Error-corrected quantum computation using transversal gates and correlated decoding

Abstract

Error-corrected quantum computation using transversal gates and correlated decoding is provided. A first and second logical qubit is encoded into physical qubits according to a quantum error correcting code. Based on the quantum error correcting code, a bipartite decoding graph is constructed corresponding to the first and the second logical qubits, the bipartite decoding graph comprising a plurality of detector nodes and a plurality of error nodes, each error node corresponding to an error mechanism. A transversal gate is applied to the first and the second logical qubits. Syndrome measurement of the first and the second logical qubits is performed. For each of the plurality of detector nodes affected by the corresponding error mechanism of one of the plurality of error nodes, an edge is generated on the bipartite decoding graph therebetween. A physical error configuration is determined from the bipartite decoding graph.

Description

HQU-01525 HU 9648 ERROR-CORRECTED QUANTUM COMPUTATION USING TRANSVERSAL GATES AND CORRELATED DECODING CROSS-REFERNCE TO RELATED APPLICATIONS

[0001] This Application claims the benefit of U.S. Provisional Application No. 63 / 604,732, filed November 30, 2023, which is hereby incorporated by reference in its entirety. BACKGROUND

[0002] Embodiments of the present disclosure relate to quantum computation, and more specifically, to error-corrected quantum computation using transversal gates and correlated decoding. BRIEF SUMMARY

[0003] According to embodiments of the present disclosure, methods of performing quantum computation are provided. At least a first plurality of physical qubits is provided. A first logical qubit is encoded into the at least first plurality of physical qubits according to a quantum error correcting code. A second logical qubit is encoded into the at least first plurality of physical qubits according to the quantum error correcting code. Based on the quantum error correcting code, a bipartite decoding graph is constructed corresponding to the first and the second logical qubits, the bipartite decoding graph comprising a plurality of detector nodes and a plurality of error nodes, each error node corresponding to an error mechanism. A transversal gate is applied to the first and the second logical qubits. A first round of syndrome measurement of the first and the second logical qubits is performed. For each of the plurality of detector nodes affected by the corresponding error mechanism of one of the plurality of error nodes, an edge is generated on the bipartite decoding graph Page 1 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 therebetween. A physical error configuration is determined from the bipartite decoding graph.

[0004] In some embodiments, at least one gate is applied to the first plurality of physical qubits and the second plurality of physical qubits to correct the physical error configuration.

[0005] In some embodiments, the quantum error correcting code is a surface code.

[0006] In some embodiments, the transversal gate is a Clifford gate. In some embodiments, the Clifford gate is a CNOT gate.

[0007] In some embodiments, a third logical qubit is encoded into the at least first plurality of physical qubits according to the quantum error correcting code. In some embodiments, the transversal gate is a non-Clifford gate. In some embodiments, the non-Clifford gate is a CCZ gate.

[0008] In some embodiments, determining the physical error configuration comprises maximizing an error probability on the bipartite decoding graph. In some embodiments, maximizing the error probability comprises solving a mixed-integer programming problem corresponding to the error probability.

[0009] In some embodiments, determining the physical error configuration comprises determining one or more subgraphs of the bipartite decoding graph corresponding to the physical error configuration. In some embodiments, determining the one or more subgraphs comprises defining a subgraph for each detector node having a detected error and expanding each such subgraph until it encompasses error nodes, which, if they had occurred, would result in syndrome measurements consistent with the observed syndrome.

[0010] In some embodiments, each qubit of the first plurality of physical qubits and the second plurality of physical qubits is a neutral atom.

[0011] In some embodiments, the at least first plurality of physical qubits comprises a second plurality of physical qubits, and the first logical qubit is encoded in the first plurality of Page 2 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 physical qubits and the second logical qubit is encoded in the second plurality of physical qubits.

[0012] In some embodiments, the at least first plurality of physical qubits comprises a second plurality of physical qubits and a third plurality of physical qubits, and the first logical qubit is encoded in the first plurality of physical qubits, the second logical qubit is encoded in the second plurality of physical qubits, and the third logical qubit is encoded in the third plurality of physical qubits.

[0013] In some embodiments, applying the transversal 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.

[0014] In some embodiments, one or more additional transversal gates is applied to the first and the second logical qubits alternately with applying one or more additional rounds of syndrome measurement of the first and the second logical qubits.

[0015] According to embodiments of the present disclosure, quantum processors are provided. A first array of optical traps is disposed in an active zone. A second array of optical traps is disposed in a readout zone. A first laser is configured to illuminate the active zone and to drive a transition to a Rydberg state. A second laser is configured to illuminate the active zone and to drive a transition between hyperfine states. A third laser is configured to illuminate the readout zone. A fourth laser is configured to adiabatically move neutral atoms between the optical traps of the active zone and the readout zone. A camera is configured to capture an image of the readout zone. The quantum processor is configured to: provide at least a first plurality of neutral atoms in the active zone, each in a respective optical trap of the first array; encode a first logical qubit into the at least first plurality of Page 3 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 neutral atoms according to a quantum error correcting code by the first and second lasers; encode a second logical qubit into the at least first plurality of neutral atoms according to the quantum error correcting code by the first and second lasers; based on the quantum error correcting code, constructing a bipartite decoding graph corresponding to the first and the second logical qubits, the bipartite decoding graph comprising a plurality of detector nodes and a plurality of error nodes, each error node corresponding to an error mechanism; place the first and second pluralities of neutral atoms in the active zone such that each neutral atom of the first plurality of neutral atoms is within a blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms; illuminate the first plurality of neutral atoms and the second plurality of neutral atoms while in the active zone by at least the first or second laser, thereby applying a transversal gate to the first and second logical qubits; performing a first round of syndrome measurement of the first and the second logical qubits; for each of the plurality of detector nodes affected by the corresponding error mechanism of one of the plurality of error nodes, generating an edge on the bipartite decoding graph therebetween; and determining a physical error configuration from the bipartite decoding graph. BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0016] Fig. 1 is a schematic view of a quantum information architecture according to embodiments of the present disclosure.

[0017] Fig. 2 is a level diagram showing key87Rb atomic levels according to embodiments of the present disclosure.

[0018] Fig. 3 is a schematic view of the implementation of the toric code according to embodiments of the present disclosure. Page 4 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0019] Fig. 4 is a schematic view of a quantum processing unit (QPU) according to embodiments of the present disclosure.

[0020] Fig. 5 is a schematic view of logical qubits, illustrating efficient control according to embodiments of the present disclosure.

[0021] Fig. 6 is a schematic view of logical qubits, illustrating the application of a transversal CNOT gate according to embodiments of the present disclosure.

[0022] Fig. 7 is a schematic view of a portion of a processor core according to embodiments of the present disclosure.

[0023] Fig. 8 is a schematic view of a portion of a processor core suitable for use in implementing a repetition code according to embodiments of the present disclosure.

[0024] Fig. 9 is a schematic view of a portion of a processor core suitable for use in implementing a surface code according to embodiments of the present disclosure.

[0025] Fig. 10 is a schematic view of a method of active, feedforward QEC according to embodiments of the present disclosure.

[0026] Fig. 11 is a schematic view of an apparatus for quantum computation according to embodiments of the present disclosure.

[0027] Fig. 12 is an illustration of logical operation time for lattice surgery and transversal CNOTs according to embodiments of the present disclosure.

[0028] Figs. 13A-C are illustrations of error propagation in a transversal CNOT and the resulting detector error model according to embodiments of the present disclosure.

[0029] Figs. 14A-F illustrate the reduction in spacetime cost of logical algorithms according to embodiments of the present disclosure.

[0030] Figs. 15A-B illustrate a logical S gate according to embodiments of the present disclosure. Page 5 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0031] Fig. 16 illustrates a magic state distillation circuit according to embodiments of the present disclosure.

[0032] Figs. 17A-F illustrate an exemplary neutral atom layout for magic state distillation according to embodiments of the present disclosure.

