also present during the many-body dynamics, turning the Rydberg drive Ω off allows the system to freely accumulate phase and makes it particularly sensitive to dephasing errors.

[0353] The above error mechanisms are mitigated as follows. To minimize errors from (1), మmany-body dynamics are performed with ஐଶ^బమ ≈ 0.01. This minimizes the probability of anatom to violate blockade to the of order of 1%. To help minimize errors from (2), theamplitude of the 420-nm laser is increased for the final ^^ pulse by a factor of 2 ×, such that^^ಿಿಿ ଶஐ ^ = 0.005 (where ^^ேேே are the interactions with next-nearest neighbors), reducingpulse errors from long-range interactions to the order of 1%. Finally, to reduce errors from (3), a fast Raman ^^ pulse is performed, leaving only 150 ns between ending the many-body Rydberg dynamics and beginning the Rydberg ^^ pulse. The 150-ns gap is comparably shortrelative to the ^^∗ଶ ≈ 3 − 4^^^^ of the {|^^^, |^^^} basis, leading to a random phase accumulationon the order of ∼ 0.02 × 2^^ ^^^^^^ per particle, but is further compounded by having entangledstates of N particles in one copy accumulating a random phase relative to entangled states of N particles in the second copy.HQU-01425 HU 9476

[0354] The global Raman beam induces a light-shift-induced phase shift of ≈ ^^ on |0^, |1^relative to |^^^ during the Raman ^^ pulse. Similarly, the global 420-nm laser also induces alight-shift-induced phase shift of ≈ ^^ between |0^ and |1^ during the Rydberg ^^ pulse.While the measurements performed here are interferometric (in other words, the singlet state measured is invariant under global rotations), and thus not affected by these global phase shifts, these phase shifts can be measured and accounted for where relevant.

[0355] Formation of Array of Particles Using Optical Tweezers

[0356] Optical trapping of neutral atoms is a powerful technique for isolating atoms in vacuum. Atoms are polarizable, and the oscillating electric field of a light beam induces an oscillating electric dipole moment in the atom. The associated energy shift in an atom from the induced dipole, averaged over a light oscillation period, is called the AC Stark shift. Based on the AC Stark shift induced by light that is detuned (i.e., offset in wavelength) from atomic resonance transitions, atoms are trapped at local intensity maxima (for red detuned, that is, longer wavelength trap light), because the atoms are attracted to light below the resonance frequency. The AC Stark shift is proportional to the intensity of the light. Thus, the shape of the intensity field is the shape of an associated atom trap. Optical tweezers utilize this principle by focusing a laser to a micron-scale waist, where individual atoms are trapped at the focus. Two-dimensional (2D) arrays of optical tweezers are generated by, for example, illuminating a spatial light modulator (SLM), which imprints a computer-generated hologram on the wavefront of the laser field. The 2D array of optical tweezers is overlapped with a cloud of laser-cooled atoms in a magneto-optical trap (MOT). The tightly focused optical tweezers operate in a “collisional blockade” regime, in which single atoms are loaded from the MOT, while pairs of atoms are ejected due to light-assisted collisions, ensuring thatHQU-01425 HU 9476 the tweezers are loaded with at most single atoms, but the loading is probabilistic, such that the trap is loaded with a single atom with a probability of about 50-60%.

[0357] To prepare deterministic atom arrays, a real-time feedback procedure identifies the randomly loaded atoms and rearranges them into pre-programmed geometries. Atom rearrangement requires moving atoms in tweezers which can be smoothly steered to minimize heating, by using, for example, acousto-optic deflectors (AODs) to deflect a laser beam by a tunable angle which is controlled by the frequency of an acoustic waveform applied to the AOD crystal. Dynamic tuning of the acoustic frequency translates into smooth motion of an optical tweezer. A multi-frequency acoustic wave creates an array of laser deflections, which, after focusing through a microscope objective, forms an array of optical tweezers with tunable position and amplitude that are both controlled by the acoustic waveform. Atoms are rearranged by using an additional set of dynamically moving tweezers that are overlaid on top of the SLM tweezer array.

[0358] Exemplary Hardware

[0359] Optical tweezer arrays constitute a powerful and flexible way to construct large scale systems composed of individual particles. Each optical tweezer traps a single particle, including, but not limited to, individual neutral atoms and molecules for applications in quantum technology. Loading individual particles into such tweezer arrays is a stochastic process, where each tweezer in the system is filled with a single particle with a finite probability p<1, for example p~0.5 in the case of many neutral atom tweezer implementations. To compensate for this random loading, real-time feedback may be obtained by measuring which tweezers are loaded and then sorting the loaded particles into a programmable geometry. This may be performed by moving one particle at a time, or in parallel.HQU-01425 HU 9476

[0360] Parallel sorting may be achieved by using two acousto-optic deflectors (AODs) to generate multiple tweezers that can pick up particles from an existing particle-trapping structure, move them simultaneously, and release them somewhere else. This can include moving particles around within a single trapping structure (e.g., tweezer array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezer array and another type of optical / magnetic trap). This sorting is flexible and allows programmed positioning of each particle. Each movable trap is formed by the AODs and its position is dynamically controlled by the frequency components of the radiofrequency (RF) drive field for the AODs. Since the RF drive of the AODs can be controlled in real time and can include any combination of frequency components, it is possible to generate any grid of traps (such as a line of arbitrarily positioned traps), move the rows or columns of the grid, and add or remove rows and columns of the grid, by changing the number, magnitude, and distribution of the frequency components in the RF drive fields of the AODs.

[0361] In an exemplary embodiment, an optical tweezer array is created using a liquid crystal on silicon spatial light modulator (SLM), which can programmatically create flexible arrangements of tweezers. These tweezers are fixed in space for a given experimental sequence and loaded stochastically with individual atoms, such that each tweezer is loaded with probability p ~ 0.5. A fluorescence image of the loaded atoms is taken, to identify in real-time which tweezers are loaded and which are empty.

[0362] After detecting which tweezers are loaded, movable tweezers overlapping the optical tweezer array can dynamically reposition atoms from their starting locations to fill a target arrangement of traps with near-unity filling. The movable tweezers are created with a pair of crossed AODs. These AODs can be used to create a single moveable trap which moves one atom at a time to fill the target arrangement or to move many atoms in parallel.HQU-01425 HU 9476

[0363] Referring to Fig.30, a schematic view is provided of an apparatus 3000 for quantum computation according to embodiments of the present disclosure. As shown in Fig.30, using a beam generated by a light source 3002 (for example, a coherent light source, in some example embodiments – a monochromatic light source), SLM 3004 forms an array of trapping beams (i.e., a tweezer array) which is imaged onto trapping plane 3008 in vacuum chamber 3010 by an optical train that, in the example embodiment shown in Fig.30, comprises elements 3006a, 3006c, 3006d, and a high numerical aperture (NA) objective 3006e. Other suitable optical trains can be employed, as would be easily recognized by a person of ordinary skill in the art. Using a beam generated by light source 3012 (for example, a coherent light source; in some example embodiments - a monochromatic light source), a pair of AODs 3014 and 3016, having non-parallel directions of acoustic wave propagation (for example, orthogonal directions) creates dynamically movable sorting beams. By using the optical train, such as the one depicted in Fig.30 (elements 3017, 3006b, 3006c, 3006d, and 3006e), the sorting beams are overlapped with the trapping beams. It is understood that other optical train can be used to achieve the same result. For example, source 3002 and 3012 can be a single source, and the trapping beam and the sorting beam are generated by a beam splitter.

[0364] The dynamic movement of the steering beams is accomplished by employing two non-parallel AODs 3014, 3016, arranged in series. In the example embodiment depicted in Fig.30, one AOD defines the direction of “rows” (“horizontal” – the ‘X’ AOD) and the other AOD defines the direction of “columns” (“vertical” – the ‘Y’ AOD). Each AOD is driven with an arbitrary RF waveform from an arbitrary waveform generator 3020, which is generated in real-time by a computer 3022 which processes the feedback routine after analyzing the image of where atoms are loaded. If each AOD is driven with a single frequency component, then a single steering beam (“AOD trap”) is created in the same planeHQU-01425 HU 9476 3008 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap, and the frequency of the Y AOD drive determines the vertical position; in this way, a single AOD trap can be steered to overlap with any SLM trap.

[0365] In Fig.30, laser 3002 projects a beam of light onto SLM 3004. SLM 3004 can be controlled by computer 3022 in order to generate a pattern of beams (“trapping beams” or “tweezer array”). The pattern of beams is focused by lens 3006a, passes through mirror 3006b, and is collimates by lens 3006c on mirror 3006d. The reflected light passes through objective 3006e to focus an optical tweezer array in vacuum chamber 3010 on trapping plane 3008. The laser light of the optical tweezer array continues through objective 3024a, and passes through dichroic mirror 3024b to be detected by charge-coupled device (CCD) camera 3024c.

[0366] Vacuum chamber 3010 may be illuminated by an additional light source (not pictured). Fluorescence from atoms trapped on the trapping plane also passes through objective 3024a, but is reflected by dichroic mirror 3024b to electron-multiplying CCD (EMCCD) camera 3024d. In this example, laser 3012 directs a beam of light to AODs 3014, 3016. AODs 3014, 3016 are driven by arbitrary wave generator (AWG) 3020, which is in turn controlled by computer 3022. Crossed AODs 3014, 3016 emit one or more beams as set forth above, which are directed to focusing lens 3017. The beams then enter the same optical train 3006b…3006e as described above with regard to the optical tweezer array, focusing on trapping plane 3008.

[0367] It will be appreciated that alternative optical trains may be employed to produce an optical tweezer array suitable for use as set out herein.HQU-01425 HU 9476

[0368] Fast and Parallelizable Logical Computation with Homological Product Codes

[0369] Quantum error correction is required for realizing large quantum devices, but requires extremely large overheads in both space and time. High-rate quantum low-density-parity- check (qLDPC) codes promise a route to reduce qubit numbers, at the expense of additional time cost. In the following example, the total space-time cost of quantum computation is lowered through fast and parallelizable logical gadgets for qLDPC codes that are implemented for key algorithmic subroutines such as the quantum adder. These components are built upon transversal logical CNOTs between two, potentially different, qLDPC codes. These transversal logical CNOTs then enable generalized Steane measurements in parallel on a selected subset of logical qubits in a data qLDPC code using a smaller ancilla qLDPC code.

[0370] It is shown that for homological product codes, such an ancilla can be constructed by simply modifying the base classical codes of the data code. Applying the general construction to hypergraph product (HGP) codes, a gadget performing parallel Pauli product measurements (PPMs) on any subgrid of the logical qubits is obtained. Such a gadget enables efficient logical computations, such as preparing a ^^-qubit GHZ state anddistilling / injecting ^^ magic states with ^^(1) space overhead in ^^(1) and ^^(√^^ log ^^) logicalcycles, respectively. The latter, in particular, enables key algorithmic subroutines involving parallel non-Clifford gates, such as the quantum adder.

[0371] Gadgets for 3D and 4D homological product codes are constructed that feature even faster PPMs with a constant depth. These gadgets can be efficiently implemented in platforms with moveable qubits, such as reconfigurable atom arrays.

[0372] Quantum error correction (QEC) is essential for realizing large-scale, fault-tolerant quantum computation. Paradigmatic QEC schemes based on surface codes are very costly in terms of the space overhead, requiring millions of qubits for problems at scale. High-rateHQU-01425 HU 9476 quantum low-density-parity-check (qLDPC) codes, in both coding theories and practical implementations, promise a route to significantly reduce the qubit numbers.

[0373] However, although the qLDPC codes are promising for hardware-efficient quantum memories, processing the information stored in these codes becomes generically more difficult due to the overlapped support of all the logical qubits in the same code block. Consequently, logical computations based on these codes generally involve additional time overhead. For instance, exemplary approaches for implementing selective logical operations involve interfacing the qLDPC codes with rateless ancillae (codes with asymptotically bashing rate), e.g., the surface codes, via lattice surgeries. To maintain the low space overhead, only a few ancillae can be used, and consequently, logical computations have to be serialized. In contrast, computations using only surface codes can be maximally parallelized when assuming long-range connectivies. As such, when rateless ancillae are needed, computations with qLDPC codes seem to be only trading time for space.

[0374] An alternative approach for implementing logical operations while offering logical parallelism involves transversal gates. For example, transversal inter-block CNOTs give logical CNOTs between every inter-block pair of encoded logical qubits for any two identical CSS codes. However, although parallel, these transversal gates are not selective and they act on all the logical qubits homogeneously.

[0375] An alternative direction is the homomorphic CNOT, which generalizes the transversal CNOTs between two identical codes to two distinct codes. Using a smaller code as an ancilla, it is possible to perform selected operations in parallel on a subset of logical qubits of a data block. For example, it has been shown how to perform a measurement on only one of the logical qubits in a toric code using a surface-code ancilla. However, such a homomorphic CNOT relies on finding a nontrivial homomorphism between two quantum codes, which is aHQU-01425 HU 9476 challenging task for generic codes, and, in particular, it remains unclear how to generalize the constructions from the topological codes to algebraically constructed qLDPC codes.

[0376] In embodiments of the present disclosure, homomorphic inter-block CNOTs are constructed for a family of qLDPC codes, the homological product codes, which have been considered as leading candidates for practical fault tolerance. Utilizing the structure of these codes as the tensor product of classical codes, quantum-code homomorphisms are constructed by simply taking the tensor product of classical-code homomorphisms. Applying these constructions to the hypergraph product (HGP) codes, which are 2D homological product codes, parallel logical computations are realized with low space-time overhead. It is shown that any layer of Θ(^^) Clifford gates can be implemented (consisting of Hadamards, ^^ gates, and intra-block CNOTs) on an HGP block with ^^ logical qubits with a constant spaceoverhead in sublinear (< ^^(^^)) logical cycles. Furthermore, this logical depth can be furtherreduced by compiling specific algorithms with more structured layers of gates. By way of example, it is shown how to prepare a block of logical GHZ states, distill and consume magic states in parallel, and implement the quantum adder, an important subroutine for many useful quantum algorithms, with low space-time overhead. In addition, when applying the constructions to higher-dimensional homological product codes, which support single-shot logical state preparation, even faster logical gadgets with a constant gate depth are obtained.

[0377] Since the logical gadgets developed in this example are built upon transversal inter- block physical CNOTs, they are natural to implement in reconfigurable atom arrays by overlapping two code blocks and shining global Rydberg laser pulses.

[0378] Parallelizable logical computations may be realized based on concatenated codes as well as color codes on hyperbolic manifolds. This example considers more practically relevant product qLDPC codes with concrete implementations and easy fault-tolerant protocols. The constructions in this example can be generalized to other product codes, suchHQU-01425 HU 9476 as the lifted product codes, generalized bicycle codes, fiber-bundle codes, and good qLDPC codes.

[0379] The basic tools for logical-gadget construction are described herein, including the homomorphic CNOTs, which generalize the transversal CNOTs between two identical CSS codes to two distinct codes, and the homomorphic measurement gadget, a generalization of the Steane measurement gadget utilizing the homomorphic inter-block CNOTs. These homomorphic gadgets rely essentially on finding nontrivial homomorphisms between two quantum codes.

[0380] The present disclosure provides a class of homomorphic gadgets for generic homological product codes (at any dimension) that are induced by classical-code homomorphisms. It is shown that some of the well-known techniques for modifying classical codes, such as puncturing and augmenting, are structure-preserving, i.e. one can naturally find a homomorphism between the modified code ^^ᇱand the original code ^^. Moreover, such modifications can preserve the distances of classical codes.

[0381] Thus, structure-preserving and distance-preserving modifications of a homological product quantum code are obtained by simply modifying their base classical codes. This naturally induces a quantum-code homomorphism, which is essentially a tensor product of the classical-code structure-preserving maps, between the original data code and the modified ancilla code.