[0033] Figs. 18A-B illustrate circuits for the ripple-carry adder and QROM according to embodiments of the present disclosure. DETAILED DESCRIPTION

[0034] Large-scale quantum computers have the potential to solve problems that are intractable for classical processors. While exciting progress has been made on developing small- and medium-scale quantum processors and applying them to study physical phenomena in complex quantum systems in a regime that is difficult to simulate classically, it is unclear if and how truly large scale quantum processors can be constructed and applied to solving general purpose, computationally hard problems, whose value exceeds the cost of construction. For example, current estimates for the resources required to realize one of the most prominent high value applications, Shor’s factoring algorithm, require around five thousand logical qubits, and a few billion non-Clifford gates with error probability below 10-12to break 2048-bit RSA encryption. Alternatively, using conventional error correction methods, 20 million superconducting qubits with realistic gate errors (0.1%) could be used, which exceeds the scale of currently available, well-controlled systems by nearly six orders of magnitude.

[0035] This necessitates the development of novel, unconventional approaches to utility-scale quantum computation, likely involving a synergistic combination of new hardware architectures that significantly reduce the costs of error correction and computation, new resource-efficient approaches to algorithm development co-designed with the hardware, as Page 6 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 well as large scale engineering efforts to construct practical systems. Furthermore, any such large-scale development will also require extensive efforts to verify and validate.

[0036] To address these and other shortcomings of alternative approaches, the present disclosure provides systems and methods for reducing overhead in error-corrected quantum computing using transversal gates and correlated decoding.

[0037] In particular, the present disclosure describes how an error correction architecture based on transversal gates can lead to substantial reductions in the space-time volume of fault-tolerant quantum computation, from ^^^^^in alternative lattice-surgery-based schemes, to potentially ^^^^^ in a transversal-gate-based architecture. Algorithmic building blocks are described, with a particular focus on joint, correlated decoding across multiple logical qubits that allows the use of fewer than ^ rounds of quantum error correction per transversal gate. An exemplary implementation of this architecture in neutral atom array systems, one of the leading fault-tolerant quantum computing architectures, is also provided.

[0038] 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.

[0039] 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’.

[0040] The term qudit (quantum digit) denotes the unit of quantum information that can be realized in suitable ^-level quantum systems. A collection of qubits that can be measured to ^ states can implement an ^-level qudit. Page 7 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0041] 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 to different orientations of electron spins, nuclear spins, electron orbits, and molecular rotations / vibrations.

[0042] 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. Quantum mechanically, the same error may occur: |0^ may randomly flip to |1^ after some characteristic 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.

[0043] 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 superposition state (|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- Page 8 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0044] 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.

[0045] 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. Page 9 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0046] An alternative approach is based on stimulated Raman transitions. In this case, a laser field 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. 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.

[0047] 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.

[0048] 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 (CNOT) 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. Page 10 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0049] 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, or trapping ions individually with local electric fields with a more complex on-chip electrode structure.

[0050] 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.

[0051] 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.

[0052] 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). Page 11 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0053] 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.

[0054] 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. An example of high fidelity Rydberg gates is 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, which is hereby incorporated by reference.

[0055] 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 Page 12 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0056] 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.

[0057] 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).

[0058] 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 Page 13 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 the acceleration of the subject atom is not greater than a predetermined value, the movementis considered adiabatic. Typically, adiabatic movement occurs when ^^^^ <^^^^^ ^^ ^^^^^ × ^^^^^ ^^^^^^^^ ^^. In physics, jerk or jolt is the term given to the rateat which an object's acceleration changes with respect to time.

[0059] 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.

[0060] 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.

[0061] 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.

[0062] Current fault-tolerant quantum computing architectures are primarily based on surface codes and lattice surgery, due to the convenience of such operations to be laid out in a 2D- local planar architecture. However, neutral atom arrays, trapped ions, silicon quantum dots, photonics, and other systems may allow non-local connectivity, opening up new architectural possibilities. This application describes how transversal gates in such an architecture can lead to substantial reductions in the space-time volume of fault-tolerant quantum computation. Page 14 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0063] Conventional lattice surgery schemes with 2D topological codes require ^^^^^space per logical qubit, and each logical operation typically requires ^^^^time to be fault-tolerant. Here, ^ is the code distance, characterizing the error correcting capability of a given code. For example, both a CNOT gate and an S gate in the surface code require this ^^^^time cost when implemented with lattice surgery. On the other hand, transversal gates, such as the transversal CNOT in CSS codes, only require a single time step yet are still fault-tolerant. This suggests a reduction of the time cost of running a logical operation from ^^^^ to ^^1^, and a corresponding space-time overhead reduction of ^^^^^to ^^^^^. In the deep fault-tolerant regime, code distances on the order of ^ ∼ 30 are expected to obtain sufficient errorsuppression, which means that this method can give rise to a 30x reduction in required resources. In the present application, various building blocks are provided that support and enable this architecture. This includes: • Reduction of error-correction overhead from ^^^^^to ^^^^^by using transversal gates. • Correlated decoding techniques to enable this in the context of transversal algorithm execution. • The use of such techniques in key algorithmic gadgets such as magic state factories and quantum arithmetic. • The direct logical qubit connectivity enabled by this approach, which can reduce routing overhead. • The hardware-efficient implementation of this architecture in neutral atoms.

[0064] Referring to Fig. 1, a quantum information architecture enabled by coherent transport of neutral atoms is illustrated. Qubits are transported to perform entangling gates with distant Page 15 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 qubits, enabling programmable and nonlocal connectivity. Atom shuttling is performed using optical tweezers, with high parallelism in two dimensions and between multiple zonesallowing selective manipulations. The inset shows the atomic levels used: the |0^, |1^ qubitstates refer to the ^% = 0 clock states of 87Rb, and |^^ is a Rydberg state used for generatingentanglement between qubits, which are further described with regard to Fig. 2.

[0065] Fig. 2 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.

[0066] 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.

[0067] 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.

[0068] 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 Page 16 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 by two waveforms in a 2D acousto-optic deflector (AOD), and qubit operations are realized 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.

[0069] 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.

[0070] Fig. 3 illustrates implementation of the toric code state encoding two protected qubits obtained using mobile ancilla qubit arrays. The top illustrates a graph state realizing the two logical-qubit product stateof the toric code upon projective measurement of the ancilla qubits in the X-basis. The bottom includes images showing the movement steps implemented in creating and measuring the toric code state. Shading in the final image represents a local rotation on the data qubit zone.

[0071] Referring to Fig. 4, 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.

[0072] 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 Page 17 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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-scale 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.

[0073] QPU 400 is segmented into several key zones: a storage zone 411, entangling zone 412, readout zone 413, atom loading zone 404, and remote entangling zone 405. Storage zone 411, entangling zone 412, and readout zone 413 form processor core 401, which in some embodiments contains 104to 106qubits in a footprint of 0.5-5mm. Fresh atoms are continuously reloaded from distant atom loading zone 404, and a distant remote entangling zone 405 (using optical interconnects and / or lattice transport) delivers remote Bell pair entanglement resources.

[0074] In storage zone 411, 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 seconds, 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. Page 18 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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 approximately 1 / ^21^^^in the dense storage zone, and densities of approximately 1 / ^101^^^in the active zone.

[0075] The active logical qubits are manipulated in active zone 412. 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 calibrating 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.

[0076] Readout zone 413 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.

[0077] Outside of the core processor 401, atoms are constantly reloaded from loading zone 404 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 405, and are shuttled into the core processor 401 for creating remote logical entanglement. This allows interconnection of 10-100 single processing units into one error- corrected, utility-scale quantum computer.

[0078] 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. Page 19 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0079] To connect multiple units, many high-fidelity, long-distance Bell pairs may be generated in parallel, using lattice transport and / or photonic links.

[0080] 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 by the ancilla atoms.

[0081] 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.

[0082] 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 Page 20 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0083] It will be appreciated that while certain operating parameters are provided below by way of example, increased fidelity in two-qubit gate errors may be achieved through various 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.

[0084] 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.

[0085] In various embodiments, cross-talk during readout is suppressed by moving the ancilla atoms away from the data qubits.

[0086] 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 Page 21 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0087] Referring to Fig. 5, a schematic view is provided of logical qubits, illustrating efficient control of single logical qubits by parallelized optical control of the physical qubit blocks that constitute the logical qubits. Logical qubits 501, 502, 503, 504 are each made up of 13 atomic qubits (shown as circles). It will be appreciated that the number and arrangement of qubits is purely exemplary. A variety of qubit blocks are known in the art and are suitable for use as described herein. For example, the 2D surface code and the 2D color code are particularly suitable due to their high thresholds and simplistic 2D structure. However, a variety of other codes are available that include the transversal CNOT, such as the 3D color code and 3D toric code. In various embodiments, a single laser beam is configured to illuminate a given logical qubit when positioned in an active zone of a processor (e.g., active zone 412). This is illustrated by beam 511 illuminating logical qubit 501. In some embodiments, a single laser beam is configured to illuminate a plurality of logical qubits when positioned in an active zone of a processor (e.g., active zone 412). This is illustrated by beam 513 illuminating logical qubits 503, 504, 505.

[0088] This structure allows for the application of transversal logical gates. To apply a transversal logical gate on one logical qubit, the corresponding physical qubit gate is performed on each physical qubit in the block making up the logical qubit.

[0089] For example, to do a transversal single-qubit gate, the same single-qubit rotation is applied to each physical qubit in the block by illuminating that entire spatial block (e.g., 501) with one beam that covers all physical qubits (e.g., 511). In various embodiments, this is realized by creating a grid of Raman beams using a crossed AOD device in order to Page 22 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 illuminate a grid of one surface code. This single-qubit example is shown with beams 511, 512, where surface code blocks 501, 502 (the connected grid of 13 atoms) are illuminated with beams that come out of a microscope objective and into the plane of the atoms. In this example, two logical blocks 511, 512 are illuminated in parallel. This is advantageous in various use cases, but one code block may also be illuminated at a time.

[0090] This structure allows for the application of transversal logical multi-qubit gates between two or more logical qubits. A transversal logical multi-qubit gate is a logical gate wherein each atomic (i.e., physical) qubit of one logical qubit is coupled to only one atomic qubit of another logical qubit, and therefore errors do not spread to other atomic qubits by propagation. Referring to Fig. 6, a schematic view is provided of logical qubits, illustrating the application of a transversal controlled-NOT (CNOT) gate. As in Fig. 5, each of a plurality of logical qubits is made up of 13 atomic qubits (shown as circles). To perform a transversal CNOT on two logical qubits (e.g., 601, 602) a physical qubit CNOT is performed on each pair of the two logical blocks. The architectures provided herein enable moving groups of atoms in parallel in order to efficiently perform logical transversal CNOTs between any two logical qubit blocks.

[0091] In particular, a logical qubit block is picked up with a crossed AOD, moved to interlace with another logical qubit within the same 2D plane, which is stored in a different set of optical tweezers (e.g., a backbone spatial light modulator (SLM) grid). When interlaced, each atomic qubit of one logical qubit is within a blockade radius of exactly one corresponding atomic qubit of the other logical qubit. A single pulse of a global Rydberg laser is applied (e.g., beam 611). This realizes a transversal CNOT between the two logical qubits in a single, parallel step. Transversal CNOTs may be performed in parallel on multiple logical qubits at the same time, as is shown in Fig. 6. Page 23 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0092] Transversal CNOTs are allowed, fault tolerant operations, between any two Calderbank-Shor-Steane (CSS) codes, which is a broad class of codes encompassing surface codes, color codes (e.g., Steane code), hypergraph product low density parity check (LDPC) codes, etc. The key intuition is that a CNOT propagates X on a first qubit to X on a second qubit, and Z on the first qubit to Z on the second qubit. For a CSS code, the logical qubit operators are products of X and Z, and the logical CNOT is formed of products of physical qubit CNOTs. Accordingly, logical X on the first logical qubit will propagate to logical X on the second logical qubit, and logical Z on the first logical qubit will propagate to logical Z on the second logical qubit. This follows the rules of a CNOT on the logical qubit level. Accordingly, this implements a transversal logical CNOT.

[0093] More particularly, a controlled-NOT can be performed bitwise on any CSS code. Consider the operations on 2^4 and 4^2. In the first case, if 2 is an 5 generator, it becomes 2^2. Since both the first and second blocks have the same stabilizer, this is anelement of 6 × 6. If 2 is a 7 generator, 2^4 becomes 2^4 again. Similarly, if 2 is an 5generator, 4^2 becomes 4^2, and if 2 is a 7 generator, 4^2 becomes 2^2, which isagain in 6 × 6. For an arbitrary CSS code, the 598 operators are formed from the product ofall 5s, and the 798 operators are formed from the product of all 7s. Therefore:Equation 1 Page 24 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0094] Thus, the bitwise CNOT produces an encoded CNOT for every encoded qubit in the block.

[0095] Without a transversal CNOT, entangling operations between logical qubits often have to be done with braiding or lattice surgery. These are significantly less efficient than the transversal CNOT. For example, both braiding and lattice surgery require doing ^ rounds of stabilizer measurement in order for them to be actually fault-tolerant, whereas no rounds of stabilizer measurement are required to make the transversal CNOT fault-tolerant — its fault- tolerance is already guaranteed by the fact that that the CNOT is transversal. Transversal gates are inherently fault-tolerant, because, as described above, no error can spread from one qubit in the block to another qubit in the same block. Braiding and lattice surgery thus are much more resource intensive in requiring multiple rounds of stabilizer measurement, being slower and also being lower in their threshold.

[0096] The threshold of a two-qubit (2Q) gate when doing a transversal CNOT is given by the threshold for perfect syndrome extraction and such should be roughly 10% (surface code), whereas the 2Q gate threshold for doing repeated syndrome extraction is roughly 1% (surface code). The ability to perform this transversal CNOT efficiently by only using a few classical controls, in a way that is independent of code size, is key to simplifying the classical controls required for building a large-scale quantum computer.

[0097] The atom movement and parallel optical control of logical qubit blocks greatly simplifies the controls required for realizing logical quantum computation. In various embodiments, logical qubits are multiplexed into grids, and each logical qubit block behaves much like one large atom. To perform a single-qubit rotation on a logical qubit block, it is illuminated with one beam. To perform a CNOT between two logical qubit blocks, they are moved together and pulsed with one Rydberg beam. With this highly efficient parallelized control, logical qubit algorithms can be performed on logical qubits. Page 25 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0098] In exemplary embodiments, a two-qubit CZ gate is implemented by two global Rydberg pulses, with each pulse at detuning Δ and length τ, and with a phase jump ξ 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.

[0099] Referring to Fig. 7, a schematic view of a portion of a processor core, such as processor core 401, is provided with exemplary measurements and qubit arrangements. Inthis example, storage zone portion 701 measures 145 × 401^, active zone portion 702measures 145 × 401^, and readout zone portion 703 measures 145 ×Storage zoneportion 701 is separated from active zone portion 702 by a 201^ buffer. Active zone portion 702 is separated from readout zone portion 703 by a 201^ buffer.

[0100] In this example, active zone portion 702 has 50 positions, separated by 161^ in one dimension and 101^ in the other dimension. Each dot represents an atomic qubit, and so, in this example, qubits are located proximate to each other when interlaced to perform a bitwise operation. Storage zone portion 701 has 250 positions, separated byin one dimension and 41^ in the other dimension. Accordingly, storage zone portion 701 has a higher density than active zone portion 702.

[0101] Alternative arrangements are provided in Figs. 8-9. For example, the arrangement in Fig. 8 is suitable for use in implementing a repetition code. In another example, the arrangement in Fig. 9 is suitable for use in implementing a surface code.