[0382] Referring to Figs.31A-B, illustrations of the homomorphic CNOT and the homomorphic measurement gadgets for 2D (Fig.31A) and 3D (Fig.31B) homological product codes are shown including classical bits 3101, classical checks 3102, physical qubits 3103, Quantum X checks 3104, Quantum Z checks 3105, and logical qubits 3106. Each ^^- dimensional homological product code ^^ 3107, encoding a set of logical qubits ^ഥ^ 3109, is constructed by taking the tensor product of ^^ classical base codes. An ancilla code ^^ᇱ3108HQU-01425 HU 9476is constructed that only encodes a subset of the logical qubits ^ഥ^ᇱ3110 of ^ഥ^ by puncturing on(removing) a subset of bits of the base classical codes of ^^. Then, applying physical transversal CNOTs between the qubits of ^^ and ^^ᇱgives rise to logical transversal CNOTsbetween ^ഥ^ and ^ഥ^ᇱ. Pauli product measurements on a selected subset of data logical qubits in^ഥ^ can be implemented using the ancilla logical qubits ^ഥ^ᇱ via the generalized Steanemeasurement circuit shown in Fig.31C.

[0383] Referring to Fig.31C, the circuit 3112 generalizes the standard Steane measurementcircuit by allowing a different ancilla code ^^ᇱ(with logical qubits ^ഥ^ᇱ) from the data code ^^(with logical qubits ^ഥ^). As shown in Figs.31A-C, deleting (puncturing) a subset of bits of the base classical codes of the ancilla code removes a subset of the physical qubits. When choosing an appropriate logical operator basis, this also removes a subset of the encoded logical qubits and thus one obtains a smaller ancilla code ^^ᇱthat only encodes a subset of the logical qubits of ^^, but with a preserved distance. The natural inclusion map from the undeleted physical qubits of ^^ᇱto those of ^^, as the constructed quantum-codehomomorphism, indicates that transversal CNOTs between the remaining physical qubits of^^ and ^^ᇱ give transversal logical CNOTs between the remaining logical qubits. Then, usingthe generalized Steane measurement circuit Fig.31C which utilizes the homomorphic inter- block logical CNOTs between ^^ and ^^ᇱ, a subset of logical qubits of ^^ is measured in parallel.

[0384] Referring to Figs.32A-C, an illustration of parallel Grid Pauli product measurements (GPPMs) for HGP codes is shown. Fig.32A shows an example for measuring the single- qubit ^^ operator of the top-left logical qubit in ^ഥ^. Fig.32B shows another example for measuring the weight -four ^^ operator supported on the top-left four logical qubits. Fig.32C illustrates a GPPMs gadget that measures a grid pattern of PPMs on a subgrid of the logicalHQU-01425 HU 9476 qubits. The PPMs (the 3D diamonds, e.g., 3201) are specified by the product of two collections of hyperedges, where each hyperedges (the empty squares, e.g., 3202) is a set of rows or columns. A set of Pauli products of ^^ can be measured in parallel by adding (augmenting) checks to the base classical codes of the ancilla as shown in Figs.32A-C. The constructed homomorphic CNOTs and measurements are applied to HGP codes and new schemes for logical computation are presented using only HGP codes with low space-time overhead. A "Grid PPMs" (GPPMs) gadget is first put together that measures a selected pattern of PPMs on the logical qubits of an HGP code. As shown in Figs.32A-C, the logical qubits of an HGP code form a 2D grid and the GPPMs gadget measures a product pattern of PPMs on any subgrid of the logical qubits in parallel in one logical cycle (consisting of ^^ code cycles for a distance-^^ code). It is shown that this GPPMs gadget, when combined with the known transversal and fold-transversal gates, generates the full Clifford group for the HGP codes. More importantly, parallel logical computations are executed with not only a constant space overhead but also lower space-time cost compared to surface-code computations.

[0385] Referring to Fig.33, a table is presented showing the comparison of the space-time cost of Θ(^^) Clifford gates on ^^ logical qubits using (1) surface codes (2) HGP codes using lattice surgeries with rateless ancillae, and (3) HGP codes using GPPMs, assuming long- range connectivity. Referring to Fig.33 and Theorem 12, a generic layer of Θ(^^) Clifford gates (consisting of Hadamards, ^^ gates, and CNOTs) acting on ^^ logical qubits on a HGPblock can be implemented in at most ^^(^^ଷ / ସ ) logical cycles. This sublinear logical depth (<Θ(^^)) is crucial for outperforming the surface-code computations in terms of the total space-time overhead, assuming long-range connectivities. Achieving such a sublinear logical depth and low space-time overhead is fundamentally enabled by the parallelism of the GPPMsHQU-01425 HU 9476 gadget, as existing selective gadgets involving lattice surgeries with rateless codes would have to execute computations sequentially and result in the same space-time cost compared to the surface codes.

[0386] It is further shown that the logical parallelism can be further enhanced (and consequently, the time overhead can be further reduced) by compiling specific algorithms with more structured layers of gates. As examples, it is shown that one can prepare a ^^- logical-qubit GHZ state (see Fig.34) and distill / consume ^^ magic states (see Fig.35A-D andFig. 36A-B) in parallel with ^^(1) space overhead in ^^(1) and ^^(√^^ log ^^) logical cycles,respectively, using HGP codes encoding ^^ logical qubits. The latter, in particular, enables many practical algorithms involving parallel non-Clifford gates. An efficient implementation of the quantum adder is shown with HGP blocks as such an example (see Fig.37A-B).

[0387] The GPPMs gadget for 2D HGP codes are generalized to higher-dimensional (3^^ and4^^) homological product codes, which support single-shot logical state preparation in thecomputational basis. As a result, selected PPMs using the generalized Steane measurements can be performed in parallel and in constant depth. Moreover, since some of these codes, e.g., the 3D surface code, support transversal non-Clifford gates, this opens new possibilities for exploring a richer set of constant-depth logical circuits with these high-dimensional codes.

[0388] Bold symbols are used to denote a set of objects. [^^] is denoted as the set of integers{1, 2, ⋯ , ^^} for some ^^ ∈ ℤା, {^^^ → ^^ଶ} the set of integers {^^^, ^^^ + 1, ⋯ , ^^ଶ} with ^^ଶ ≥^^^.

[0389] Given a vector ^^ ∈ ^^^ over some field ^^ and a subset of indices ^^ ⊆ [n] , ^^|^^ isdenoted as the restriction of ^^ on ^^, i.e., a subvector of ^^ with only entries indexed by ^^.HQU-01425 HU 9476Similarly, given a matrix ^^ ∈ ^^^× ^ over some field ^^ and a subset of column indices ^^, ^^|^^is denoted as the restriction of ^^ on ^^, i.e. a submatrix of ^^ with only columns in ^^.

[0390] ^ ^^^ప⃗ is denoted as a unit column vector with the ^^-th entry being 1. The dimension ofthe vector in this notation is not specified, which should be clear from the context.

[0391] ^^^is denoted as the ^^-qubit Pauli group.

[0392] The basics of classical linear codes, quantum stabilizer codes, and their representation as chain complexes are reviewed below.

[0393] A [^^, ^^, ^^] classical linear code ^^ over ^^ ^ଶ is a ^^-dimensional subspace of ^^ଶ. It canbe specified as the row space of a generator matrix ^^ ∈ ^^^×^ଶ , or the kernel of a check matrix^^ ∈ ^^(^ି^)×^, with ^^^^்ଶ = 0. The distance ^^ of the code is the minimum Hamming weightof all codewords.

[0394] A [[^^, ^^, ^^]] quantum stabilizer code ^^ is a 2^ dimensional subspace of the 2^-dimensional ^^-qubit Hilbert space. It is specified as the common +1 eigenspace of an Abelian subgroup ^^ of ^^^that does not contain −^^. The non-trivial logical operators of the code are given by ^^(^^)\^^, where ^^ denotes the normalizer with respect to ^^^. The distance of the code is the minimum Hamming weight of all nontrivial logical operators.

[0395] For a CSS stabilizer code, the stabilizer generators can be divided into ^^-typeoperators and ^^-type operators, represented by the ^^- and ^^-check matrix ^^ ^^×^^ ∈ ^^ଶ and^^^ ∈ ^^^ೋ× ^ଶ , respectively. Each row ^^ of ^^^ (^^^) represents a ^^ (^^) type ^^-qubit Paulioperator ^^^ୀ^^^^^^ (^^^ୀ^^^^^^ ), where ^^^(^^^) denotes the Pauli operator on the ^^-th qubit. The commutativity of the stabilizers requires that ^^^^^^்= 0 mod 2, which is also called the CSS condition.HQU-01425 HU 9476

[0396] A length-^^ chain complex (over ^^ଶ) ℬ:Equation 28is a collection of ^^ଶ vector spaces {^^^}^^ୀ^ and linear boundary maps between them {^^^: ^^^ →^^^ି^}^^ୀ^ that satisfy ^^^ି^^^^ = 0. Let ^^ ∶= {^^}^^ୀ^ . Informally, one can write ^^ଶ = 0.

[0397] A classical code ^^ with a check matrix ^^ can be represented by a length-1 chain complex: Equation 29 where the basis of ^^^and ^^^are associated with classical bits and checks, respectively.

[0398] A CSS quantum stabilizer code ^^ with check matrices ^^^and ^^^can be represented by a length-2 chain complex: Equation 30 where the basis of ^^ଶ, ^^^, and ^^^are associated with ^^ checks, qubits, and ^^ checks, respectively. Note that the maps are valid boundary maps due to the CSS condition: ^^^^^^்= 0 mod 2.

[0399] Because of the above relation between codes and chain complexes, classical code ^^ or a quantum code ^^ are referred to herein interchangeably as their representing chain complex.

[0400] The homological product codes are reviewed below and it is shown how they are constructed from the tensor product of chain complexes. The terminology "homological product codes" can refer to different constructions that cover the product of quantum codes. In this example, only a subset of them are considered, the product of classical codes, orHQU-01425 HU 9476 equivalently, length-1 chain complexes, (albeit referred to as high-dimensional hypergraph product codes).

[0401] Given ^^ base chain complexes, one can obtain a product complex by taking the tensor product of them:

[0402] Definition 1 (D-dimensional complex). For any ^^ ∈ ℤା, a D-dimensional complex ^^

[0403] A ^^-dimensional complex is a high-dimensional generalization of the chain complex (see Equation 28), which is a one-dimensional complex. Intuitively, it can be viewed as a ^^- dimensional hypercube, and its projection at a point ^⃗^ along the ^^-th direction resembles achain complex. One can define the ^^-th type boundaries as ^^^ ∶=Informally, onecan write Equation 31 asEquation 32 indicating that the linear maps along any direction form valid boundary maps and maps along different directions commute.

[0404] Note that in Definition 1, for simplicity, the indices ^^^of the base chain complexes are allowed to take any integer values, making the projection of ^^ along any direction infinite-length chain complexes. In practice, some cutoff ^^ is applied to the indices by, e.g.setting ^^௫⃗ = 0 for any

[0405] Importantly, one can obtain a ^^-dimensional complex by taking the tensor product of^^ 1dimensional complexes (or chain complexes):HQU-01425 HU 9476

[0406] Definition 2 (Product complex).D one dimensional complexes,the range of ^^ ). A ^^-dimensional complex ^^ = Prodproduct of these 1D complexes:

[0407] It is straightforward to check that the product complex in Definition 2 is a valid ^^- dimensional complex.

[0408] Finally, since a quantum code is defined on a 1D chain complex (see Equation 30), one needs to derive a 1D chain complex from the product complex, which is called the total complex:

[0409] Definition 3 (Total complex of a product complex) Let ^^ be a product complex out ofchain complex ^^ = {{^^^}^, {^^^}^}: = Tot(^^) is defined as follows:and the boundary maps satisfyHQU-01425 HU 9476Equation 36for any ^^௫⃗ ∈ ^^௫⃗ andEquation 37

[0410] Intuitively, the total complex is obtained by projecting the ^^-dimensional complex along the ''diagonal'' direction.

[0411] Once a total chain complex from a product complex (which can have a length longer than 2) is obtained, a quantum code from a length-2 subcomplex can be defined. In this example, product complexes with length-1 base complexes {^^^}^∈ [^](classical codes) are thefocus. In this case, a total complex ^^ = Tot(Prod({^^^}^∈ [^])) is length ^^:

[0412] Furthermore, the primary focus is on ^^ = 2,3,4. For ^^ = 2, the standardhypergraph product code is obtained, with the planar surface code being a special instance.For ^^ = 3 and ^^ = 4, 3^^ and 4^^ homological product codes are obtained, with 3^^ and4^^ surface / toric code being special instances, respectively.

[0413] The general framework of the homomorphic CNOT and the homomorphic measurement gadget is briefly reviewed below.

[0414] Definition 4 (Homomorphic CNOT). Let ^^ and ^^′ be two quantum codes associatedwith two chain complexes {{^^^}ଶ ଶ ᇱ ଶ ᇱ ଶ^ ୀ ^ , {^^^}^ ୀ ^ }, and {{^^^}^ ୀ ^ , {^^^}^ୀ^ }, respectively. Let ^^ =HQU-01425 HU 9476{^^^ ∶ ^^ᇱ^ → ^^^ }ଶ^ ୀ ^ be a homomorphism between the two chain complexes, i.e., thefollowing diagram is commutative:Equation 39 Then physical ^^-controlled CNOTs specified by ^^^, i.e. a physical CNOT controlled by the ^^-th qubit of ^^ and targeted the j-th qubit of ^^′ is applied if and only if ^^^[^^, ^^] = 1, give some^^-controlled logical CNOT gates between ^^ and ^^′. Such a logical gadget is referred to as ahomomorphic CNOT associated with the homomorphism ^^.

[0415] The above homomorphic CNOT gadget is a valid logical operation since the condition that the stabilizers are preserved under such a gadget is equivalent to that the diagram in Equation 39 is commutative. In other words, finding a homomorphism between two quantum codes directly leads to a logical gadget that implements some inter-block logical CNOTs.

[0416] Using the homological CNOT in Definition 4, one can implement a homomorphic measurement gadget on a data quantum code ^^ by constructing a specific ancilla code ^^′ and implementing the generalized Steane measurement (see Fig.31C) that utilizes the inter-block homomorphic CNOTs. Specifically, by initializing the logical qubits of ^^′ in the ^^ (^^) basis, applying the ^^- (^^′-) controlled homomorphic CNOTs, and measuring ^^′ in the ^^ (^^) basis, products of Pauli ^^ (^^) logical operators of ^^ can be measured.

[0417] In general, the homomorphic measurement gadget implements ^^ Pauli product measurements (PPMs) on the data code non-destructively and in parallel, where ^^ equals the number of logical qubits in ^^′. In addition, one can guarantee the fault tolerance of the gadget by using a large-distance ancilla, constant-depth homomorphic CNOTs, and fault-HQU-01425 HU 9476 tolerant ancilla state preparation and measurement (which can be done by performing ^^ QEC cycles and transversally measuring the qubits, respectively, for any distance-^^ CSS code). An alternative approach would be to construct homomorphic measurement gadgets for performing one PPM on a toric code or a hyperbolic surface code using an ancilla code that encodes a single logical qubit. However, this construction relies on the notion of covering spaces and the topological properties of codes, and it was not clear how to generalize their constructions to algebraically constructed qLDPC codes.

[0418] Here, techniques for modifying classical codes are reviewed, which are utilized to induce structure-preserving modifications on the quantum homological product codes for implementing the homomorphic gadgets.

[0419] Definition 5 (Puncturing and shortening). Let ^^ be a [^^, ^^, ^^] classical code with aindices. Puncturing ^^ on ^^ gives a new check matrix ^^^^, which is defined as a submatrix of^^ with columns in ^^ deleted, i.e.Equation 40 Moreover, the generator matrix of ^^^^is ^^^^, which is obtained by shortening ^^ on ^^:

[0420] Puncturing and shortening are dual to each other: puncturing the check matrix of a code corresponds to shortening its generator matrix (on the same set of bits). This gives a way of constructing a new code ^^′ from an old code ^^ by puncturing its check matrix on aHQU-01425 HU 9476 subset of bits. Such a transformation is referred to as puncturing a code on some set of bits,for simplicity. In general, the new code ^^′ has ^^ᇱ ≤ ^^ and ^^ᇱ ≤ ^^. For generic puncturing,there is no guarantee on the new code distance, which could either increase, decrease, or remain the same.

[0421] Definition 6 (Augmenting and expurgating). Let ^^ be a [^^, ^^, ^^] classical code with anew checks. Augmenting ^^ with ^^^gives a new check matrix ^^ାுబ, which is defined as appending the new checks in ^^^to ^^, i.e.Equation 41 The generator matrix of ^^ାு_^is ^^ିுబ, which is obtained by removing the codewords in rs(G) that do not satisfy the extra constraints imposed by ^^^, a process called expurgating:

[0422] Augmenting and expurgating are also dual to each other: augmenting the check matrix of a code corresponds to expurgating its generator matrix (with respect to the same set of extra checks). This also gives a way of constructing a new code ^^ᇱfrom an old code ^^ by adding some extra checks. Such a transformation is referred to as augmenting a code withsome new checks, for simplicity. In general, the new code ^^ᇱ has ^^ᇱ = ^^, ^^ᇱ ≤ ^^ and ^^ᇱ ≥^^.

[0423] For the purpose of this example, classical codes are modified in a way that preserves the code distance. The augmenting-expurgating operation trivially satisfies this requirement since it only removes a subset of codewords. However, the same does not hold for the puncturing-shortening operation in general since some bits are removed during such anHQU-01425 HU 9476 operation. Nevertheless, it is shown that the distance is preserved if one only punctures on a specific subset of bits for any given code.

[0424] Given any [^^, ^^, ^^] classical code ^^ with a check matrix ^^ ∈ ^^(^ି^)× ^ଶ and a set ofbits labeled by the column indices of ^^, i.e. B ≃ [^^], one can find a subset of ^^ − ^^ bits ^^୍such that the columns of ^^ indexed by ^^୍are linearly independent. Without loss ofgenerality, it can be assumed that ^^୍ are the last ^^ − ^^ bits since otherwise one can simplypermute ^^୍ to the last ^^ − ^^ bits. Then, by performing row elementary operations, one cantransform ^^ to a canonical form:∈^^(^ି^)× ^ଶ . With this canonical ^^^, the generator matrix is easily obtained, whichis also in the canonical form:Equation 44

[0425] Note that in this canonical form, the first ^^ bits are referred to as the information bits in the classical code literature.

[0426] Now, it is shown that puncturing on any set of bits ^^ ⊆ ^^୍ that are information bitsdoes not reduce the code distance. Again, without loss of generality, it is assumed that ^^୍ =[^^] and ^^ = [|^^|]. Based on Definition 5, it is known that puncturing ^^ on ^^ corresponds toshortening ^^^on ^^, i.e., the new generator matrix can be written asHQU-01425 HU 9476Equation 45

[0427] Since the ^^ − |^^| rows of ^^^^ are the same as the last ^^ − |^^| rows of ^^^, up to someextra zeros, the distance of the new code is equal or greater than ^^.

[0428] Moreover, the transformation from ^^^to ^^^^entails a concise transformation on thecodewords by the puncturing operation. Let ^ഥ^ ≃ [^^] denote the ^^ codewords of ^^, whichare referred to as logical bits, for simplicity. Puncturing on the bits indexed by ^^ simply corresponds to removing the logical bits indexed by ^^ (up to shortening other logical bits by some zero entries).

[0429] The following describes constructing nontrivial logical homomorphic CNOT and measurements for generic homological product codes.

[0430] Constructing a homomorphic CNOT between two quantum codes ^^ and ^^ᇱimpliesessentially finding a homomorphism ^^: ^^ᇱ → ^^. However, finding a nontrivialhomomorphism between two generic, non-topological quantum codes is challenging. Fortunately, the homological product codes have a special structure in that they are essentially products of classical codes. Utilizing such a structure, one can reduce the task of finding homomorphisms between two homological product codes to finding homomorphisms between their base classical codes, which is shown to be a much easier task.

[0431] Let ^^^^^^∈ [^]be the two sets of base classical codes for constructingtwo ^^-dimensional homological product codes ^^ and ^^ᇱ, respectively. Let ^^^^: ^^ᇱ^ → ^^^^ bea set of homomorphisms between the classical codes. In the following, it is shown that aHQU-01425 HU 9476homomorphism ^^: ^^ → ^^ᇱ can be essentially constructed by taking tensor products of theclassical-code homomorphisms ^^^.

[0432] Recall that to construct a homological product ^^, one first constructs a productcomplex ^^ = Prod({^^^}^^ୀ^ ) as the tensor product of all the base classical codes (seeDefinition 2). Then ^^ is projected onto a length-^^ chain complex ^^ = Tot(^^) called thetotal complex of ^^ (see Definition 3). Finally, for ^^ ≥ 2, the quantum code ^^ ⊆ ^^ isdefined as a length-2 subcomplex of ^^. The same construction is applied to ^^ᇱ, i.e. ^^ᇱ ⊆^^ᇱ ∶= Tot(^^ᇱ), where ^^ᇱ ∶= Prod

[0433] To construct a homomorphism between ^^ and ^^ᇱ, a homomorphism between ^^ and^^′ is first constructed by simply taking the tensor product of the classical-codehomomorphisms:

[0434] Lemma 7 (Classically induced homomorphism for product complexes). The linearmap ^^^^ = {^^^௫⃗ : ^^ ᇱ௫⃗ → ^^௫⃗ } from ^^ᇱ to ^^, whereis a homomorphism between ^^ᇱand ^^.

[0435] Proof. One needs to show that ^^^^preserves the boundary maps of the ^^-dimensional complexes. More concretely, it suffices to show that the following square is commutative:Equation 47

[0436] Based on Equation 34 and Equation 46, it is shown thatHQU-01425 HU 9476where the assumption that ^^^: ^^^ᇱ → ^^ ^ is a homomorphism is utilized, i.e.,

[0437] With the classically induced homomorphism {^^^^}: ^^ᇱ → ^^, one can transform it to ahomomorphism between the total complexes ^^^^: ^^′ → ^^:

[0438] Proposition 8 (Classically induced homomorphism for homological product codes).The linear map ^^ = {^^ ᇱ^: ^^^ → ^^^} from ^^′ to ^^, whereEquation 50 with {^^^௫⃗} defined in Equation 46, is a homomorphism from ^^′ to ^^. ^^ also serves as ahomomorphism from ^^ᇱ ⊆ ^^ᇱto ^^ ⊆ ^^.

[0439] Proof. It is proven that the following diagram is commutative:Equation 51

[0440] Let ^^ᇱ =⊕ ᇱ ᇱ|௫⃗|ୀ^ ^^௫⃗ ∈∈ ^^௫⃗ . According to Equation 36 and Equation50, it is shown thatHQU-01425 HU 9476

[0441] To prove Equation 52 equals Equation 53, one only needs to show ^^ ^ ᇱ௬^⃗ ,௫⃗^^௫⃗^^௫⃗ =37), andEquation 54according to Lemma 7 (see Equation 48); Otherwise, ^^ ^ ^ ᇱ௬^⃗ ,௫⃗^^௫⃗ = ^^௬^⃗^^௬^⃗ ,௫⃗ = 0 since ^^௬^⃗ ,௫⃗ =^^ᇱ௬^⃗ ,௫⃗ = 0.

[0442] According to Proposition 8, finding a homomorphism between two homological product codes can be reduced to finding a set of homomorphisms between their base classical codes. In following, useful classical-code homomorphisms are constructed.

[0443] It is first shown that the puncturing (see Definition 5) and the augmenting (see Definition 6) operations on a classical code are structure-preserving and they naturally induce a homomorphism between the old code and the modified code, which is called the puncturing-augmenting homomorphism:

[0444] Proposition 9 (Puncturing-Augmenting homomorphism for classical codes) Let^^^× (^ ି |^^|)ଶ a set of extra checks. Without loss of generality, assume that ^^ = [|^^|]. LetHQU-01425 HU 9476 is a homomorphism from ^^′ to^^, i.e.Equation 56 is commutative.

[0445] Proof. Based on the definition of ^^^^,ାுబ, it is shownfor some ^^ ∈ |^^|. Then, according to Equation 55,

[0446] It is assumed that ^^ are the first |^^| bits for simplicity. In general, for any ^^ ⊆ [^^], thebits [^^ − |^^|] of ^^′ are identified with the bits [^^]\^^ of ^^. Then, ^^^ can be defined as theinclusion map from [^^ − |^^|] to [^^] with such an identification. Algebraically, ^^^ =^^ form some ^^ × ^^ permutation matrix that permutes ^^ to the first |^^| bits.

[0447] If a homomorphism in Proposition 9 only involves puncturing (augmenting) of the classical code, it is referred to as a puncturing (augmenting) homomorphism.

[0448] The homomorphic measurement gadget that can perform selected and parallel PPMs on a generic homological product code is sketched below. Then, as an example, concrete constructions for the HGP codes are constructed, which are 2D homological product codes.HQU-01425 HU 9476

[0449] To perform a selected set of PPMs on a data code ^^, a carefully designed ancilla code^^′ is prepared, which is another homological product code of the same dimension.According to Definition 4, one can construct a homomorphic CNOT gadget from ^^ to ^^′ if a homomorphism between ^^ and ^^′ is found. According to Proposition 8, such a homomorphism can be constructed by essentially taking the tensor product of the homomorphisms between the base classical codes. Such classical-code homomorphisms are constructed by using the puncturing-augmenting homomorphisms presented in Proposition 9. Under such a construction, the base codes of ^^′ are constructed by puncturing and / or augmenting the base codes of ^^, which can remove some classical codewords. It is shown that the logical operators of ^^ and ^^′ are essentially given by the product of the base classical codewords. As such, a smaller ancilla code ^^′ that encodes fewer logical qubits is obtained. Then, using the Steane measurement gadget (see Fig.31C), one can achieve selected measurements on a subset of logical qubits of ^^ using ^^′.

[0450] For example, as shown in Figs.31A-B, by puncturing the classical codes on selectedsubsets of bits, a subset ^^^^ of the qubits ^^ of ^^ are removed when constructing ^^′. Let ^^ ={^^ଶ, ^^^, ^^^}: ^^ᇱ → ^^ be the homomorphism induced by the classical puncturing-augmentinghomomorphisms (see Proposition 9). The qubits ^^′ are identified with ^^\^^^^ under ^^^, i.e.^^^(^^ᇱ^) = ^^^ for ^^ ∈ [|^^\^^^^|], where the qubits are labeled such the ^^^^ are the last |^^^^|qubits of ^^. Deleting ^^^^also deletes a subset of logical qubits in ^^ when constructing ^^′. Specifically, let ^̅^ be the set of nontrivial logical operators of ^^. One can choose arepresentation of the logical qubits ^ഥ^ = {^ത^ ^^}^ୀ^ , such that the logical ^^ and ^^ operators(^ത^^, ^̅^^) (^ത^^, ^̅^^ ∈ ^̅^) of ^ത^^ form conjugating pairs, i.e. ^^ത^^, ^ത^^൧ ==^^^,^. A similar basis for the logical qubits ^^′ of ^^′ is found with a logical operator basisHQU-01425 HU 9476{(^ത^ᇱ^, ^̅^ᇱ^)}^ି|^ഥ^బ|^ୀ^ where ^ഥ^^ ⊆ ^ഥ^ denotes the set of logical qubits that are removed. The logicaloperators of ^ഥ^′ are identified with those of ^ഥ^\^ഥ^^ under ^^^, i.e.Equation 59for ^^ ∈ [^^ − |^ഥ^^|]. Note that, again, the logical qubits of ^^ are labeled such that ^ഥ^^ are thelast |^ഥ^^| logical qubits. With a slight abuse of notation, ^^^(^ത^ ᇱ^ ) = ^ത^^ is written.

[0451] The homomorphic CNOT gate (specified by ^^^) between ^^ and ^^′ is implemented by physical transversal ^^-controlled CNOTs between ^^\^^^^and ^^′. This amounts to applyinglogical transversal ^^-controlled CNOTs between ^ഥ^\^ഥ^^ and ^ഥ^′. Specifically, the physicalCNOTs transform the logical operators of the two codes as follows:for any ^^ ∈ [^^ − |^ഥ^^|], andfor any ^^ ∈ {^^ – |^ഥ^^| + 1 → ^^}. Then, using the generalized Steane measurement circuitshown in Fig.31C, one can perform a parallel, non-destructive measurement of the logical ^^operators of ^ഥ^\^ഥ^^ using the ancilla ^ഥ^′.

[0452] It is shown that one can also measure certain products of logical ^^ operators of ^ഥ^ byalso augmenting new checks to the base codes of ^^ when constructing ^^′.

[0453] Note that the above homomorphic measurement gadget can be both parallel and selective since, depending on the puncturing and augmenting pattern performed on the base codes of ^^, one can obtain different ancilla codes with different logical qubits, and therebyHQU-01425 HU 9476 different patterns of PPMs on ^^. The number of parallel PPMs being performed equals the number of logical qubits in ^^′.

[0454] Following the general description for generic homological product codes, homomorphic measurement gadgets for the HGP codes are concretely constructed, as an example.

[0455] To find useful logical gadgets, one needs to first identify a canonical logical operator basis that defines a canonical set of logical qubits to work with. డ

[0456] Let ^^ :→భభ డమ^ ^^^^ and ^^ଶ: ^^ଶ^ →భ ^^ଶ^ be two length-1 chain complexes. ReferringFig.38, a chain complex of an HGP code is shown. As shown in Fig.38, a 2D homological ு^ ுproduct code ^^: ^^ ೋ ^ଶ ^^ ^^^ ^^ ^^^, also called an HGP code, is constructed as the totalcomplex of the tensor product of ^^^and ^^ଶ.

[0457] Specifically, it is shown: ^^ଶ = ^^^^ ⊗ ^^ଶ^, whose basis are associated with the ^^checks ^^^^; ^^^ = ^^^^ ⊗ ^^ଶ^ ⊕ ^^^^ ⊗ ^^ଶ^, whose basis are associated with the qubits ^^; ^^^ =^^^^ ⊗ ^^ଶ^, whose basis are associated with the ^^ checks ^^^^.

[0458] One assigns ^^ଶ with a [^^ଶ, ^^ଶ, ^^ଶ] classical code with a check matrix ^^ଶ ∈associated with the bits ^^the checks ^^ ଶ^^ of the code, respectively, and ^^^ = ^^ଶ. However, to be consistent with theliterature, ^^^ is assigned with the transpose of another [^^^, ^^^, ^^^] classical code with a checkmatrix ^^ ∈ ^^(^భି ^భ)× ^భ. In this case, the basis ^ ^^ ଶ of ^^^ and ^^^ are associated with the checks^^ ^^^ and the ^^^^, respectively, and ^^^ = ^^^் .

[0459] Under such an assignment, it is shown thatHQU-01425 HU 9476

[0460] Now, the quantum bits and checks of ^^ on a 2D grid are arranged and labeled with their coordinates. As shown in Fig.31A, by laying the bits / checks of the first and the second classical code vertically and horizontally, respectively, the following identification of the quantum bits and checks with the Cartesian product of the classical bits and checks are shown:where ^^^ ∶= ^^^ − ^^^ and ^^ଶ ∶= ^^ଶ − ^^ଶ. With these coordinates, each basis element in ^^ଶ,^^^, or ^^^ is associated with the corresponding object (^^ check, qubit, or ^^ check) placed attheir assigned coordinate. For instance, the basis element ^^^ ⊗ ^^^ in ^^^ is associated with aqubit at the coordinate (^^, ^^), which are denoted as ^^^,^.

[0461] A complete set of logical ^^ and ^^ operators of ^^ are given bywhere rs(^^ఈ)∙ denotes the complementary space of rs(^^ఈ) infor ^^ = 1, 2.