[0102] Referring to Fig. 10, a schematic view of a method of active, feedforward QEC is provided. In this example, a portion of a quantum processor (such as depicted in Fig. 4) is depicted, including a reservoir 1001 (such as loading zone 404), an active zone 1002 (such as active zone 412), and a readout zone 1003 (such as readout zone 411). Ancilla qubits are continually replenished 1004 from reservoir 1001 to replace ancilla qubits that are moved Page 26 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 1005 from active zone 1002 to readout zone 1003 to be measured. Using the zoned architecture provided herein enables a complete QEC round within 1ms.

[0103] Transversal Logical Gates

[0104] Transversal logical gates, which act by applying the same gate on each individual physical qubit of a single code patch or between multiple code patches, are widely regarded as the simplest and best-performing technique to achieve fault-tolerant entangling gates. This is because by acting only on individual physical qubits within a code patch, they do not spread errors within each patch and are inherently fault tolerant. Moreover, unlike lattice surgery, which often requires ^ rounds of repetition in time to achieve fault-tolerance, transversal CNOTs only require a single time step (see Fig. 12). This reduces the number of physical gates applied, potentially reducing the number of errors incurred. Thus, the use of transversal CNOTs has the potential to both reduce the space-time resources required and the logical error rate for FTQC.

[0105] Referring to Fig. 12, an illustration is provided of logical operation time for lattice surgery and transversal CNOTs. The latter does not require ^ rounds of repetition to be fault- tolerant.

[0106] There is little detailed analysis of transversal CNOT gate performance, primarily because of implementation challenges in conventional 2D planar architectures. However, breakthroughs in neutral atom array and trapped ion technologies have enabled the possibility of long-range connectivity via dynamic reconfigurability, making it possible to directly bring two code patches together and execute a transversal CNOT gate.

[0107] Of particular interest are dynamically-reconfigurable neutral atom arrays, where qubits are encoded in long-lived hyperfine states with second-long coherence times, and entangling gate operations are mediated by transient excitation into strongly-interacting Page 27 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 Rydberg states. Parallel shuttling of atoms using acousto-optic deflectors (AODs) then allows the efficient reconfiguration of atom locations. Qubit numbers as large as 289 have been experimentally demonstrated, and can be readily scaled to the thousands by increasing laser power. Using the dynamical reconfigurability of neutral atoms, a variety of quantum error correction codes are possible, including the toric code on a torus.

[0108] A key feature of this platform is its high degree of parallelism and low control overhead, which is naturally suited to quantum error correction: two control wires are sufficient to program a pair of AODs and generate a large grid of traps, allowing the manipulation of an entire logical qubit consisting of hundreds of physical qubits with only a handful of control channels. With this approach, performing a transversal CNOT is as simple as interleaving two code patches using parallel AOD control and shining a global Rydberg beam that performs a physical CNOT between each pair of qubits in the two code patches. This highly resource efficient implementation of the transversal CNOT may significantly simplify the realization of FTQC, and reduce the overhead of quantum algorithms.

[0109] These considerations enable reducing the unit space-time cost of logical operations from ^^^^^in lattice surgery, to ^^^^^with transversal gates, a factor that can be on theorder of 30 for typical code distances ^ ≈ 30 assumed in the fault-tolerant regime.

[0110] However, a key question is how to appropriately decode such a circuit to maintain the full code distance and have good logical error performance. The correlated decoding methods described below, which jointly decode multiple logical qubits in a quantum algorithm, allow minimal degradation in the threshold and obtain promising logical error rates even when there are a relatively small number of rounds of error correction (much less than ^) between each pair of transversal operations, thus fulfilling the promise of large overhead reductions. Page 28 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0111] Heuristic Scaling Analysis

[0112] In this section, a heuristic estimate is provided of the expected logical error rate as a function of the number of CNOTs per syndrome extraction round. This heuristic estimate depends on several assumptions, but provides some intuition for the expected performance and overhead reduction of the architectures provided herein.

[0113] For a regular surface code memory, the logical error rate per round of syndrome extraction can be well-approximated by the scaling formulaEquation 2 where ^ is the code distance (assumed to be odd for simplicity), ^@Ais the error threshold, ^ is the physical error rate, and > is a constant.

[0114] Separating this expression in terms of underlying CNOT gate error rates ^EFGHand other operation error rates ^Iallows rewriting the above expression asEquation 3

[0115] The number 4 reflects the fact that 4 CNOTs are performed on each data qubit in one round of syndrome extraction, and the prefactors J and K are scaling factors that can be obtained from numerical simulations.

[0116] This analysis is now generalized to the case of transversal CNOTs with interleaved rounds of syndrome measurement. Assuming L CNOTs are performed per syndrome measurement round, the following ansatz is used for the logical error per logical qubit per CNOT:Page 29 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 Equation 4

[0117] The factor N is added to account for the extra error due to the decoding problem being a bit more complex. Note that the constants may be modified from the memory setting, and some of the coefficient dependencies may be more complicated than are described here. However, as a heuristic estimate, this is a good starting point. This formula can be understood as follows: the rate of syndrome measurements sets how often one is extracting entropy out of the system, so one may expect to have a comparable threshold but an elevated effective physical error rate, as captured by the NL term. As one performs L CNOTs per round of syndrome measurement, one divides L outside to estimate the logical error per logical qubit per CNOT. The spacetime cost per CNOT can then be estimated to beEquation 5

[0118] The alternative method of fault-tolerance, which involves a full ^ rounds of syndromemeasurements for each transversal gate, corresponds to the case where L = .C.

[0119] Having established a heuristic scaling formula, the following questions may be posed: for a given target logical error rate, what is the smallest spacetime cost per CNOT? How many rounds of QEC does this correspond to? If the optimal number of rounds of QEC is less than ^, then this would provide an indication that one can indeed substantially reduce the resource overhead required.

[0120] One can perform the above optimization problem by solving for the code distance that provides sufficient error suppression for a given set of parameters, and then calculate thespacetime cost. For a particular choice of J = 1 and K = 12, motivated by counting of errorchannels in some error models, one finds the following condition for the optimal number of CNOTs per round: Page 30 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648Equation 6

[0121] This analysis will also hold for other, more accurate parameter choices. To simplify the above expression, the following definitions are madeEquation 8

[0122] Although these are in principle functions of L, since they are inside a logarithm, they may be treated as roughly constant. One then hasEquation 9

[0123] To understand the asymptotics, it is noted that increasing the logical error requirements will scale ^, so one wants to match the linear coefficients of that. Thus, the condition is observed that asymptotically,Equation 10 Page 31 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0124] For fixed error rates, this is just a fixed equation for L that does not scale with the code distance or target logical error rate, indicating that performing a constant number of CNOTs per round of error correction minimizes the spacetime volume. Thus, this heuristic analysis provides evidence for ^^^^^spacetime overhead with the use of transversal gates. It also suggests that as one lowers the physical error rates, one can do more transversal CNOTs per round of syndrome extraction.

[0125] Referring to Figs. 13A-C, an illustration is provided of error propagation in a transversal CNOT, and the resulting detector error model, which has weight 3 hyperedges that make decoding more challenging. Fig. 13C illustrates the expansion process in the generalized union-find decoder described herein.

[0126] Correlated Decoding to Enable Low-Overhead Algorithm Execution

[0127] One challenge of applying and decoding transversal gates is error propagation between code patches. As illustrated in Fig. 13A, an 5 error on the control code patch is propagated to a pair of 5 errors on both patches through the CNOT, while a 7 error on the target patch propagates to a pair of 7 errors on both patches. Consequently, this leads to additional hyperedges in the decoding graph (also known as the detector error model), which can be challenging to decode. Indeed, a naïve decoding strategy that independently performs matching on the two code patches is not expected to yield a threshold, because many accumulated errors on one side may be copied over, causing the decoder to fail.

[0128] Although this error propagation increases the density of errors, crucially, it happens in a deterministic fashion set by the logical gates applied. Thus, by decoding qubits in a way that accounts for physical error propagation in the specific implemented algorithm, the effects of such spreading could possibly be reduced or even utilized, as physical errors on a given Page 32 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 logical qubit contain information about which physical errors occurred on other logical qubits.

[0129] To understand the challenges of decoding in the presence of transversal gates better, consider the detector error model in more detail, which describes how individual error events trigger detectors (checks), which are products of stabilizer measurement results that are deterministic in the absence of errors. For example, Fig. 13B illustrates an 5 error before the transversal CNOT, which creates two 5 errors across the two code patches and triggers 4 detectors. As another example, Fig. 13C illustrates the pattern of detectors triggered by a single ancilla measurement error on the control patch, which triggers weight 3 errors. The weight 3 errors are particularly challenging to deal with, as they are odd weight and cause the error decomposition heuristics commonly employed by software packages such as Stim to fail. This is because the heuristics try to decompose errors into lower weight existing errors, but, in the bulk of the detector error model, there are no weight-1 errors (all errors in the bulk produce pairs of syndromes), so there is no clear way to perform the decomposition.

[0130] To address this challenge, several custom decoders are provided that differ substantially from alternative minimum-weight perfect-matching decoders employed in various surface code simulations and experiments. This discussion focuses on two decoders, one based on phrasing the decoding problem as an optimization problem, thereby allowing use of mature integer programming solvers such as Gurobi to perform decoding, and the other based on a generalization of union-find. A few additional decoders are also considered, including a decoding strategy that first performs matching of 5 errors on the control patch and then updates the detectors on the target patch, as well as a strategy based on belief propagation + ordered statistics decoding, a decoder for quantum low-density parity-check (qLDPC) codes. However, these additional decoders each suffered from some downsides so far; the first strategy only works when there are no loops of CNOTs with at most d gates in Page 33 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 between, while the latter strategy ran too slowly for the particular open-source implementation used.

[0131] The decoders are now described in more detail. The first algorithm uses a state-of- the-art mixed-integer programming solver to exactly or approximately solve for the most- likely error consistent with the measured syndrome. The goal of a mixed-integer programming problem is to maximize an objective function, subject to constraints. The objective function being optimized encodes the total probability of a candidate set of physical error mechanisms occurring in the logical circuit, and the constraints will ensure that the candidate error set is consistent with the measured syndrome. Concretely, each physical error source XYin the circuit is associated with a binary variable that is equal to one if that error occurred, and zero otherwise. Each error source XYoccurs with some known probability ^Ydetermined by the circuit noise model. The goal will be to find an assignment of errorvariables such that the resulting error probabilityis maximized, subjectto the constraint that the error is consistent with the measured syndrome. To be consistent with the measured syndrome, the parity of the detector must match the parity of the errors connected to that detector by a hyperedge in the decoding graph. Concretely, let ^ be a map from each detectoràto the subset of error mechanisms that flip its parity, the most-likely error is given by the solution to the following mixed-integer program: maximize ∑cYd. Q^R]^Y^XY + Q^R]1 − ^Y^]1 − XY^subject to ∑[\∈i^jk^ XY − 2ea = à ∀^ = 1, … , ^XY ∈ l0, 1m ∀^ = 1, … , 2ea ∈ ℤop ∀^ = 1, … , ^

[0132] Each variable eaensures that the parity of the error variables associated withàmatchesà. One can verify that the objective function evaluates to the logarithm of the Page 34 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 probability of the assigned error configuration. One may then solve the mixed-integer program to optimality using Gurobi, a state-of-the-art solver, and apply the correction stringassociated with the error indices ^ for which XY = 1 in the optimal assignment. For worst-case circuits, this algorithm can have an exponential runtime. This is consistent with the fact that finding the most likely correlated error is NP-hard and thus unlikely to admit a polynomial time algorithm. Nevertheless, in many practical cases, the algorithm still has a very fast run time.

[0133] For the generalized union-find (UF) decoder, the termination condition is replaced by a linear-system-of-equations condition. The UF decoder proceeds in two steps: first, it starts from existing syndromes and attempts to grow an envelope of errors that contains an error configuration consistent with the observed syndromes; ideally, the resulting envelope is separated into multiple disjoint clusters. Second, it applies the peeling decoder to each of the clusters and finds a correction that is consistent with the observed syndrome. A key component of this decoder is the termination condition; for the UF decoder on the surface code, this is typically chosen to be that the total error weight is even since errors in the surface code generate pairs of detector events. However, for transversal CNOTs, the presence of high-weight errors means that this condition must be generalized. In the approached provided herein, this condition is replaced by the satisfiability of a linear system of equations, which allows direct checking of whether a satisfied solution exists for the current cluster. An additional benefit of this approach is that solving the linear system of equations automatically provides the corrections to be applied in the second step of the algorithm.

[0134] The generalized union-find decoder expands clusters on decoding graphs. The decoding graph is a bipartite graph whose vertices are divided into two types: detector nodes and error nodes. Each detector event corresponds one-to-one to a detector node in the Page 35 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 decoding graph, and each weight-^ error event corresponds one-to one to a degree-^ error node in the decoding graph. Each edge in the decoding graph indicates that the corresponding detector event can be triggered by the corresponding error event. An edge is weighted according to the error rate of its connecting error event. Clusters are defined as subsets of vertices in the decoding graph. Given a sample of detector events, a cluster is called satisfiable if the inner error nodes in this cluster give an error configuration consistent with the given detector events in the cluster. Otherwise, it is called unsatisfiable. The satisfiability of a cluster can be checked by a linear system solver over the binary field.

[0135] The generalized union-find decoder initializes clusters to individual subsets of vertices, each of which contains only one triggered detector event. When there exist any unsatisfiable clusters, one of the minimal sizes is chosen and it is expanded by adding one boundary node into the cluster according to the weight of the boundary edge. If the boundary node belongs to another cluster, these two clusters are merged into a larger cluster. Once all clusters are satisfiable, a decoded error configuration is obtained.

[0136] Numerical experiments are conducted to demonstrate the performance of the decoders on the transversal CNOT decoding problem. The logical error rate is estimated at different physical error rates in the circuit. First, the generalized union-find decoder is simulated. The circuit consists of only one transversal CNOT, along withC^ rounds of syndromemeasurement before and after it. The weight of an edge is set to be T = Q^R q r._rswhere ^ is the error rate of the connecting error and u is a hyperparameter of the decoder.Different u values give different thresholds: when u = −1^0^, the threshold is around0.51%^0.76%^. This difference originates from the different behavior of the cluster expansion. A cluster has a higher priority to expand through high-rank errors at lower u, making it easier to become larger. A larger cluster makes it more likely for the decoding to Page 36 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 fail because a global solution is less likely than a local solution to provide a high-weight solution in the decoding graph.

[0137] The performance of these decoders is also studied in the full context of a quantum circuit involving multiple operations, as shown in Fig. 14. By studying the number of rounds of error correction that are performed following each CNOT gate, one finds that the logical error rate appears to be minimized for the integer programming decoder with a constant number of rounds following each CNOT, suggesting that lower spacetime volume is indeed attainable. The generalized union find decoder also has an optimal number of rounds that is smaller than the code distance ^, although further optimizations and use of belief-propagation pre-processing are needed to further improve the performance.

[0138] The present disclosure demonstrates that these decoders substantially improve the performance of algorithm decoding, and serve as a key enabling piece to the low-overhead transversal architecture.

[0139] Referring to Fig. 14, the reduction in spacetime cost of logical algorithms in various embodiments is illustrated. Fig. 14A shows that when a transversal CNOT is performed between rounds of noisy syndrome extraction, measurement errors on 5 (7) stabilizers directly before the CNOT generate order-three hyperedges on the decoding hypergraph of the control (target) logical qubit. Fig. 14B shows that because these order-three hyperedges cannot be decomposed into existing edges, naively applying MWPM leads to a reducedthreshold (^@A ≃ 0.49%) compared to hypergraph MLE (^@A ≃ 1.0%), whose threshold isunaffected by the transversal CNOT. Because the runtime of hypergraph MLE is exponential in the worst-case, a modified hypergraph UF algorithm is also implemented that runs inpolynomial time but maintains a similar threshold (^@A ≃ 0.81%). Fig. 14C explores deeplogical Clifford circuits with 32 layers of random transversal Pauli and CNOT operations interlaced with ^ of noisy syndrome extraction between adjacent layers. Fig. 14D shows Page 37 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648that, because transversal operations are inherently fault-tolerant, ^ = 1 rounds of noisysyndrome extraction are sufficient to obtain a thresholds of ^@A ≃ 0.45% and 0.80% inhypergraph MLE and UF, respectively. Fig. 14E shows that when optimizing the logicalfidelity over the number of rounds of syndrome extraction per CNOT, ^ = 1 − 2 rounds areoptimal for hypergraph MLE. Fig. 14F shows that, similarly, ^ ≪ ^ rounds are optimal forhypergraph UF.