[0462] A canonical basis for the logical operators in Equation 64 are determined such that they form conjugate pairs and they admit a canonical form analogous to that of a classical code in Equation 44. Without loss of generality, it is assumed that each ^^ఈcan be row-HQU-01425 HU 9476reduced to their canonical forms ^^ ఈఈ,^ =ℎ^^ , ^^^^షೖ^ ) via elementary rowoperations, where ℎఈ^ ∈ ^^^^ ି^^ଶ . Note that this is always the case if one permutes theinformation bits of each classical code to the first few bits. Using the canonical form of ^^ఈ, their generator matrices are derived whose rows span ker (^^ఈ):Equation 65

[0463] Let ^^^^denote a unit ^^-dimensional column vector with the ^^-th entry being 1, and letEquation 66

[0464] Then one can find a canonical basis for ^ഥ^ and ^ത^ in Equation 64 forming conjugatingpairswhere:Equation 67

[0465] The conjugation relation can be verified for the paired logical operators in Equation 67 sinceHQU-01425 HU 9476

[0466] A key feature for such a canonical basis in Equation 67 is that each pair of logicaloperators ൫^ത^^,^, ^̅^^,^൯ is supported on the ^^-th column and ^^-th row of the qubit grid,respectively, and they overlap and only overlap on one physical qubit with coordinate (^^, ^^)on the [^^^] × [^^ଶ] subgrid. See Equation 63 for the coordinate system and Fig. 31A for anexample of ^ത^^,^ and ^ത^ଶ,ଷ. As such, the set of logical qubits ^ഥ^ =arearranged on a [^^^] × [^^ଶ] 2D grid, which are in one-to-one correspondence to the[^^^] × [^^ଶ] subgrid of the physical qubits (see the upper left physical block in Fig. 31A).

[0467] Let ^^ = HGP(^^^, ^^ଶ) be an HGP code with a [^^^, ^^^, ^^^] base vertical code and a[^^ଶ, ^^ଶ, ^^ଶ] base horizontal code. Recall that in the canonical logical operator basis (seeEquation 67), the logical qubits of ^^ are arranged on a [^^^] × [^^ଶ] grid, i.e. ^ഥ^ =Let ^^^^ = {^^^^} be a collection of disjoint horizontal hyperedges, i.e., ^^^^ ⊆[^^ଶ] and ^^^^ ∩ ^^^^ = ∅ for ^^ ≠ ^^. A gadget for measuring ^^^ × |^^^^| logical Pauli products ispresented, which are horizontal Pauli products specified by ^^^^ across all the ^^^ rows:⋃^∈[^భ]in parallel in the algorithm shown in Fig. 39.

[0468] A few useful notations are first introduced. Let ^^ be some set of qubits and ^^ besome set of coordinates. It is stated that ^^ ≃ ^^ if ^^ are assigned with coordinates ^^. Let^^^ ⊆ ^^. ^^|ைబ is referred to as the subset of qubits with coordinates ^^^. For two sets ofqubits ^^ and ^^′ assigned with the same set of coordinates ^^, one refers to transversal CNOTsbetween them as pairs of CNOTs on each coordinate, i.e. ^ ᇱ(^,^)∈ ^^CNOT(^^^,^, ^^^,^ ).

[0469] Let ^^^^^^୰^୮ denote the check matrix of |^^^^| disjoint repetition codes, each supported on^^^^ ∈ ^^^^, ^^. ^^. , ^^^^^^୰^୮ is the vertical concatenation of {^^ ^)× ^మ^}, where the rows ofare simply parities of any two bits of ^^^^. For example, for ^^ଶ = 6 and ^^^^ = {{1,2,3}, {4,5}},it is shown thatHQU-01425 HU 9476

[0470] Theorem 10. The horizontal-PPMs gadget in the algorithm shown in Fig.39code in parallel and fault-tolerantly.

[0471] Proof. Let ^^^,^ = (ℎ^, ℎଶ, ⋯ , ℎ^మ, ^^^మି^మ) be the canonical form of ^^ଶ. Thecorresponding generator matrix is then

[0472] Without loss of generality, assume that the hyperedges in ^^^^ are ordered, i.e., ^^^^ ={^^^^}௧^ ୀ ^ , where ^^^^According to Definition 6, the generator^^atrix of ^^ାு ^^m౨^౦ଶ , ^^ଶ,ିு^^^^౨^౦, is given by removing the codewords in ^^ଶthat do not satisfy theextra constraints imposed by the augmented checks ^^^^^^୰^୮. It is straightforward to findHQU-01425 HU 9476for ^^ ∈ {^^ + 1 → ^^ + |^^|}. Namely, each of the first ^^ codewords of ^^ଶ,ିு^^^^౨^౦ is given bymerging the codewords of ^^ଶindexed by ^^^, and the last |^^| codewords of ^^ଶare preserved.Finally, according to Definition 5, the generator matrix of ^^ᇱ ᇱଶ^^ଶ , is given byshortening ^^ଶ,ିு^^^^౨^౦ on ^^, which simply corresponds to removing the last |^^| codewords of^^ and deleting the columns indexed by ^^, i.e., ^^ᇱଶ ∈

[0473] Let kerAccording to Equation 73, it is shown

[0474] In addition, one can derive that

[0475] Then, according to the canonical logical operator basis in Equation 67, the logical operators of ^^ are identified with those of ^^ via their shared coordinates:

[0476] According to Proposition 8 and 9, a homomorphism ^^ = {^^ଶ, ^^^, ^^^}: ^^ᇱ → ^^ inducedby the classical puncturing-augmenting homomorphism fromHQU-01425 HU 9476^^ᇱ to ^^ is constructed, where ^^^ identifies qubits with the same coordinates, i.e.^^^: ^^ᇱ|^^\^^^^ → ^^|^^\^^. Finally, the logical action associated with the transversal CNOTsspecified by ^^^can be calculated:for any (^^, ^^) ∈ [^^^] × [^^], andup to some stabilizers, for any (^^, ^^) ∈ [^^^] × [^^] and ^^ ∈ ^^^^}. ^^^,^,^ is some stabilizer of ^^′.Other logical operators of ^^′ and ^^ remain unchanged.

[0477] Equation 77 and Equation 78 indicate that the constructed logical gate implements thefollowing logical CNOTs between each logical qubit ^ത^^ᇱ,^ and ^^ത^^,^^^∈ ^^^^. Therefore, theSteane measurement circuit utilizing such logical CNOTs implements the logical PPMs⋃^∈ [^భ] {^^∈ ^^^̅^^,^}^^∈^^^^ in parallel.

[0478] Finally, it is shown that the entire protocol is fault-tolerant. Since only ^^ଶispunctured on the information bits, the distance of ^^ᇱ ᇱ ᇱଶ is preserved, i.e. ^^ଶ ≥ ^^ଶ. Since ^^ଶ isstill full-rank, the distance of ^^′ satisfies ^^ᇱ = min{^^ ᇱ^, ^^ଶ } ≥ ^^. Now, the logical Steanemeasurement circuit is clearly fault-tolerant as it uses a large-distance ancilla code and only involves fault-tolerant gadgets: computational-state preparation with ^^ code cycles, transversal logical gates, and transversal readout.

[0479] In following, new schemes for realizing parallel logical computations are presented using only HGP codes. The new schemes leverage the transversal (homomorphic) CNOTs between two distinct HGP codes, and the related selective homomorphic measurements, that are constructed.HQU-01425 HU 9476

[0480] The coding-theoretical, and physical-level results are first merged together and a new fault-tolerant logical gadget is obtained for a generic HGP code that performs selective PPMs on any subgrid of the logical qubits in parallel. Then, logical-level computations with HGP codes are considered by utilizing the new parallel PPMs gadget, combined with known logical transversal / fold transversal gates.

[0481] Here, the construction of a homomorphic measurement gadget for HGP codes is presented that measures Pauli products on any subgrid of the logical qubits selectively and in parallel. Such a gadget is referred to as a "Grid PPMs" (GPPMs) gadget (see Definition 11) and used as an elementary gadget for executing logical computations with HGP codes later in this example.

[0482] The GPPMs gadget is essentially built upon the homomorphic measurement gadget introduced above (see algorithm 1 shown in Fig.39) that measures horizontal ^^-PPMs specified by a set of horizontal hyperedges ^^^^across all the rows. One can similarly construct a vertical PPMs gadget that measures vertical ^^-PPMs specified by a set of vertical hyperedges ^^^^across all the columns. One can simply achieve this by performing the samepuncturing-augmenting operation on the vertical codeof ^^, instead of the horizontal code^^. Note, however, that the quantum code homomorphism ^^ = {^^ଶ, ^^^, ^^^}: ^^ → ^^′ nowreverses the direction since the base vertical complex ^^^is associated with the transpose of the corresponding vertical codes, and the puncturing-augmenting operation ofactually induces a puncturing-augmenting homomorphism from ᇱto ^^^ . This is because if^^′ is obtained by puncturing (augmenting) ^^, then ^^் is obtained by augmenting(puncturing) ^^ᇱ். Consequently, the homomorphic CNOT between ^^ and ^^′ now reverses the direction and becomes ^^′-controlled both physically and logically. As such, one should prepare the logical qubits of ^^′ in the ^^ basis, apply ^^′-controlled homomorphic CNOTs, andHQU-01425 HU 9476 finally measure ^^′ transversely in the ^^ basis. Then, this gadget implements the logicalparallel.

[0483] In following, it is shown how to construct a GPPMs gadget that measures PPMs onany subgrid of the logical qubit [^^^] × [^^ଶ], which could contain PPMs across differentcolumns and rows, by combining the horizontal ^^-PPMs gadget and the vertical ^^-PPMs gadget.

[0484] As the simplest example, it is shown how to perform a ^^ measurement for only the top-left (1,1) logical qubit. As illustrated in Fig.32B, by puncturing the horizontal code on all the first ^^^bits but the very first bit, one can simultaneously remove all but the first column of the logical qubits of ^^ when constructing the ancilla code ^^′. Applying thehorizontal PPMs gadget with horizontal hyperedges ^^^^ = {{1}} would then perform single-qubit ^^ measurements on the first-column logical qubits of ^^.

[0485] To only measure the ^ത^^,^, another code ^^ᇱᇱis introduced, which is called the maskcode, that is obtained by puncturing the vertical code of ^^′ on the first bit. As shown in Fig. 32A, doing so removes the first logical qubit of ^^′. One can then perform single-qubit ^^ measurements on all but the first logical qubit of ^^′ with the mask code ^^ᇱᇱby utilizing thevertical PPMs gadget with hyperedges ^^^^ = {{^^}}^∈[^భ]\[^]. Combining these elementstogether, a two-step protocol for measuring the (1,1) logical qubit of ^^ is obtained: 1. Prepare all the logical qubits on ^^′ and ^^ᇱᇱinത|ത0തത^ andത|+തതത^ states, respectively.Then perform vertical ^^-PPMs on ^^′ with ^^^^ = {{^^}} ᇱᇱ^∈[^భ]\[^] using ^^ , which resetsall but the first logical qubit of ^^′ inത|+തതത^ states.2. Perform a horizontal ^^-PPMs on ^^ with ^^^^ = {{1}} using ^^′. Note that only ^ത^^,^is measured since the rest of the first-column logical qubits of ^^ are not entangledHQU-01425 HU 9476 with the corresponding logical qubits in ^^′, which are reset toത|+തതത^ by the firsthomomorphic measurement, via the homomorphic CNOT gate.

[0486] The above scheme can be readily generalized to performing parallel single-qubit ^^ measurements on a subgrid of the logical grid, i.e., logical qubits supported on theintersection of any subset of rows Rows= {^^^} and columns Cols= {^^^}. This can be done bypuncturing ^^ଶon [^^ଶ]\Rows and puncturingon Cols when constructing ^^ᇱand ^^ᇱᇱ, respectively.

[0487] Moreover, one can generalize the above scheme to measure some pattern of Pauli product measurements (PPMs) on a subgrid of the logical grid. As an example, it is shown how to measure a weight-4 Pauli product ^̅^^,^^̅^^,ଶ^̅^ଶ,^^̅^ଶ,ଶsupported on the intersection of the first two columns and rows in Fig.32B. One first constructs an ancilla patch ^^′ by puncturing ^^ଶon all but the first two bits and then augmenting an extra check that checks the first two bits. Doing so removes all but the first two columns of logical qubits of ^^. In addition, the remaining two columns of logical qubits merge into one column by the augmented checks, i.e., the logical operators of each pair of logical qubits in each row are now equivalent up to some stabilizers in ^^′. Now, applying the horizontal ^^-PPMs gadgetwith ^^^^ = {{1,2}} between ^^ and ^^′ would perform weight-2 ^̅^^,^^̅^^,ଶ measurements acrossall the rows on ^^.

[0488] To obtain the desired weight-4 measurement, another mask code ^^′′ is introduced.^^′′ is constructed by augmenting an extra check tothat checks the first two bits. Thismerges the first two logical qubits in ^^′′. Then, applying the vertical ^^-PPMs gadget with^^ = {{1,2}, {3}, {4}, ⋯ , {^^ }} be തᇱ തᇱ^^ ^ tween ^^′ and ^^′′ performs a ^^^,^ ^^ଶ,^ measurement on thefirst two logical qubits of ^^′ and single-qubit ^^ measurements on the remaining logicalqubits. This creates a Bell pair for the first two while masking (resetting toത|+തതത^ states) theHQU-01425 HU 9476 rest of the logical qubits. Finally, the horizontal ^^-PPMs gadget between ^^ and ^^′ performs the desired ^̅^^,^^̅^^,ଶ^̅^ଶ,^^̅^ଶ,ଶmeasurement.

[0489] Finally, the above protocol is generalized and the following pattern of PPMs is performed in parallel on an HGP code:

[0490] Definition 11 (Grid PPMs). Given a canonical representation of logical qubits in anHGP code on a 2D grid [^^^] × [^^ଶ], and two sets of disjoint hyperedges ^^^^ and ^^^^, whereeach hyperedge is a collection of rows or columns, respectively, then a pattern of Grid PPMsof a type ^^ ∈ {^^, ^^} is defined as:where ^ത^^ denotes the logical Pauli operator ^ത^ (^̅^) of the logical qubits ^ത^^ with coordinates ^^for ^^ = ^^ (^^).

[0491] For completeness, the concrete protocol for implementing ^^-type GPPMs in algorithm 2 is provided and shown in Fig.40. Note that the step 2 of algorithm 2 merges columns of logical qubits of ^^′ according to ^^^^and the step 4 further prepares these merged logical qubits into vertical disjoint GHZ states according to ^^^^across all the columns. Together, they make sure that the final step measures the desired PPMs on disjoint gridsaccording to ^^^^ × ^^^^.

[0492] For a constant-rate HGP code encoding ^^ logical qubits, each pattern of GPPMs is implemented in parallel with a constant space overhead since the data, the ancilla, and themask code are all of size ^^(^^). Meanwhile, it only takes ^^ + ^^(1) code cycles (only thestep 3 of algorithm 2 takes ^^ cycles while other steps are of constant depth). Let a logical cycle refer to ^^ code cycles, each GPPM can then be implemented efficiently with ^^(1) space overhead in ^^(1) logical cycles.HQU-01425 HU 9476

[0493] Note that it is also possible to implement PPMs with other patterns, e.g., PPMs with a mixture of ^^ and ^^ Paulis, if combining the homomorphic CNOT with other Clifford operations such as the fold-transversal H-SWAP gate in the table shown in Fig.41 during the Steane measurement. In this example, the focus is on only these CSS-type GPPMs as elementary gadgets and executing logical computation by combining them with other transversal / fold-transversal gates.

[0494] In Theorem 12, it is shown that the GPPMs developed in this work (see Definition 11), when combined with the fold-transversal gates, generate the full Clifford group on ^^ logical qubits encoded in an HGP code. All the HGP gadgets that are used in this example for logical computation are listed in the table shown in Fig.41.

[0495] Referring to Fig.41, a table summarizes the HGP logical gadgets utilized for logicalcomputation. All the gadgets but the inter-block CNOTs are applied on a [[^^, ^^, ^^]]symmetrical HGP(^^^, ^^^) code ^^ with physical qubits^^, where coordinates ^^ ∶=logical qubitsthe coordinates ofthe physical and logical qubits). ^^ା ∶= {(^^, ^^) ∈ ^^ ∣ ^^ > ^^} are denoted as the upper blocksof ^^, and similarly for ^^ା. The translation gadget, which translates the logical qubit alongany direction under periodic boundary conditions, i.e.,^భ,further requires that the base codeis quasi-cyclic. The inter-block CNOTs are applied between ^^ and another HGP code ^^′ that is obtained by puncturing / augmenting the basecodes of ^^, during which a subset of physical qubits ^^^^ and logical qubits ^ഥ^^^ of ^^ areremoved (see Fig. 31A). Transversal physical CNOTs between pairs of qubits identified by^^\^^^^ give transversal logical CNOTs between pairs of logical qubits identified by ^ഥ^\^ഥ^^^. Inthe case of ^^ᇱ ≃ ^^ and ^^^^ = ^ഥ^^^ = ∅, the standard transversal logical CNOTs between twoHQU-01425 HU 9476 blocks of CSS codes are recovered. All the gadgets can be implemented with a constant space overhead, i.e. using ^^(^^) physical qubits and the gadget times are listed in unites ofcode cycles (one round of QEC syndrome extraction). The inter-block CNOTs (with ^^ᇱ ≄^^) and the GPPMs (see Definition 11) are introduced in this work, under the generalframework of homomorphic gates and measurements; the translation gadget is explicitly constructed in this work, under the general framework of autonomorphism gates; the state preparation and transversal measurements are standard logical operations for any CSS codes; the H-SWAP and CZ-S gates are introduced under the general framework of fold-transversal gates.