[0140] Basic Building Blocks: Gates and Gadgets

[0141] The following considers the basic building blocks employed in fast transversal gate architectures according to the present disclosure. These building blocks and capabilities are directly inspired by the hardware capabilities of neutral atom platforms.

[0142] This discussion considers logical qubits that are encoded in individual rotated surface codes, although other quantum error-correcting codes can also be employed. The square layout of such logical qubits makes them very easy to manipulate with optical tools, such as tweezer arrays generated with crossed acousto-optic deflectors (AODs).

[0143] To perform syndrome extraction, one can bring in a set of ancilla qubits, perform the desired physical CNOT gates for syndrome extraction (either via local Rydberg addressing or atom moving), and then move the ancilla qubits out to a separate readout zone. With appropriate choice of gate ordering, this procedure can be made fault-tolerant and realizes the full circuit-level code distance.

[0144] For logical gate operations, the following basic operations are realized: Hadamard gate, S gate, CNOT gate, and state injection. This will be further supplemented by the operation of resizing a logical code patch. With suitable decoding techniques, and by making use of perfect syndrome information when performing transversal readouts, one can reduce the number of rounds of quantum error correction following each logical gate. To maintain Page 38 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 code distance at the algorithmic level, any undetectable error that anti-commutes with any logical operators has a weight of at least ^. Since errors in the bulk always extend the syndrome, the only place where errors can terminate are on spatial boundaries and time boundaries. Thus, one may expect to require time boundaries to be separated by at least distance ^, although for specific circuits, these undetectable operations may commute with all logical operators, thereby further relaxing the separation requirements between boundaries.

[0145] The logical Hadamard gate can be implemented transversally for the surface code with physical Hadamards and a 90 degree rotation of the code patch. While rotations themselves are not a native operation with existing fast optical tools, by using a recursive decomposition into 2-by-2 blocks at each layer, one can perform a rotation in log depth.

[0146] The logical S gate is usually implemented via lattice surgery, requiring ^^^^^spacetime volume. However, by folding the code patch on top of itself, as shown in Figs. 15A-B, executing a CZ gate between the marked qubit pairs, and S gates on the ^ qubits along the fold axis, one can implement a logical S gate on the rotated surface code. This approach has the advantage of nearly preserving the distance (during the operation, the code distance is reduced by 1) and not requiring any deformation of the code patch, thereby avoiding initialization of new stabilizer values during deformation that may take multiple logical cycles.

[0147] The logical CNOT gate can be readily implemented by interlacing two logical qubits and pulsing an entangling operation between them with a global laser pulse.

[0148] State injection can be performed, which can then be used to build magic state factories, as discussed in the following section.

[0149] One can also create bridging Bell pairs to connect multiple logical subcircuits, thereby allowing connections between different circuit fragments that can now be executed out of time order. The Bell pair creation and Bell basis measurements can in principle be done in a Page 39 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 single-shot fashion, although similar to the above, more detailed analysis is required to understand the lowest weight logical error behavior.

[0150] Referring to Fig. 15, an illustration is provided of a fold-transversal S gate for the rotated surface code. This implementation does not require any patch deformation to be applied, which makes it much faster. In Fig. 15A, the data qubits in the bottom-right (dashed box) are mirrored over the top-right qubits to execute CZ gates. This is easier to perform with a diagonally-oriented AOD. In Fig. 15B a few S gates along the diagonal complete the logical gate.

[0151] Algorithmic Building Block: Magic State Distillation Factory

[0152] One of the most important sub-routines of FTQC is magic state distillation (MSD). This key subroutine allows the preparation of high-fidelity T states or CCZ states, which enable the implementation of non-Clifford gates. It is also one of the most expensive subroutines in quantum computation, as distillation requires a much larger space-time footprint than Clifford gates.

[0153] Referring to Fig. 16, a magic state distillation circuit is illustrated.

[0154] Referring to Fig. 17, an exemplary neutral atom layout for magic state distillation is illustrated, showing the high degree of parallelism available.

[0155] Consider the 15-to-1 magic state distillation circuit shown in Fig. 16. As shown, the circuit is highly structured, involving 4 layers of transversal CNOTs, each layer connecting logical qubits at a fixed distance from each other, and finally an additional T gate layer on a subset of the qubits that can be executed via gate teleportation. As mentioned above, each transversal gate only involves a single circuit layer, rather than the ^ layers required for lattice surgery, significantly reducing the time cost. Page 40 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0156] This circuit has a particularly natural implementation with neutral atom array systems, as illustrated in Fig. 17. This panel shows how the logical qubits can be laid out in a 4-by-4 grid, and how movements can be executed to perform the desired circuit. The movement is highly parallel, naturally allowing optical multiplexing with very few control channels. The longest distance that any codeblock needs to travel within the magic state distillation factory is only the span of two logical qubits, indicating that the required movement time will still remain very short.

[0157] In the circuit shown, to guarantee fault-tolerance of the Clifford codeword preparation portion by itself, each logical qubit requires ^ rounds of syndrome measurement for fault- tolerant state preparation. Each transversal layer involves a single CNOT layer that is executed in parallel between all logical qubits. Since one round of syndrome measurement involves 4 entangling gates and 1 measurement and reset, the time cost will be substantially larger than the CNOT operation. Thus, the MSD factory based on transversal gates requires atime cost on the order of ^ + 1, substantially lower than the 6^ or 13^ numbers required formagic state distillation based on lattice surgery.

[0158] The time cost may be further reduced to ^ / 2 or less: to ensure that the smallest error chain has weight at least ^, it is sufficient to ensure that the distance between two boundaries where error strings can terminate is at least ^. Since time-like error strings that cause a logical error can't terminate on a transversal measurement, which provides perfect measurement results, they can only join two of the state preparation boundaries, thereby reducing the distance requirement to ^ / 2. If such error chains do not cause a logical flip, the spacetime cost can be further reduced and only a constant distance may be necessary.

[0159] In terms of spatial overhead, a feature of the neutral atom architecture is that unlike conventional lattice surgery, which requires additional rows and columns for logical qubit access, parallel atom movement via AODs can mediate transversal gates without the need of Page 41 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 any additional ancilla qubits, thus saving space overhead by around 2x compared to existing schemes.

[0160] Algorithmic Building Block: Quantum Adder and Quantum Read-Only Memory

[0161] The preceding analysis can also be generalized to various algorithmic gadgets, such as the ripple-carry adder and quantum read-only memory (QROM).

[0162] The circuits for the ripple-carry adder and QROM are shown in Fig. 18. By rewiring the circuit in spacetime using bridge qubits, one can separate it into smaller gadgets that act on a few qubits; for example, one can separate out the first 3 qubits of the adder and implement the subcircuit, with some additional bridge qubits on the parts that connect to other qubits, so that different circuit gadgets can be stitched together. Similar to the discussions in the previous sections, by replacing the circuit components with transversal gates, one can significantly speed up their execution.

[0163] For the single control, multi-target CNOT shown in the QROM circuit in Fig. 18, one can employ constant-time Pauli product measurement methods, where an appropriate ancilla is prepared and then directly split up and interacted with each of the target blocks of the CNOT. Although this discussion has focused on particular well-structured gadgets, the ability to move different codeblocks without needing to worry about their position and possible intersection of paths can also significantly simplify routing challenges that can occur in planar architectures.