[0496] To realize logical computation with low space-time overhead, it is further required that Clifford operations on different qubits can be implemented in parallel. Here, the implementation of a layer of Θ(^^) Clifford gates is considered, consisting of Hadamards, ^^gates, and intra-block CNOTs, acting on ^^ logical qubits of a [[^^^, ^^^, ^^^]] HGP code and itsspace-time cost is compared to that using surface codes with the same distances. Note that such layers of dense but low-weight Clifford operations can be used for performing active- volume computation with minimized idling locations when assuming long-range qubit connectivities. As shown in the table shown in Fig.33, although implementing such a sequence of PPMs using HGP data codes and lattice surgeries with rateless ancillae, e.g., surface codes, has a lower space overhead compared to using only surface codes, their space- time cost is essentially the same since the Θ(^^) gates would have to be executed sequentially in Θ(^^) logical cycles to maintain the constant space overhead. As such, the former scheme is essentially trading time for space. In contrast, introducing the GPPMs gadget for the HGP codes enables the parallel implementation of these gates in less than ^^(^^) logical cycles and outperforms surface codes also in terms of the total space-time cost.HQU-01425 HU 9476

[0497] The ability to implement generic Clifford gates with a sublinear depth for HGP codes utilizing the GPPMs is summarized in Theorem 12.

[0498] Theorem 12 (Parallizable Clifford gates for HGP codes). The gadgets in Fig.41, excluding the translation gadget, generate the full Clifford group on a generic HGP code. Furthermore, a layer of Θ(^^) random Clifford gates acting on ^^ logical qubits of a quasi- cyclic HGP code can be implemented with ^^(1) space overhead and in ^^(^^ଷ / ସ) logical cycles using the gadgets in Fig.41.

[0499] One can also distill and inject magic states for implementing parallel non-Clifford gates utilizing these parallel Clifford operations. In combination, one can realize logical computation with an asymptotically lower space-time cost using HGP codes, enabled by the new GPPMs gadget.

[0500] Note that the upper bound ^^(^^ଷ / ସ) in Theorem 12 on the circuit depth obtained is for implementing a generic (or worst-case) layer of Clifford operations. For many practical computational tasks / subroutines, one can compile them into layers of structured Clifford operations such that they can be implemented in much fewer, even a constant number of, logical cycles. Some task-specific and algorithm-tailored examples are presented below.

[0501] Referring to Fig.34, an illustration of a protocol for generating a GHZ state on all logical qubits of an HGP code using four sequences of GPPMs is shown. Each sequence of GPPMs is indicated by the 3D diamonds. As shown in Fig.34, a GHZ state across all logicalqubits on a HGP code block with a ^^ × ^^ grid of logical qubits can be generated with ^^(1)space overhead and ^^(1) logical cycles by transversally preparing all the logical qubits in the|+^ and performing the following sequence of GPPMs:HQU-01425 HU 9476where the four GPPM steps 3401, 3402, 3403, 3404 measure the PPMs in Fig.34.

[0502] As another example, distilling magic states in parallel is considered using only qLDPC codes. At the high level, one can distill a block of ^^ magic states in parallel encodedin a [[^^, ^^, ^^]] code patch using ^^ identical patches. As each qLDPC patch has a constantencoding rate, such a parallel distillation scheme is still space efficient as long as ^^ is not too large. Moreover, doing so only requires inter-block Clifford operations, which are generally easier than intra-block operations. Most of the operations are transversal, i.e., the same operation is applied to the same type of qubits across ^^ patches. However, the conditional Clifford corrections (or equivalently, reactive measurements) can break the transversal structure of the circuit. As such, it is necessary for the GPPMs gadget to perform selected operations on certain code patches. Therefore, the entire distillation scheme is also time- efficient as long as these selected operations can be implemented in parallel using the GPPMs.

[0503] Referring to Fig.35A-D, an illustration of one round of parallel magic state distillation on diagonal logical qubits of HGP codes with ^^(1) space overhead and ^^(1) logical cycles is shown. Each thick grey line indicates a block HGP code encoding ^^ logical qubits. An operation involving different thick lines refers to transversal operations on all the logical qubits of the corresponding HGP codes, unless specially noted.

[0504] Referring Fig.35A, an illustration of the 8-to-CCZ distillation logical circuit 3501 that converts 8 blocks of noisy |^^^ states into one block of less noisy |CCZ^ state 3502 is shown. As shown in Fig.35A, the "8-to-CCZ" distillation circuit is considered that convertsHQU-01425 HU 9476 8 noisy |^^^ states into one less noisy |CCZ^ state. The input is set as four HGP code patches 3503, three of which encoding |+^ states 3504 transversally while one of which encoding |^^^ states 3505 transversally. Then, the distillation circuit is implemented by a sequence of transversal ^^-type Pauli product rotations across the four patches and transversal ^^ measurements on the |^^^ patch. Each of the transversal Pauli product rotations can be implemented by introducing another code patch that encodes |^^^ transversally 3506, performing a joint transversal PPM, and finally measuring the introduced |^^^ patch reactively depending on the PPM outcomes (see Fig.35B). Since each qubit on the |^^^ patch is measured in ^^ or ^^ basis reactively, such measurements are not transversal, and thus are the most expensive components of the distillation task.

[0505] Referring to Fig.35B, each of the Pauli product rotations in Fig.35A can be implemented by supplementing an extra block of |^^^ states, performing joint transversal PPMs, and finally measuring the |^^^ block reactively in ^^ or ^^ basis, depending on the previous PPMs. Referring to Fig.35C, the reactive measurements of the |^^^ block in Fig. 35B can be implemented by introducing another ancilla block, whose logical qubits are initialized reactive in |+^ or |^^^ states, and then performing transversal Bell measurements.

[0506] As shown in Fig.35C, one can further reduce the problem of reactive measurements in ^^ / ^^ basis in Fig.35B to reactive state preparation in the ^^ / ^^ basis. By introducing another ancilla patch 3507 with qubits reactively initialized in ^^ / ^^ basis, and performing a transversal Bell measurement between the two patches, one can obtain the ^^ / ^^ reactive measurement outcomes of the data patch from the Bell measurement outcomes.

[0507] Referring Fig.35D, the reactive state preparation of the ancilla block in Fig.35C can be implemented by (1) initializing the logical qubits transversely in |+^ states 3508 (2) applying the fold-transversal CZ-S gate 3509 to convert the diagonal qubits to |^^^ states 3510HQU-01425 HU 9476 (3) performing a pattern of GPPMs 3511 to reset some of the diagonal qubits to |+^ states 3512.

[0508] One can then implement the reactive state preparation subroutine efficiently using the GPPMs gadget. As shown in Fig.35D, only the diagonal logical qubits in a symmetric HGPcode are first considered. One can prepare the diagonal qubits in an arbitrary pattern of|+^ / |^^^ states in three steps:1. Prepare all the diagonal qubits in |+^ states. 2. Apply the fold-transversal CZ-S gate (see Fig.41) to convert all the diagonal |+^ states to |^^^ states. 3. Apply a GPPMs(X, ^^^^, ^^^^), where ^^^^ = {{^^}}^^∈^^ and ^^ are the indices of thediagonal qubits to be prepared in the ^^ basis.

[0509] The above procedure implements the reactive state preparation subroutine on the diagonal √^^ qubits with ^^(^^) physical qubits and in ^^(1) logical cycles. Therefore, eachround of the distillation circuit in Figs. 35A-D can also be run on the diagonal √^^ qubits with^^(^^) physical qubits and ^^(1) code cycles.

[0510] Note that the main obstacle of performing the above distillation task on all logical qubits instead of the diagonal ones boils down to preparing an entire block of logical qubitsreactively in ^^ / ^^ basis (see Fig. 35D). It is shown how to realize it in ^^(√^^ log ^^) logicalcycles by imposing an additional translational symmetry of the HGP codes. At a high level, by using a family of quasi-cyclic classical codes, one can obtain HGP codes with a translational automorphism, i.e., the logical qubit grid can be translationally shifted with periodic boundary conditions by simply permuting the physical qubits. Then, one can repeatedly generate √^^ |^^^ states using the diagonal logical qubits (utilizing the CZ-S gate) and distribute them to other qubits via the translational automorphism. Finally, one canHQU-01425 HU 9476 selectively reset a subset of logical qubits to |+^ states efficiently using the ^^-type GPPMs. Therefore, by using HGP codes with additional translational symmetry, one can performmagic state distillation in Figs. 35A-D on ^^ logical qubits with ^^(1) space overhead in^^(√^^ log ^^) logical cycles per distillation round.

[0511] The same protocol can be applied to distilling |^^^ states in parallel using, e.g., a 15- to-1 distillation protocol, with the same scaling of the space and time overhead.

[0512] With Θ(^^) magic states distilled on qLDPC patches in parallel, one can also consume them and perform parallel non-Clifford gates. For instance, as shown in Fig.36A, one canimplement parallel Toffoli gates on three [[^^, ^^, ^^]] HGP patches by consuming three patchesof distilled |CCZ^ states. The |CCZ^ states are consumed via the transversal ^^-type PPMs together with the data patches, and the transversal ^^ measurements that follow. Then, reactive CZ gates on the data patches are applied depending on the PPMs outcomes. These reactive CZs are converted into reactive PPMs by introducing six ancilla patches 3601 as shown in Fig.36A, interacting them with the data patches via the transversal CNOTs, and reactively measuring them in either ^^ / ^^ basis or a Bell basis in a pairwise fashion. These reactive measurements are, again, not transversal operations since qubits on the same patch need to be measured in a different basis. Thus, one needs to implement them using the selective GPPMs gadget.

[0513] Referring to Figs.36A-B, illustrations of parallel inter-block Toffoli gates on HGP codes by injecting |CCZ^ states 3602 in parallel on diagonal logical qubits with ^^(1) space overhead and ^^(1) logical cycles are shown. Referring to Fig.36A, an illustration of a |CCZ^ injection circuit by performing joint PPMs between the data blocks and the |CCZ^ blocks and transversally measuring the |CCZ^ blocks, followed by reactive CZs on the data blocks is shown. The reactive CZs are implemented by coupling the data blocks via transversalHQU-01425 HU 9476 CNOTs to extra ancilla blocks, which are then reactively measured in a single-qubit or Bell basis. Referring to Fig.36B, the reactive measurements on each pair of the ancilla blocks in Fig.36A can be implemented by two GPPMs 3603 on each block followed by transversal Bell measurements.

[0514] As shown in Fig.36B, one can realize the reactive measurement on a pair of the ancilla patches in Fig.36A by first performing non-destructive ^^ / ^^-type GPPMs selectively on the subset of qubits to be measured in the single-qubit basis, and then performing the transversal destructive Bell measurements. Both the GPPMs in Fig.36B and the transversal Hadamard gates in Fig.36A can be implemented in parallel in ^^(1) logical cycles if the task is restricted again only on the diagonal logical qubits of symmetrical HGP codes (the fold- transversal H-SWAP gate implements transversal Hadamards on the diagonal qubits). Therefore, a protocol is obtained that implements√^^ Toffoli gates in parallel on diagonal logical qubits by consuming√^^ |CCZ^ states with ^^(^^) physical qubits in ^^(1) logical cycles.

[0515] Similar to the magic state distillation task, one can also extend the magic state consumption task to all ^^ logical qubits (beyond the diagonal ones) with ^^(^^) physical qubitsin ^^(√^^ log ^^) logical cycles by using HGP codes with additional translation symmetries.The trick is to implement the transversal Hadamards in Fig.36A on an HGP patch by first implementing the fold-transversal H-SWAP gate, and then using the translational automorphism, combined with GPPMs, to cancel the extra swaps on the non-diagonal qubits.

[0516] By using HGP codes with additional translational symmetries, one can not only distill^^ magic states but also consume them and implement non-Clifford gates in parallel with^^(1) space overhead in ^^(√^^ log ^^) logical cycles.HQU-01425 HU 9476

[0517] With the ability to distill and consume magic states in parallel, one can explore computation subroutines that require many parallel non-Clifford gates. As an example, the quantum adder is considered, which is an important subroutine of many useful quantum algorithms such as the factoring algorithm.

[0518] The adder inputs two ^^-qubit registersrepresenting two integers ^^ and ^^, respectively, and outputs a ^^-qubit register|(^^ + ^^)^^. Note that the adder is a unitary quantum circuit, so the input can also be asuperposition of integers and the corresponding output will be a superposition of the added integers.

[0519] Referring to Figs.37A-B, illustrations of an efficient parallel implementation of a quantum addder using HGP codes are shown. Referring to Fig.37A, the Gidney adder circuit 3701 is shown with temporal-AND Toffolis and repeated sectors, each sandwiched by two dashed lines. Different sectors are connected by the bridge qubits that are teleported across different blocks back in time. Specifically, each output carry qubit ^^^is teleported back in time as the input carry qubit ^^^ ା ^(of the next bit) utilizing the Bell measurements (BM) and the Bell state preparation (BSP). Referring to Fig.37B, one can implement all the sectors of Fig.37A approximately in parallel by encoding different qubits of the same type, e.g. {^^^}, into an HGP code and performing mostly inter-block operations. In particular, thetemporal-AND Toffolis in different sectors can be implemented in parallel using transversal|CCZ^ distillation and consumption (see Figs. 35A-D and Figs. 36A-B).

[0520] As in the magic state consumption circuit in Figs.36A-B, the ancilla blocks 3701 first interact with the ^^, ^^, and ^^ blocks via the transversal inter-block CNOTs (with details omitted here) and are finally measured reactively in a pairwise fashion. Different from Figs. 36A-B, here, the ancilla reactive measurements need to be executed sequentially and qubitsHQU-01425 HU 9476 on the same block need to be measured in order, depending on the previous outcomes (the details of the classical controls are omitted here).

[0521] One follows the circuit, which is a variant of the Gidney ripple-carry adder, thatperforms a ^^-qubit addition using ^^ − 1 temporal-AND Toffolis. As shown in Fig. 37A, thecircuit consists of ^^ − 2 repeated segments (the first and the last segments are different), eachcomputing the addition of the ^^-th bit of ^^ and ^^. The ^^-th and the (^^ + 1)-th segments areconnected by a shared carry bit that is simultaneously the output of the ^^-th segment and theinput of the (^^ + 1)-th segment. Because of these shared carry bits, the computation isgenerically sequential. Nevertheless, as shown in Figs.37A-B, one can parallelize the computation on different segments by introducing bridge qubits for the carry bits. More specifically, an output carry qubit ^^^for the ^^-th segment and a pair of input carry qubits ^^^ା^and ^^^ା^ in a Bell state for the (^^ + 1)-th segment are introduced. As shown in Figs. 37A-B,all the segments can be executed in parallel and finally, a Bell measurement can be performedbetween ^^^ and ^^^ା^ to effectively teleport ^^^ back in time as the input carry bit ^^^ା^ of the(^^ + 1)-th segment. Such a circuit can also be easily verified using the ZX calculus.

[0522] The circuit in Fig.37A is now almost parallel, except that there is a reactive CZ on each pair of the ^^ and ^^ bits depending on the measurement of the next ^^ bit. These reactive CZs come from the uncompute step of the temporal-AND Toffolis, and can be combined with the reactive CZs that come after consuming the |CCZ^ states at the computing step of the temporal-AND Toffolis. Thus, to perform these reactive CZs, one can introduce some ancilla qubits, let them interact with the ^^, ^^, and ^^ qubits via transversal CNOTs, and finally remove them via reactive measurements (same as Fig.36A). Now, the entire circuit can be implemented exactly in parallel, except for the final reactive measurement of the ancilla qubits.HQU-01425 HU 9476

[0523] Given this repeated and parallel structure, one can implement the adder using qLDPCpatches. As shown in Fig. 37B, one can encode the five types of qubits into five identical[[^^, ^^, ^^]] HGP patches, which are called the ^^-, ^^-, ^^-, ^^-, and ^^-patch, respectively. Thenthe circuit in Fig.37A can be implemented using HGP patches in Fig.37B that involve mainly inter-block and transversal operations, except for the final reactive measurements of the six ancilla patches (blocks 3701). These reactive pairwise measurements are the same as those in Fig.36A, except that they now also depend on the ^^ measurements of the ^^ patch. Note that these reactive measurements (RMs) need to be executed sequentially, i.e., the basis of the next RMs on the same block will depend on the outcome of the previous RMs, fundamentally because the adder task is generically sequential. Nevertheless, these are simply ancilla patches to be consumed and the data patches (^^- and ^^- patches) are free to perform other computational tasks in parallel. Moreover, these RMs, although sequential, are expected to be faster than the magic state distillation and consumption for practical adder sizes. As such, the entire qLDPC-based adder scheme in Fig.37B can be implementedapproximately in parallel with ^^(√^^ log ^^) logical cycles, predominantly limited by theparallel magic state distillation and consumption.

[0524] Finally, as shown in Fig.37B, in addition to the RMs, there are two extra operations that are not transversal. The first operation is the ^^-measurement on the first qubit of the ^^-block after the first transversal CNOT. This is due to the fact that the addition of the first bits^^^ and ^^^ does not have any input carry bit, which requires a reset of the carry qubit ^^^ to |0^via a ^^ measurement. This can be easily implemented using a ^^-type GPPMs selecting only the first logical qubit of the ^^-block. Finally, as shown in Fig.37A, the final Bellmeasurements between the ^^ and ^^ blocks are performed between the ^^-th ^^ qubit and the(^^ + 1)-th ^^ qubit, which also breaks the transversal structure. Fortunately, one can realizeHQU-01425 HU 9476 such a mismatched Bell measurement by simply cyclically shifting the ^^ block and then performing a transversal Bell measurement. One can implement such a cyclic shift of the ^^ block in ^^(1) logical cycles by combining the translational automorphism of a translational- symmetrical HGP code and the GPPMs gadget.

[0525] To conclude, one can implement a ^^-bit quantum adder in parallel using qLDPCcodes in ^^(√^^ log ^^) logical cycles by leveraging the parallel magic state distillation andconsumption protocols that are developed herein.

[0526] There are many other algorithms / subroutines that demand the parallel implementation of non-Clifford gates, such as the quantum random-access memory and the quantum state preparation. Techniques that leverage the parallel qLDPC-encoded non-Clifford gates can be exploited for a broad class of computational tasks with low space-time overhead.

[0527] As described, the homomorphic measurement gadgets developed for the homologicalproduct codes work generally for any dimension ^^ ≥ 2 by simply puncturing / augmentingtheir base classical codes. In particular, one can generalize the GPPMs gadget for the 2^^ HGP code measuring a grid pattern of PPMs to a ^^-dimensional gadget for a ^^-dimensional code measuring a ^^-dimensional-hypercube pattern of PPMs selectively and in parallel.

[0528] Although higher-dimensional (3^^ or 4^^) homological product codes have larger block sizes, they have two appealing features: 1. They have redundant check matrices inherently due to their higher-dimensional product construction, i.e. the syndromes satisfy extra linear constraints referred to as metachecks. These metachecks could help reduce the number of repeated syndrome measurements in the presence of measurement errors. For example, the 4D homological product codes, whose checks satisfy the soundness property, supportHQU-01425 HU 9476 single-shot preparation of computational-basis states. This enables homomorphism measurements with constant depth. 2. Some of these high-dimensional codes, e.g. the 3D surface codes, support transversal non-Clifford gates.

[0529] Here, in Definition 13, a parallel PPMs gadget for 3D homological product codes is explicitly presented by generalizing the GPPMs gadget for the HGP codes straightforwardly, and the construction for 4D codes follows. As shown, the logical qubits of a 3D homological product codes can be arranged on a cube (see also Fig.31B), and one can construct a ''Cube'' PPMs (CPPMs) gadget that measures a pattern of PPMs in parallel on any subcube of the logical qubits:

[0530] Definition 13 (Cube PPMs). Given a canonical representation of logical qubits in a3D homological product code on a 3D cube [^^^] × [^^ଶ] × [^^ଷ], and three sets of hyperedges^^^^, ^^^^ and ^^^^, where each hyperedge is a collection of indices in the ^^, ^^, and Z direction,respectively, a pattern of Cube PPMs of a type ^^ ∈ {^^, ^^} is defined as:where ^ത^^ denotes the logical Pauli operator ^ത^ (^̅^) of the logical qubits ^ത^^ with coordinates ^^for ^^ = ^^ (^^).

[0531] Since the new homomorphic CNOTs and measurements are built upon transversal physical CNOTs between two alike code patches, one can efficiently implement them by overlapping the two patches and then applying pairwise CNOTs on the same coordinates. Such a scheme is mostly favored by platforms, e.g. the reconfigurable atom arrays, where the qubits can be dynamically rearranged, and parallel two-qubit gates can be applied with global controls.HQU-01425 HU 9476

[0532] In the description below, the logical gadgets used for the HGP codes are presented in addition to the GPPMs gadget. The logical translation gadget listed in Fig.41 for HGP codes with quasi-cyclic base classical codes is presented. Then, it is shown how to implement transversal logical Hadamard gates (without extra swaps) and transversal |^^^-state preparation utilizing the translation gadget. Such gadgets enable implementation of parallel magic state distillation and injection on all logical qubits of HGP codes, extending from the diagonal qubits.

[0533] The logical translation gadget is based on the automorphism of a quantum code ^^ ={^^ଶ, ^^^, ^^^}: ^^ → ^^ such that the following diagram commutes:

[0534] This is a special case of the homomorphism between two quantum codes ^^: ^^ᇱ → ^^with ^^ᇱ = ^^. Again, one works in the standard basis where each each basis vector of ^^ଶ, ^^^,and ^^^represents a ^^ check, a qubit, and a ^^ check, respectively. If ^^^is a permutation matrix, the physical permutation of the qubits according to ^^^preserves the stabilizer group and thus implements a logical gate, which is called an automorphism gate. In the description below, a special type of automorphism gate for HGP codes is constructed that implements a translation of all the logical qubits in the canonical basis.

[0535] Based on Proposition 8, one can also construct an automorphism for a homological product code by taking the tensor product of automorphisms of its base classical codes. In the particular case of a HGP code ^^, which is constructed from two base length-1 chainHQU-01425 HU 9476complexes ^^^ and ^^ଶ (see Fig. 38), one can construct an automorphism ^^ =where is an automorphism for ^^^ (^^ = 1, 2) such that the following diagramcommutes:Equation 84

[0536] Note that the direct sum in Equation 83 indicates that ^^^takes a block-diagonal form with respect to the two blocks of qubits in Fig.38.

[0537] For a symmetrical HGP code with a base check matrix ^^ ^ ் ଶ^, ^^^ = ^^^ = ^^^ areassigned. Furthermore, a quasi-cyclic base code is considered, whose check and generator matrix are in the following form:where ℝ^ ∶= ^^^[^^] / (^^^ − 1) denotes the quotient polynomial ring, ^^ represents a circulantmatrix of size ^^ that shifts the entries by 1, and ^^ represents the ^^ × ^^ identity matrix. ^^ isalso denoted as the lift size in the literature. Based on the form of ^^^, this code encodes ^^ codewords, which can be cyclically shifted by cyclically shifting each of the ^^^blocks of bits. Moreover, the same block shifts of the bits correspond to cyclically shifting the ^^^HQU-01425 HU 9476blocks of checks. Thus, it admits an automorphism {^^^,^, ^^^,^}, where ^^^,^ and ^^^,^ denote theblock-cyclic-shift matrix by ^^ for the checks and the bits, respectively. For example, for ^^^ =Finally, an automorphism for the HGP code is constructed asEquation 86 where the two blocks of shifts in Equation 86 are applied to the two blocks of qubits given by the product of the classical bits and checks, respectively (see Equation 63 and Fig.42).Moreover, the qubits can be further divided into ^^ × ^^ blocks, each undergoing the sametranslation by (^^, ^^) under periodic conditions (see Fig. 42). It is easy to verify that thecanonical logical qubits ^^ത^௫,௬^௫∈ [^],௬∈ [^] (there are ^^ × ^^ logical qubits in total since each basecode encodes ^^ logical bits) are also translated by (^^, ^^) under periodic conditions, i.e.,87for ^^ = ^^, ^^. ^ത^^,^ is denoted as such a gadget. An example for ^ത^^,^ with ^^ = 2 is shown inFig.42. Referring to Fig.42, an illustration of the logical translation gadget for HGP codes with quasi-cyclic base codes is presented. Physical block-translations corresponding to products of horizontal and vertical block-cyclic-shifts result in translation of the logical block under periodic boundary conditions.

[0538] Finally, it is noted that not all quasi-cyclic classical codes are in the required form of Equation 85. Generic quasi-cyclic codes might have multiple rows in their block generator matrix ^^^, corresponding to disjoint blocks of codewords, where only cyclic permutation within each block is permitted; their block generator matrix might not be in the canonical form, i.e., starting with an identity matrix, which is required to define the canonical logicalHQU-01425 HU 9476 operator basis used in this work. Generator matrices that satisfy Equation 85 are referred to as one-generator systematic-circulant (OGSC), and algebraic conditions for obtaining codes with OGSC generator matrices have been explored in the literature. In Fig.43, several examples are presented through numerical search that feature even better parameters than those based on random expander graphs. Referring to Fig.43, a table of the parameters and code matrices of finite-size OGSC classical codes is shown.

[0539] Here, a gadget is presented that teleports any subset ^ഥ^^ of the logical qubits ^ഥ^ of ageneric HGP code ^^ to the corresponding logical qubits ^ഥ^^(with the same coordinates) ofanother identical code ^^′. Let cw(^ഥ^^) (rw(^ഥ^^)) denote the number of columns (rows) of thelogical qubit grid that ^ഥ^^ are supported on. Such a teleportation gadget Tel(^ഥ^^ → ^ഥ^′^) ) ispresented in algorithm 3 shown in Fig. 44 in ^^(min{cw(^ഥ^^), rw(^ഥ^^)}) logical cycles. Thegadget performs the teleportation in either a column-by-column fashion or a row-by-row fashion using generalized versions of the GPPMs gadget.

[0540] Here, a gadget is presented for performing a cyclic logical shift on a [[^^^, ^^^, ^^^]]HGP code ^^ with OGSC base classical codes by combining the logical translation gadget andthe GPPMs gadget. Let, where ^^^^ = ^^^, be the 2D grid of the logicalqubits. They are labeled in a zigzag pattern, i.e. ^ത^^ . Here, to keep theேnotation simple, ^^ mod ^^ = ^^ is set for any ^^ ∈ ℤ. Then, one can perform a cyclic shift,ା ^)୫୭^ ^బ, in the following two steps:1. Perform a horizontal logical translation ^ത^^,^. This realizes most of the cyclic shifts, except for the rightmost column of logical qubits {^ത^ே^}^∈ [ெ], which are permuted to the leftmost column. It thus remains to do a vertical cyclic shift of this leftmost column.HQU-01425 HU 9476 2. Teleport the first column of logical qubits of ^^ to another identical code ^^′, perform a vertical logical translation ^ത^^,^ on ^^′, and then teleport the first column oflogical qubits back to ^^.

[0541] Here, a gadget is presented for preparing |തത^^ത^⊗ ^బ for a [[^^^, ^^^, ^^^]] HGP code. Twoidentical HGP codes ^^^ and ^^^ , both initialized inത|ത0തത^⊗ ^బstates, are prepared. Then, thefollowing sequence of operations iteratively for ^^^is applied:for ^^ ∈ [^^^^ / 2], and the following sequence of operation iteratively for ^^^:for ^^ ∈ [^^^^ / 2 − 1]. ^^^^ denotes a subroutine for measuring all diagonal logical qubits inthe ^^ basis non-destructively, using algorithm 4 shown in Fig. 45A in ^^(log ^^^) logicalcycles.

[0542] Referring to Fig.45B, an illustration for measuring all the non-diagonal qubits of an HGP code in log depth for the step 3 of algorithm 4 shown in Fig.45A at t=1 is presented. Performing the single-qubit measurements can be viewed as “filling” an empty patch except for the diagonal. The first time step fills the solid 4501 and the hashed squares 4502 with two GPPMs gadget. This leaves two subsquares 4503 to be filled. This procedure can then be applied recursively to fill the entire square in log depth. Note that later steps might fill the regions that have already been filled by the previous steps, but importantly, the diagonal line will not be filled.

[0543] Referring to Fig.46, an illustration of one sequence of operations of the iterativegadget for preparingis shown. The filled region 4601 and the white region 4602HQU-01425 HU 9476 represent logical qubits in 0തand ^^̅states, respectively. As shown in Fig.46, each sequence inEquation 88 or Equation 89 generates ^^^^ത|ത^^ത^ states on the diagonal qubits utilizing the CZ-Sgate and then distributes them to non-diagonal qubits utilizing the logical translation gadget.The two sequences of operations then fill ^^^ and ^^^ with two complementary half blocks ofത |ത^^ത^ states, respectively. Finally, the ത |ത^^ത^ states in ^^^ are merged into ^^^ by performing thefollowing transversal operations: initially, each pair of qubits in ^^ and ^^ are either stabilizedby ^ ^ത^^, ^̅^^^ or ^ ^̅^^, ^ത^^^; performing transversal ^ത^^ ^ത^^ measurements projects all the pairsinto the same entangled state stabilized by ^^ത^^ ^ത^^, ^^^ ^̅^^, ^̅^^, ^ത^^^ (up to some Paulicorrections); final transversal ^^ measurements on B projects each pair into a product statestabilized by ^ ^^^, ^^^ ^, which indicates that the ^^ states in ^^ are all merged into ^^.

[0544] Since each sequence in Equation 88 or Equation 89 takes ^^(log ^^^) logical cycles andthere are sequences in total, the entire gadget takes ^^(^^^^ log ^^^) logical cycles.

[0545] Here, a gadget is provided for implementing logical Hadamard gates transversely onlogical qubits of a [[^^^, ^^^, ^^^]] symmetrical HGP code. The fold-transversal H-SWAP gateis first applied, which applies the desired transversal Hadamards up to extra swaps along the diagonal (see Fig.41), then the extra logical swaps are cancelled by utilizing the GPPMsgadget in a similar fashion as that for transversely preparing theത|ത^^ത^ states.

[0546] To implement the logical swaps on a code ^^, two identical ancilla codes ^^ᇱand ^^′′are prepared. Let ^^^ denote a line of logical qubits ^^ത^^,^ ା ^^^∈ [^^బ] (only including qubitswith valid coordinates ∈ ^^^^^൧ × [^^^^]) with an offset ^^ to the diagonal. The swap gadgetamounts to swapping the "twin" lines ^^^ ↔ ^^ି^ for ^^ ∈ [^^^^ − 1].

[0547] Referring to Fig.47, an illustration of a sequence of operations of the gadget for swapping logical qubits of a symmetrical HGP code along the diagonal is shown. ReferringHQU-01425 HU 9476to Figs. 48A-C, illustrations of circuits for implementing teleportation (Fig. 48A), teleported^^ gate (Fig. 48B), and measurement-based CNOT (Fig. 48C) are shown.As illustrated in Fig.47, each such swap can be implemented by the following sequence of operations:where each teleportation between two identified lines of qubits across two codes, e.g.,Tel(^^^ → ^^ᇱ) from ^^^ to ^^ᇱ^ , can be implemented by the teleportation circuit in Fig. 48A,where the ^^^^ measurements on ^^^and ^^ᇱ^ are implemented using another identical ancillacode ^^ᇱᇱᇱ, whose logical qubits ^^ᇱ^ᇱare initialized inത|ത0തത^ while the rest are initialized inത|+തതത^.Such a selective initialization can be implemented by measuring the diagonal qubits in ^^basis in ^^(log ^^^) cycles using algorithm 4 shown in Fig. 45A (with ^^ and ^^ flipped and upto extra ^^ measurements using a single ^^-GPPMs gadget) and then distributing theത|ത0തത^ statesto ^^ᇱ^ᇱᇱby performing a logical translation ^ത^^ᇱ,ᇱ^ᇱ. The ^^ measurements in Fig.48A on ^^^can beimplemented similarly using ancilla code ^^ᇱᇱᇱwhose logical qubits ^^ᇱ^ᇱare initialized inത|+തതത^while the rest are initialized inത|ത0തത^.