[0164] Physical Implementation

[0165] The flexible connectivity required to implement transversal gates is in principle accessible to a variety of physical platforms, including neutral atoms, trapped ions, silicon Page 42 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 quantum dots via qubit shuttling, or photon-based schemes via switch networks. However, this discussion focuses on neutral atom array systems, which have experimentally realized system sizes in the hundreds to thousands and high-fidelity single-qubit and two-qubit gate operations. The flexibility and power of qubit shuttling in neutral atom arrays enables complex logical qubit algorithms consisting of at least 48 logical qubits, 228 logical two- qubit gates and 48 logical CCZ gates.

[0166] Particularly powerful in this approach is the high degree of parallelism: despite the complexity of the quantum circuit being executed, the number of independent control channels used in this experiment was around 5, thanks to the high degree of parallelism afforded by optical multiplexing and gate operation via global laser excitation.

[0167] This naturally extends to the execution of large-scale fault-tolerant algorithms, significantly ameliorating the control challenges. For example, controlling individual logical code patches consisting of hundreds to thousands of physical qubits will still only require one pair of crossed AODs. Moreover, as illustrated in Fig. 17, at the algorithmic level there is also significant structure that allows exploitation of the parallelism of optical tools. In that example, the logical qubit moves are all performed in groups of rows and columns, and therefore again can be carried out using a single pair of crossed AODs.

[0168] The time required to move a logical code patch across a certain distance scales as the square root of the move distance, similar to a bounded acceleration profile. Thus, longer moves do have higher cost, but the cost grows relatively slowly. The architecture provided herein make extensive use of moves of intermediate distance, spanning a few logical qubits, thus making use of the slightly more favorable movement times at intermediate distances while avoiding bottlenecks related to longer-range shuttling and tweezer handoff if a large number of very long range moves are required. Moreover, by making use of well-structured circuits as shown in the previous section, this can also help address challenges associated Page 43 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 with the qubit routing time overhead when performing fast transversal gates, ensuring that qubit routing can keep pace with the computer clock speed.

[0169] Dynamic reconfiguration in 2D tweezer arrays

[0170] 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.

[0171] The crossed AOD system is composed of two independently controlled AODs (AA Opto Electronic DTSX-400) for L 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 arbitrary waveforms are composed of multiple frequency tones corresponding to the L and positions of columns and rows, which are independently changed as a function of time for steering Page 44 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 around the AOD-trapped atoms dynamically; the full L 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 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.

[0172] 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 homogenize by varying the amplitude of each frequency component; during motion between two locations the amplitude of each individual frequency component is interpolated.

[0173] 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! × 162{^, with radial trapfrequencies of ∼ 2! × 80^{^, and when running quantum circuits the trap depths are ∼2! × 42{^, with radial trap frequencies of ∼ 2! × 40^{^.

[0174] Raman laser system

[0175] 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 is Page 45 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648used 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 provides direct frequency and phase control over the hyperfine qubit drive.

[0176] The Raman laser illuminates the atom plane from the side in a circularly polarized elliptical beam with waists of 401^ and 5601^ 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 × 10_} (i.e., 1 scattering event per 15000 !pulses).

[0177] Qubit coherence and dynamical decoupling

[0178] In the 830-nm traps, hyperfine qubit coherence is characterized by ~∗^ = 4^^ (notshown), ~^ = 1.5^ (XY16 with 128 total ! pulses), and ~. = 4 s (including atom loss). Theexperiments described 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, the tweezer differential lightshifts decrease as 1 / Δ and 1 / ~.decreases as 1 / Δ^) and shielding against magnetic field fluctuations. For practical QEC operation, atom loss can be Page 46 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 detected in a hardware-efficient manner and the atom then replaced from a reservoir, which could in principle be continuously reloaded by a MOT for reaching arbitrarily deep circuits.

[0179] 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.

[0180] Movement effects on atom heating and loss

[0181] 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 11 where ^^^^p^is the Fourier transform of ^^^^evaluated at the trap frequency ^p, and the zeropoint size of the particle L^ri ≡ ^ℏ / ^2^^p^. Δ^ is the same for all initial levels of thePage 47 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648oscillator. Experimentally, an acceleration profile ^^^^ = ^^ is applied to the atom, from time−~ / 2 to +~ / 2 to move a distance ` with constant jerk ^. Calculating|^^^^^|^, simplifyusing ^p~ ≫ 1, and assume a small range of trap frequencies to average the oscillatoryterms, resulting inEquation 12

[0182] 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 ~ ∝ ^_^ / ^p . Moreover, toperform a large number of moves ^ for a deep circuit, Δ^ ∝ ^ / ~^ can be estimated,suggesting that the number of moves can be increased from, e.g., 5 to 80 by slowing each move from 2001^ to 4001^. 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.

[0183] Equation 12 is now compared to experimental observations. Atom loss is observed with movement of 551m in 2001^ under a constant negative jerk. This speed limit isconsistent with the above estimates: using ^p = 2! × 40^{^ and L^ri = 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 ^ and variance ^ and integrate the population above some critical ^^^Supon which the atom will leave the trap. From this analysis, atom retention is given

[0184] 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- Page 48 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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 ^C^Ir by 2 ×, together would increase the numberof possible CZ gates per atom to thousands.

[0185] Two-qubit CZ gates implementation

[0186] 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 pulse parameters are: Δ= −0.377371Ω^ = −0.621089 × ^2!^^ = 0.683201 / [Ω / ^2!^]

[0187] Exemplary experiments are operated with a two-photon Rydberg Rabi frequency ofΩ / 2! = 3.62{^, giving a theoretical ^ = 190^^ and a theoretical Δ / ^2!^ = −1.362{^.The negative detuning sign is chosen to help minimize excitation into the ^Y = +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 ^Y = −1 / 2 state due to reducedClebsch-Gordan coefficients). In this work, strong blockade between adjacent qubits is provided, with Rydberg-Rydberg interactions Pp / 2! ranging from 200 MHz to 1 GHz. Page 49 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0188] Managing spurious phases during CZ gates

[0189] 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 (> 82{^), yielding phase accumulations on the hyperfinequbit of ≈ 6!. Small, percent-level variations of the 420-nm intensity can thus lead tosignificant qubit dephasing.

[0190] 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.

[0191] 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 each 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 fidelites 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 such Page 50 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648as a ≈ 0.01 rad phase induced by pulsing the traps off for 500 ns for the two-qubit gate. Theecho for the final CZ gate is performed in instances where the number of CZ gates is odd.

[0192] 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=^ / ^transition. 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.

[0193] Sensitivity to axial trap oscillations

[0194] 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 estimated axial temperature of ∼ 251e and axial oscillation frequency of6^{^, an axial spread ^〈^^〉 ≈is estimated. For 20-micron-waist beams, the effectof 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 by echoing out thephase induced by CZ gates separated by ∼ 2001^. When using 20-micron-waist beams, anda 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.

[0195] Rydberg beam shaping and homogeneity

[0196] 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 Page 51 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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 40-501m in the atom plane.

[0197] Coherent mapping protocol

[0198] A coherent mapping protocol is provided to transfer a generic many-body state in thel|1^, |^^m basis to the long-lived and non-interacting l|0^, |1^m 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^.

[0199] Even for perfect Raman and Rydberg ! pulses (on isolated atoms), there are three key sources of infidelity associated with this mapping process: (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 the Page 52 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 Rydberg 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 the system particularly sensitive to dephasing errors.

[0200] The above error mechanisms are mitigated as follows. To minimize errors from (1), ¡ many-body dynamics are performed with ^¢¡£ ≈ 0.01. This minimizes the probability of anatom to violate blockade to be of order 1%. To help minimize errors from (2), the amplitudeof the 420-nm laser is increased for the final ! pulse by a factor of 2 ×, such that  =0.005 (where PFFF are the interactions with next-nearest neighbors), reducing pulse errorsfrom 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 short relative tothe ~∗^ ≈ 3 − 41^ of the l|R^, |^^m basis, leading to a random phase accumulation of order ∼0.02 × 2! ^^^ per particle, but is further compounded by having entangled states of Nparticles in one copy accumulating a random phase relative to entangled states of N particles in the second copy.