[0548] Overall, the transversal Hadamard gates, implemented by an H-SWAP gate andsequences of line swaps in Equation 90, take ^^(^^^^ log ^^^) logical cycles.

[0549] In the description below, the proof of Theorem 12 regarding the parallel implementation of arbitrary Clifford gates (consisting of Hadamards, ^^ gates, and CNOTs) is provided using the gadgets in Fig.41. It is first proven that a layer of ^^(^^^) Clifford gates can be implemented in parallel in ^^(^^ଷ / ସ^ ) logical cycles using the gadgets in Fig.41, including the translation gadget. Then, it will become apparent that the translation gadget isHQU-01425 HU 9476 essential only for the parallelism, but not necessary for generating the full Clifford group. It is noted that this task of implementing Clifford gates in parallel is a compilation problem with a restricted gate set and only provides an upper bound on the circuit depth using a constructive compilation.

[0550] In the worst-case scenario, a layer of random Clifford gates contains ^^(^^^) Hadamardgates, ^^(^^^) ^^ gates, and ^^(^^^) random CNOTs, supported on the logical qubits ^ഥ^|ு, ^ഥ^|ௌ,and ^ഥ^|େ^^^of an HGP code ^^, respectively. These three types of gates are implemented separately.

[0551] First, it is noted that one can teleport any subset ^ഥ^^of the logical qubits of ^^ transversely to the corresponding logical qubits of another identical code ^^′, and vice versa, in ^^(^^^^) logical cycles using the selective teleportation gadget.

[0552] To implement the Hadamard gates, ^ഥ^|ு is teleported transversely to ^ഥ^′|ு of anothercode ^^′. Then, the transversal Hadamard gates (without extra swaps) are applied on all thelogical qubits of ^^′ in ^^(^^^^ log ^^^) logical cycles using the subroutine described above.Finally, ^ഥ^′|ு is teleported back to ^ഥ^|ு.

[0553] The ^^ gates are implemented using teleported gates. As shown in Figs.48A-B,another identical code ^^′ is prepared, where ^ഥ^′|^ are initialized inത|ത^^ത^ states, while the rest arein ത |+തതത^ states. Then transversal CNOTs between ^ഥ^ and ^ഥ^′ followed by transversalmeasurements of ^ഥ^ teleport the logical qubits from ^^ to ^^′ with the desired ^^ gates applied.The selective initialization of ^^′ can be implemented by first preparing all the logical qubitsin ത |ത^^ത^ using the subroutine in ^^(^^^^ log ^^^) logical cycles, followed by resetting the qubits^ഥ^′\^ഥ^′|^ to ത |+തതത^ using GPPMs in a column-by-column fashion in ^^(^^^^) logical cycles.

[0554] Referring to Figs.49A-D, illustrations for implementing a layer of random CNOTs using the gadgets in Fig.41 are shown. Fig.49A shows the classification of all the CNOTsHQU-01425 HU 9476 into three types: the “Aligned”-CNOTs 4901, the “TLBR”-CNOTs 4902, and the “TRBL”- CNOTs 4903. Referring to Fig.49B, the “TRBL”-CNOTs form different clusters according to their shared ancilla 4904 (empty circles) and the clusters are partitioned into sparse clusters 4905 (the right two) and dense clusters 4906 (the left two). Referring to Figs.49C-D, shifting and “symmetrizing” the CNOTs of a dense cluster is shown, such that each CNOT ismirrored along the diagonal line ^^ = ^^. To implement the CNOTs, they are classified intothree types: the "Aligned"-CNOTs, acting on logical qubits within the same row or the same column (CNOTs 4901 in Fig.49A; the "TLBR"-CNOTs, acting on pairs of qubits oriented from the top left to the bottom right (CNOTs 4902 in Fig.49A; and the "TRBL"-CNOTs, acting on pairs of qubits oriented from the top right to the bottom left (CNOTs 4903 in Fig. 49A). The "Aligned"-CNOTs are the easiest to implement. The vertically aligned CNOTs are implemented in a column-by-column fashion. For each of the qubit pairs in a column, a distinct ancilla on the same column is introduced and the measurement-based CNOT isimplemented consisting of two Bell measurements (followed by measuring the ancilla). The^^ (and similarly, ^^-) Bell measurements required for all the vertical CNOTs in a column canbe implemented using a single GPPMs gadget as long as they do not share ancillae. One can assume that there are enough ancillae since otherwise one can simply teleport part of the vertical CNOTs (at most half) to another empty patch and implement them separately. As a result, all the vertically-aligned CNOTs can be implemented inlogical cycles and similarly, all the horizontally aligned CNOTs can be implemented in ^^(^^^^) logical cycles in a row-by-row fashion.

[0555] Finally, the proof is finished by showing that the "TRBL"-CNOTs can be implemented in ^^(^^ଷ / ସ^ ) logical cycles (same for the "TLBR"-CNOTs). As shown in Fig.49B, for any "TRBL"-CNOT acting on a pair of qubits ^ത^^,^ and ^ത^ ᇱ^ᇱ,^ᇱ, where ^^ > ^^ and ^^ <HQU-01425 HU 9476 ^^ᇱ, one implements it by introducing an ancilla ^ത^^ᇱ,^(see the empty circles in Fig.49B) and implements the circuit in Fig.48C, where the ^^- and ^^-Bell measurements are applied on horizontal and vertical pairs, respectively. The CNOTs are grouped into different clusters according to the ancilla they share. These clusters are further partitioned into two types: dense clusters with more than ^^ CNOTs and sparse clusters otherwise, where ^^ is someconstant integer that will be specified below (see Fig. 49B for an illustration with ^^ = 1).One can implement the sparse clusters in a column-by-column and row-by-row fashion, similar to that for implementing the "Aligned"-CNOTs. Specifically, they are implementedin ≤ ^^ sequences and for each sequence, one picks one CNOT from each cluster andimplements all these picked CNOTs by first performing all the required vertical ^^-Bell measurements in a column-by-column fashion inlogical cycles and then performing all the required ^^-Bell measurements in a row-by-row fashion inlogical cycles. Intotal, implementing these sparse clusters thus takes ^^ௌ =logical cycles.

[0556] Lastly, the dense clusters are implemented one-by-one and for each dense cluster with^^ qubit pairs sharing a common ancilla ^ത^ఈ,ఉ, implemented in parallel in^^(1) logical cycles. The strategy is to first shift and "symmetrize" these pairs such that eachpair is mirrored along the diagonal line ^^ = ^^ (see Fig. 49A to Fig. 49B for an illustration),and then implement these symmetrized CNOTs in parallel using the CZ-S gate (up to some Hadamards), which applies pair-wise CZs folded along the diagonal. Specifically, the cluster is first teleported to another empty code ^^ᇱin ^^(1) logical cycles by first teleporting the rowand then the column. Without loss of generality, assume that ^^ < ^^. Then, the entire clusteris shifted such that the ancillaఈ,ఉ is on the diagonal (^^, ^^) by applying the translational gate. Now, each pair (^ത^ᇱ , ^ത^ᇱ ) gets sh തᇱ തᇱିఈ,^ ^^,ఉ ఈ,^^ ifted to (^^ఉିఈା^^,ఉ , ^^ఉ,^^ ). Then, to symmetrizeHQU-01425 HU 9476these pairs, the column qubits ^ത^ᇱ ↔ തᇱఉିఈା^^,ఉ ^^^^,ఉ for ^^ ∈ [^^] are swapped. Each of the swapscan be done by introducing an ancilla within the same column and performing pairs of Bell measurements (similar to implementing a CNOT). One can assume that there are enough empty ancillae and the target qubit locations do not overlap with the original qubit locations so that the ^^ swaps can be done in ^^(1) logical cycles using the parallel GPPMs (otherwise, one can teleport at most half of the qubit pairs to another code and implement them separately and in parallel). Finally, the fold-transversal CZ-S gate is applied to implement the symmetrized CNOTs (up to some Hadamards that can be addressed separately, as describedabove) and reverse the above process to teleport the qubits back to their original position in^^. Since each of the dense clusters can be implemented in ^^(1) logical cycles and there areat most ^^^ / ^^ such dense clusters, implementing all the dense clusters takes ^^^ = ^^(^^^ / ^^)logical cycles.

[0557] To sum up, implementing all the "TRBL"-CNOTs thus takes ^^ = ^^ௌ + ^^^ =^^(^^^^^^) + ^^(^^^ / ^^) logical cycles. By choosing ^^ = Θ(^^^ / ସ^ ), it is shown that ^^ =^^(^^ଷ / ସ^ ), which completes the proof of Theorem 12.

[0558] Finally, it is noted that the full Clifford group can be generated without using the translation gadget and assuming that the base codes are quasi-cyclic. The construction for any selective ^^, ^^, or CNOT uses essentially the same ingredients as described above, although different Clifford gates might have to be executed sequentially in the absence of the translation gadget.

[0559] In the description below, more details are provided on constructing the parallel PPMs gadget in Definition 13 on any subcube of the logical qubits of a 3D homological product code.HQU-01425 HU 9476

[0560] Similar to that for the HGP code, the construction here works for a canonical basis of logical qubits that is described below. Referring to Fig.50, a chain complex of a 3D homological product code is shown. As shown in Fig.50, a 3D homological product code is constructed by taking the total complex of the tensor product of three length-1 complexes డ^^^^ →భ^^ ^^^^ൠ . The first three vector spaces ^^ଷ, ^^ଶof the total complex are^ ୀ ^,ଶ,ଷassociated with ^^ checks, qubits, and ^^ checks, respectively. The extra vector space ^^^isassociated with ^^ meta checks that check the ^^ checks: ^^^ ^^^ = 0. The base complexes areassigned with three classical codes with check matrices{^^^}^ୀ^,ଶ,ଷas follows:

[0561] Then the check matrices ^^^and ^^^and the ^^ meta check matrix ^^^are given by

[0562] Based on Equation 92, one can derive a canonical basis of logical operators using the K¨unneth formula:HQU-01425 HU 9476

[0563] Again, without loss of generality, it is assumed that each check matrix ^^ఈcan be row-reduced to its canonical form, from which one can derive the canonical form of ker (^^ఈ) andrs (^^ఈ)∙ in Equation 66. Then, one can find a canonical basis for the logical operators inEquation 93 by forming conjugate pairs[^య], where:Equation 94

[0564] The logical operators are all supported on ^^^^ × ^^^^ × ^^^^ ≃ [^^^] × [^^ଶ] × [^^ଷ] andthe logical qubits ^ഥ^ =[^భ]× [^మ]× [^య] can be arranged on a [^^^] × [^^ଶ] × [^^ଷ]cube, where the logical operator pairs of ^ത^^,^,^intersect on the physical qubit ^^^,^,^.

[0565] With this canonical logical basis, one can implement the Cube PPMs gadget on a 3D code ^^ in Definition 13 using essentially the same two-step protocol for the GPPMs for HGP codes (see Fig.39 and Fig.32A-C). The protocol is as follows: (1) Construct an ancilla ^^′ by performing puncturing and augmenting on ^^^and ^^ଶaccording to ℰ௫and ℰ௬, and then construct a mask code ^^′′ by performing puncturing and augmenting on ^^ଷaccording to ℰ௭.(2) Prepare the ancilla code ^^′ in the logical ^^ basis and reset some logical qubits toത|ത+തത^ andsome to GHZ states using ^^′′. (3) Perform the desired CPPMs on ^^ using ^^′.

[0566] Definition 14 (Reduced weight). Given a binary check matrix ^^ ∈ ^^^× ^ଶ and anerror ^^ ∈ ^^^ଶ, the reduced weight of ^^ w.r.t. ^^ is defined as |^^| ∗ ∗ு ∶= min {|^^ |, ^^ ^^ = ^^ ^^}.HQU-01425 HU 9476

[0567] Definition 15 (Soundness) Let ^^ be an integer, and ^^: ℤ → ℝ be some monotonicallyincreasing function with ^^(0) = 0. Given a binary check matrix ^^ ∈ ^^^× ^ଶ , it is said that itis (^^, ^^)-sound if for any ^^ ∈ ^^^ଶ such that |^^ ^^| < ^^, it shown that

[0568] Definition 16 (Single-shot state preparation) For a state preparation protocol that prepares a |0^^(|+^^) state by measuring one round of ^^ (^^) checks associated with ^^^(^^^) and applies corresponding correction, it is said that such a protocol is (^^, ^^)-single shotif for any syndrome error ^^^ with |^^^| < ^^ that occurs during the check measurement, thecorrected output differs from |0^^(|1^^) by an error ^^ that satisfies:

[0569] Lemma 17 (Soundness is sufficient for single-shot state preparation) For a CSS codewith a (^^, ^^)-sound ^^ (^^) check matrix ^^^ (^^^) of single-shot distance ^^ௌௌ, there exists a

[0570] Proof. The case for preparing|0^^is considered by measuring one round of ^^ checks, and the other case is mirrored. Let ^^ be a (random) measured syndrome pattern in theabsence of measurement errors, and ^^ corresponds to a ^^ error ^^^ such that ^^^ ^^^ = ^^. Let^^^ be a syndrome error that adds to ^^. Let ^^^ be the metacheck matrix for ^^^ that satisfies^^^ ^^^் = 0 mod 2. The following two-stage correction protocol is applied:• Find a minimum-weight syndrome correction ^^^ such that the corrected syndrome ^^′ ∶= ^^ + ^^^ + ^^^ passes the meta checkes, i.e., ^^ᇱ ∈ ker (^^^).HQU-01425 HU 9476 •If ^^′ is a valid syndrome, i.e. ^^ᇱ ∈ Im(^^^), find a ^^ Pauli correction ^^^ that matchesthe corrected syndrome, i.e. ^^^ ^^^ = ^^ + ^^^ + ^^^; Otherwise, declare a logicalfailure.

[0571] Now, it is proven that the residual error ^^ = ^^^ ^^^ of the above protocol satisfies^^ௌௌ}. First, it is proven by contradiction that the correctedsyndrome ^^′ is a valid syndrome. Assume ^^′ is not a valid syndrome. Since ^^ is a validsyndrome, ^^^ + ^^^ must not be a valid syndrome. However, |^^^ + ^^^| ≤ 2 |^^^| < ^^ௌௌ (bythe minimum-weight assumption of ^^^). This would imply that there exists a ^^^ ∈Ker(^^_^^)\Im(^^^) with |^^^| < ^^ௌௌ, which leads to a contradiction. Therefore, thecorrected syndrome ^^′ will be a valid syndrome. Next, it is shown that the residual error ^^has a small reduced weight. Since |^^^ ^^| = |^^^ (^^^ ^^^)| = |^^^ + ^^^| ≤ 2 |^^^| < ^^, it isshown, by the soundness property in Equation 95,

[0572] Proposition 18 (4D homological product codes support single-shot preparation) A 4Dhomological product code out of four identical [^^^, ^^^, ^^^] classical base codes supports^^(^^) = ^^ଷ / 4.

[0573] Proof. A 4D homological product code inherently has (^^, ^^)-soundness for both their^^ and ^^ checks, where ^^ = ^^ and ^^(^ ଷ^ ^) = ^^ / 4. Moreover, it has a single-shot distance^^ௌௌ = ∞. Then, according to Lemma 17, it supports (^^, ^^)-single shot state preparation inboth ^^ and ^^ basis, where ^^ = ^ଶ ^^^.HQU-01425 HU 9476

[0574] The descriptions of the various embodiments of the present disclosure have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

HQU-01425 HU 9476 CLAIMS What is claimed is:

1. A quantum computing system, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a quantum low-density parity-check (qLDPC) code; a second plurality of physical qubits, configured to encode a second plurality of logical qubits using an error-correcting code; and a plurality of ancilla qubits, wherein the quantum computing system is configured to teleport at least one of the first plurality of logical qubits into a corresponding at least one of the second plurality of logical qubits via the plurality of ancilla qubits, the quantum computing system is configured to perform at least one logical operation on the at least one of the second plurality of logical qubits, and the quantum computing system is configured to teleport the at least one of the second plurality of logical qubits into the at least one of the first plurality of qubits via the plurality of ancilla qubits.

2. The quantum computing system of claim 1, wherein the plurality of ancilla qubits is configured to encode a hypergraph product of a classical code corresponding to a logical operator of the qLDPC code and a classical code corresponding to a logical operator of the error-correcting code.

3. The quantum computing system of claim 1, wherein teleporting the at least one of the first plurality of logical qubits and / or the at least one of the second plurality of logical qubits comprises performing a measurement-based circuit with the plurality of ancilla qubits.

4. The quantum computing system of claim 3, wherein performing the measurement- based circuit comprises performing lattice surgery.HQU-01425 HU 9476 5. The quantum computing system of claim 1, further configured to perform a plurality of quantum error correction (QEC) cycles on the first plurality of physical qubits.

6. The quantum computing system of claim 5, wherein the plurality of QEC cycles comprises a plurality of intervals, and wherein performing the plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals.

7. The quantum computing system of claim 6, wherein applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph.

8. The quantum computing system of claim 5, wherein performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered-statistical decoding (OSD).

9. The quantum computing system of claim 8, wherein applying the final BP and OSD comprises constructing a decoding graph of data errors.

10. The quantum computing system of claim 1, wherein the qLDPC code is a hypergraph product (HGP) code.

11. The quantum computing system of claim 1, wherein the qLDPC code is a lifted- product (LP) code.

12. The quantum computing system of claim 1, wherein the error-correcting codes is a topological code.

13. The quantum computing system of claim 12, wherein the topological code is a surface code.

14. The quantum computing system of claim 1, wherein the first plurality of physical qubits comprises a first array of neutral atoms and the second plurality of physical qubits comprises a second array of neutral atoms, wherein each neutral atom has a first state and an excited Rydberg state, andHQU-01425 HU 9476 each neutral atom is arranged to impose a Rydberg blockade on at least its nearest neighbors in its array when in the excited Rydberg state.

15. The quantum computing system of claim 1, wherein the first plurality of physical qubits comprises a plurality of data qubits and a plurality of ancilla qubits, and wherein encoding the first plurality of logical qubits using the qLDPC code comprises: providing the plurality of data qubits, each of the plurality of data qubits disposed in a corresponding trap; providing the plurality of ancilla qubits, each of the plurality of ancilla qubits disposed in a corresponding trap; arranging the plurality of data qubits and the plurality of ancilla qubits in a plurality of rows and a plurality of columns, thereby forming a lattice; performing a plurality of permutations of the plurality of rows and the plurality of columns, each of the plurality of permutations placing each of the plurality of data qubits within an interaction radius of one of the plurality of ancilla qubits, thereby forming a plurality of proximate pairs; and subsequent to each of the plurality of permutations, applying a global control pulse to the lattice, thereby applying a gate to each of the plurality of proximate pairs, and thereby encoding a parity check matrix between the plurality of ancilla qubits and the plurality of data qubits.

16. The quantum computing system of claim 15, wherein encoding the first plurality of logical qubits using the qLDPC further comprises, prior to forming the lattice: applying a control laser pulse to the plurality of data qubits to thereby prepare them in an initial state.HQU-01425 HU 9476 17. The quantum computing system of claim 15 or 16, wherein arranging the plurality of data qubits and the plurality of ancilla qubits in the lattice comprises: moving, in parallel, the plurality of ancilla qubits into the lattice.

18. The quantum computing system of claim 15, wherein performing the plurality of permutations comprises: moving, in parallel, one or more rows within the lattice and moving, in parallel, one or more columns within the lattice.

19. The quantum computing system of claim 15, wherein the plurality of ancilla qubits comprises Z stabilizers and X stabilizers.

20. The quantum computing system of claim 15 or 19, wherein encoding the first plurality of logical qubits using the qLDPC code further comprises: removing a subset of the plurality of ancilla qubits from the lattice and performing a measurement on the subset.

21. The quantum computing system of claim 20, wherein the subset corresponds to Z stabilizers.

22. The quantum computing system of claim 20, wherein the subset corresponds to X stabilizers.

23. The quantum computing system of claim 15, wherein performing the plurality of permutations comprises: determining a collision-free path for each of the plurality of data qubits and each of the plurality of ancilla qubits for each of the plurality of permutations.

24. The quantum computing system of claim 23, wherein determining the collision-free path comprises: bipartition and recursive sorting of the plurality of data qubits and the plurality of ancilla qubits.HQU-01425 HU 9476 25. The quantum computing system of claim 23, wherein the collision-free path is a cubic spline.

26. The quantum computing system of claim 23, wherein determining the collision-free path comprises: decomposing the parity check matrix into a first product graph and a second product graph; determining a routing of the plurality of rows according to the first product graph; and determining a routing of the plurality of columns according to the second product graph.

27. The method of claim 26, wherein the plurality of permutations comprises row-specific and / or column-specific permutations.

28. The quantum computing system of claim 27, wherein performing the plurality of permutations comprises: applying a pinning beam to at least one qubit of the plurality of data qubits or the plurality of ancilla qubits, thereby maintaining the position of the at least one qubit according to the routing of the plurality of rows or columns.

29. The quantum computing system of any one of claims 15-28, wherein the gate applied to each of the plurality of proximate pairs is a CZ gate.

30. The quantum computing system of any one of claims 15-29, wherein the global control pulse is a laser pulse.

31. The quantum computing system of any one of claims 15-30, wherein the trap corresponding to each of the plurality of data qubits and to each of the plurality of ancilla qubits is an optical trap.HQU-01425 HU 9476 32. The quantum computing system of claim 31, wherein the optical traps corresponding to the plurality of data qubits and to the plurality of ancilla qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), and moving the one or more rows and moving the one or more columns comprises varying a drive frequency of the at least one AOD.

33. The quantum computing system of claim 32, wherein moving the one or more rows and moving the one or more columns further comprises applying one or more rotations during said moving.

34. The quantum computing system of claim 33, wherein applying the one or more rotations comprises applying a Raman pulse.

35. A method of performing quantum computation, the method comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits using a quantum low-density parity-check (qLDPC) code on the first plurality of physical qubits; providing a second plurality of physical qubits; encoding a second plurality of logical qubits using an error-correcting code on the second plurality of physical qubits; providing a plurality of ancilla qubits; teleporting at least one of the first plurality of logical qubits into a corresponding at least one of the second plurality of logical qubits via the plurality of ancilla qubits; performing at least one logical operation on the at least one of the second plurality of logical qubits; and teleporting the at least one of the second plurality of logical qubits into the at least one of the first plurality of qubits via the plurality of ancilla qubits.HQU-01425 HU 9476 36. A method of performing quantum error correction, comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits using a quantum low-density parity-check (qLDPC) code on the first plurality of physical qubits; and performing a plurality of quantum error correction (QEC) cycles on the first plurality of physical qubits, the plurality of QEC cycles comprising a plurality of intervals, wherein performing the plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals.

37. The method of claim 36, wherein applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph.

38. The method of claim 37, wherein performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered-statistical decoding (OSD).

39. The method of claim 38, wherein applying the final BP and OSD comprises constructing a decoding graph of data errors.

40. A quantum computing system, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a first homological product code, the first homological product code being a product of a first plurality of classical codes; and a second plurality of physical qubits, configured to encode a second plurality of logical qubits using a second homological product code, wherein the quantum computing system is configured to apply a first homomorphic gate to at least a subset of the first plurality of logical qubits and at least a subset of the second plurality of logical qubits.HQU-01425 HU 9476 41. The quantum computing system of claim 40, wherein the quantum computing system is further configured to measure a subset of the second plurality of physical qubits, thereby measuring the subset of the first plurality of logical qubits.

42. The quantum computing system of claim 40 or 41, wherein the second homological product code is determined by puncturing at least some of the first plurality of classical codes.

43. The quantum computing system of claim 40 or 41, wherein the second homological product code is determined by adding checks to at least some of the first plurality of classical codes.

44. The quantum computing system of claim 40 or 41, wherein the first homological product code and / or the second homological product code is a hypergraph product (HGP) code.

45. The quantum computing system of claim 40 or 41, further configured to perform a plurality of quantum error correction (QEC) cycles on the first plurality of physical qubits.

46. The quantum computing system of claim 45, wherein the plurality of QEC cycles comprises a plurality of intervals, and wherein performing the plurality of QEC cycles comprises applying belief propagation (BP) decoding on each of the plurality of intervals.

47. The quantum computing system of claim 46, wherein applying the belief propagation (BP) decoding comprises constructing a space-time circuit-level decoding graph.

48. The quantum computing system of claim 46, wherein performing the plurality of QEC cycles comprises applying a final belief-propagation (BP) and ordered-statistical decoding (OSD).

49. The quantum computing system of claim 48, wherein applying the final BP and OSD comprises constructing a decoding graph of data errors.HQU-01425 HU 9476 50. The quantum computing system of claim 40 or 41, wherein the first plurality of physical qubits comprises a first array of neutral atoms and the second plurality of physical qubits comprises a second array of neutral atoms, wherein each neutral atom has a first state and an excited Rydberg state, and each neutral atom is arranged to impose a Rydberg blockade on at least its nearest neighbors in its array when in the excited Rydberg state.

51. The quantum computing system of claim 50, wherein performing the first homomorphic gate comprises placing the first and second pluralities of physical qubits such that each physical qubit of the first plurality of physical qubits is within a blockade radius of exactly one corresponding physical qubit of the second plurality of physical qubits and illuminating the first and second plurality of physical qubits with a first laser.

52. The quantum computing system of claim 40 or 41, wherein the first plurality of physical qubits comprises a plurality of data qubits and a plurality of ancilla qubits, and wherein encoding the first plurality of logical qubits using the first homological product code comprises: providing the plurality of data qubits, each of the plurality of data qubits disposed in a corresponding trap; providing the plurality of ancilla qubits, each of the plurality of ancilla qubits disposed in a corresponding trap; arranging the plurality of data qubits and the plurality of ancilla qubits in a plurality of rows and a plurality of columns, thereby forming a lattice; performing a plurality of permutations of the plurality of rows and the plurality of columns, each of the plurality of permutations placing each of the plurality of data qubits within an interaction radius of one of the plurality of ancilla qubits, thereby forming a plurality of proximate pairs; andHQU-01425 HU 9476 subsequent to each of the plurality of permutations, applying a global control pulse to the lattice, thereby applying a gate to each of the plurality of proximate pairs, and thereby encoding a parity check matrix between the plurality of ancilla qubits and the plurality of data qubits.

53. The quantum computing system of claim 52, wherein encoding the first plurality of logical qubits using the first homological product code further comprises, prior to forming the lattice: applying a control laser pulse to the plurality of data qubits to thereby prepare them in an initial state.

54. The quantum computing system of claim 52 or 53, wherein arranging the plurality of data qubits and the plurality of ancilla qubits in the lattice comprises: moving, in parallel, the plurality of ancilla qubits into the lattice.

55. The quantum computing system of claim 52, wherein performing the plurality of permutations comprises: moving, in parallel, one or more rows within the lattice and moving, in parallel, one or more columns within the lattice.

56. The quantum computing system of claim 52, wherein the plurality of ancilla qubits comprises Z stabilizers and X stabilizers.

57. The quantum computing system of claim 52 or 56, wherein encoding the first plurality of logical qubits using the first homological product code further comprises: removing a subset of the plurality of ancilla qubits from the lattice and performing a measurement on the subset.

58. The quantum computing system of claim 57, wherein the subset corresponds to Z stabilizers.HQU-01425 HU 9476 59. The quantum computing system of claim 57, wherein the subset corresponds to X stabilizers.

60. The quantum computing system of claim 52, wherein performing the plurality of permutations comprises: determining a collision-free path for each of the plurality of data qubits and each of the plurality of ancilla qubits for each of the plurality of permutations.

61. The quantum computing system of claim 60, wherein determining the collision-free path comprises: bipartition and recursive sorting of the plurality of data qubits and the plurality of ancilla qubits.

62. The quantum computing system of claim 60, wherein the collision-free path is a cubic spline.

63. The quantum computing system of claim 60, wherein determining the collision-free path comprises: decomposing the parity check matrix into a first product graph and a second product graph; determining a routing of the plurality of rows according to the first product graph; and determining a routing of the plurality of columns according to the second product graph.

64. The quantum computing system of claim 60, wherein the plurality of permutations comprises row-specific and / or column-specific permutations.

65. The quantum computing system of claim 63, wherein performing the plurality of permutations comprises:HQU-01425 HU 9476 applying a pinning beam to at least one qubit of the plurality of data qubits or the plurality of ancilla qubits, thereby maintaining a position of the at least one qubit according to the routing of the plurality of rows or columns.

66. The quantum computing system of any one of claims 52-65, wherein the gate applied to each of the plurality of proximate pairs is a CZ gate.

67. The quantum computing system of any one of claims 52-66, wherein the global control pulse is a laser pulse.

68. The quantum computing system of any one of claims 52-67, wherein the trap corresponding to each of the plurality of data qubits and to each of the plurality of ancilla qubits is an optical trap.

69. The quantum computing system of claim 68, wherein the optical traps corresponding to the plurality of data qubits and to the plurality of ancilla qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), and moving the one or more rows and moving the one or more columns comprises varying a drive frequency of the at least one AOD.

70. The quantum computing system of claim 69, wherein moving the one or more rows and moving the one or more columns further comprises applying one or more rotations during said moving.

71. The quantum computing system of claim 70, wherein applying the one or more rotations comprises applying a Raman pulse.

72. The quantum computing system of claim 40 or 41, further comprising: a third plurality of physical qubits, configured to encode a third plurality of logical qubits using a third homological product code, wherein the first homomorphic gate is a CNOT,HQU-01425 HU 9476 the quantum computing system is configured to apply a second homomorphic gate to at least a subset of the second plurality of logical qubits and at least a subset of the third plurality of logical qubits, the second homomorphic gate being a CNOT, and the quantum computing system is configured to measure a subset the third plurality of physical qubits, thereby measuring the subset of the second plurality of logical qubits.

73. A quantum computing system, comprising: a first plurality of physical qubits, configured to encode a first plurality of logical qubits using a first homological product code, wherein the first homological product code has a quasi-cyclic classical base code, the quantum computing system is configured to perform a permutation of the first plurality of physical qubits, the first permutation corresponding to a block-cyclic shift of the classical base code, thereby performing a translation of the first plurality of logical qubits.

74. A method of performing quantum computation, the method comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits using a first homological product code on the first plurality of physical qubits; providing a second plurality of physical qubits; encoding a second plurality of logical qubits using a second homological product code on the second plurality of physical qubits; and applying a first homomorphic gate to at least a subset of the first plurality of logical qubits and at least a subset of the second plurality of logical qubits.HQU-01425 HU 9476 75. The method of claim 74, further comprising: measuring a subset of the second plurality of physical qubits, thereby measuring the subset of the first plurality of logical qubits.

76. The method of claim 74 or 75, wherein the first homomorphic gate is a CNOT, the method further comprising: providing a third plurality of physical qubits; encoding a third plurality of logical qubits on the third plurality of physical qubits using a third homological product code; applying a second homomorphic gate to at least a subset of the second plurality of logical qubits and at least a subset of the third plurality of logical qubits; and measuring a subset the third plurality of physical qubits, thereby measuring the subset of the second plurality of logical qubits.

77. A method of performing quantum computation, the method comprising: providing a first plurality of physical qubits; encoding a first plurality of logical qubits on the first plurality of physical qubits using a first homological product code, the first homological product code having a quasi- cyclic classical base code; and performing a permutation of the first plurality of physical qubits, the first permutation corresponding to a block-cyclic shift of the classical base code, thereby performing a translation of the first plurality of logical qubits.

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