[0201] 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.

[0202] Formation of Array of Particles Using Optical Tweezers Page 53 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648

[0203] 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 that 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%.

[0204] 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, Page 54 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0205] Exemplary Hardware

[0206] 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.

[0207] 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 Page 55 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 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.

[0208] 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.

[0209] 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.

[0210] Referring to Fig. 11, a schematic view is provided of an apparatus 1100 for quantum computation according to embodiments of the present disclosure. As shown in Fig. 11, using a beam generated by a light source 1102 (for example, a coherent light source, in some example embodiments – a monochromatic light source), SLM 1104 forms an array of trapping beams (i.e., a tweezer array) which is imaged onto trapping plane 1108 in vacuum chamber 1110 by an optical train that, in the example embodiment shown in Fig. 11, comprises elements 1106a, 1106c, 1106d, and a high numerical aperture (NA) objective 1106e. 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 1112 (for example, a coherent light source; in some example embodiments - a monochromatic light Page 56 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 source), a pair of AODs 1114 and 1116, 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. 11 (elements 1117, 1106b, 1106c, 1106d, and 1106e), 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 1102 and 1112 can be a single source, and the trapping beam and the sorting beam are generated by a beam splitter.

[0211] The dynamic movement of the steering beams is accomplished by employing two non-parallel AODs 1114, 1116, arranged in series. In the example embodiment depicted in Fig. 11, 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 1120, which is generated in real-time by a computer 1122 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 plane 1108 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.

[0212] In Fig. 11, laser 1102 projects a beam of light onto SLM 1104. SLM 1104 can be controlled by computer 1122 in order to generate a pattern of beams (“trapping beams” or “tweezer array”). The pattern of beams is focused by lens 1106a, passes through mirror 1106b, and is collimates by lens 1106c on mirror 1106d. The reflected light passes through objective 1106e to focus an optical tweezer array in vacuum chamber 1110 on trapping plane 1108. The laser light of the optical tweezer array continues through objective 1124a, and Page 57 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 passes through dichroic mirror 1124b to be detected by charge-coupled device (CCD) camera 1124c.

[0213] Vacuum chamber 1110 may be illuminated by an additional light source (not pictured). Fluorescence from atoms trapped on the trapping plane also passes through objective 1124a, but is reflected by dichroic mirror 1124b to electron-multiplying CCD (EMCCD) camera 1124d. In this example, laser 1112 directs a beam of light to AODs 1114, 1116. AODs 1114, 1116 are driven by arbitrary wave generator (AWG) 1120, which is in turn controlled by computer 1122. Crossed AODs 1114, 1116 emit one or more beams as set forth above, which are directed to focusing lens 1117. The beams then enter the same optical train 1106b…1106e as described above with regard to the optical tweezer array, focusing on trapping plane 1108.

[0214] It will be appreciated that alternative optical trains may be employed to produce an optical tweezer array suitable for use as set out herein.

[0215] 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. Page 58 of 63 FoleyHoagUS12614343.1

Claims

HQU-01525 HU 9648 CLAIMS What is claimed is:

1. A method of performing a quantum computation, the method comprising: providing at least a first plurality of physical qubits; encoding a first logical qubit into the at least first plurality of physical qubits according to a quantum error correcting code; encoding a second logical qubit into the at least first plurality of physical qubits according to the quantum error correcting code; based on the quantum error correcting code, constructing a bipartite decoding graph corresponding to the first and the second logical qubits, the bipartite decoding graph comprising a plurality of detector nodes and a plurality of error nodes, each error node corresponding to an error mechanism; applying a transversal gate to the first and the second logical qubits; performing a first round of syndrome measurement of the first and the second logical qubits; for each of the plurality of detector nodes affected by the corresponding error mechanism of one of the plurality of error nodes, generating an edge on the bipartite decoding graph therebetween; and determining a physical error configuration from the bipartite decoding graph.

2. The method of claim 1, further comprising: applying at least one gate to the first plurality of physical qubits and the second plurality of physical qubits to correct the physical error configuration.

3. The method of any one of claims 1-2, wherein the quantum error correcting code is a surface code.

4. The method of any one of claims 1-3, wherein the transversal gate is a Clifford gate. Page 59 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 5. The method of claim 4, wherein the Clifford gate is a CNOT gate.

6. The method of any one of claims 1-3, further comprising: encoding a third logical qubit into the at least first plurality of physical qubits according to the quantum error correcting code.

7. The method of claim 6, wherein the transversal gate is a non-Clifford gate.

8. The method of claim 7, wherein the non-Clifford gate is a CCZ gate.

9. The method of any one of claim 1-8, wherein determining the physical error configuration comprises maximizing an error probability on the bipartite decoding graph.

10. The method of claim 9, wherein maximizing the error probability comprises solving a mixed-integer programming problem corresponding to the error probability.

11. The method of any one of claims 1-8, wherein determining the physical error configuration comprises determining one or more subgraphs of the bipartite decoding graph corresponding to the physical error configuration.

12. The method of claim 11, wherein determining the one or more subgraphs comprises defining a subgraph for each detector node having a detected error and expanding each such subgraph until it encompasses error nodes, which, if they had occurred, would result in syndrome measurements consistent with the observed syndrome.

13. The method of any one of claims 1 to 12, wherein each qubit of the first plurality of physical qubits and the second plurality of physical qubits is a neutral atom.

14. The method of any one of claims 1-13, wherein the at least first plurality of physical qubits comprises a second plurality of physical qubits, and wherein the first logical qubit is encoded in the first plurality of physical qubits and the second logical qubit is encoded in the second plurality of physical qubits. Page 60 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 15. The method of any one of claims 6-8, wherein the at least first plurality of physical qubits comprises a second plurality of physical qubits and a third plurality of physical qubits, and wherein the first logical qubit is encoded in the first plurality of physical qubits, the second logical qubit is encoded in the second plurality of physical qubits, and the third logical qubit is encoded in the third plurality of physical qubits.

16. The method of claim 14, wherein applying the transversal 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.

17. The method of any one of claims 1-16, further comprising: alternately applying one or more additional transversal gates to the first and the second logical qubits and one or more additional rounds of syndrome measurement of the first and the second logical qubits.

18. A quantum processor, comprising: a first array of optical traps disposed in an active zone; a second array of optical traps disposed in a readout zone; a first laser configured to illuminate the active zone and to drive a transition to a Rydberg state; a second laser configured to illuminate the active zone and to drive a transition between hyperfine states; a third laser configured to illuminate the readout zone; a fourth laser configured to adiabatically move neutral atoms between the optical traps of the active zone and the readout zone; and Page 61 of 63 FoleyHoagUS12614343.1HQU-01525 HU 9648 a camera configured to capture an image of the readout zone, wherein the quantum processor is configured to: provide at least a first plurality of neutral atoms in the active zone, each in a respective optical trap of the first array; encode a first logical qubit into the at least first plurality of neutral atoms according to a quantum error correcting code by the first and second lasers; encode a second logical qubit into the at least first plurality of neutral atoms according to the quantum error correcting code by the first and second lasers; based on the quantum error correcting code, constructing a bipartite decoding graph corresponding to the first and the second logical qubits, the bipartite decoding graph comprising a plurality of detector nodes and a plurality of error nodes, each error node corresponding to an error mechanism; place the first and second pluralities of neutral atoms in the active zone such that each neutral atom of the first plurality of neutral atoms is within a blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms; illuminate the first plurality of neutral atoms and the second plurality of neutral atoms while in the active zone by at least the first or second laser, thereby applying a transversal gate to the first and second logical qubits; performing a first round of syndrome measurement of the first and the second logical qubits; for each of the plurality of detector nodes affected by the corresponding error mechanism of one of the plurality of error nodes, generating an edge on the bipartite decoding graph therebetween; and determining a physical error configuration from the bipartite decoding graph. Page 62 of 63 FoleyHoagUS12614343.1