Quantum low-density parity check code using a reconfigurable atomic array

Abstract

Quantum error correction in a reconfigurable atomic array is described. Multiple data qubits are provided, each of which is placed within a corresponding trap. Multiple auxiliary qubits are provided, each of which is placed within a corresponding trap. The multiple data qubits and the multiple auxiliary qubits are arranged in multiple rows and multiple columns, thereby forming a lattice. Multiple sortings of the multiple rows and multiple columns are performed, each of which sorting places each of the multiple data qubits within the interaction radius of one of the multiple auxiliary qubits, thereby forming multiple proximity pairs. After each of the multiple sortings, a global control pulse is applied to the lattice, thereby applying a gate to each of the multiple proximity pairs, and thereby encoding a parity check matrix between the multiple auxiliary qubits and the multiple data qubits.

Description

Technical Field

[0001] Cross - reference to Related Applications This application claims the benefit of U.S. Provisional Application No. 63 / 499,325, filed May 1, 2023, and U.S. Provisional Application No. 63 / 530,026, filed Jul. 31, 2023, each of which is hereby incorporated by reference in its entirety.

[0002] Description of Research and Development Sponsored by the Federal Government This invention was made with government support under NSF Grants No. 1855879, 2100013, 2106189, and 1941583, and USAF / AFOSR Grant No. FA9550 - 19 - 0360. The government has certain rights in this invention.

[0003] Embodiments of this disclosure relate to quantum computing, and more particularly, to a dynamically reconfigurable architecture for parallel quantum operations in an array of neutral atoms.

Summary of the Invention

Means for Solving the Problems

[0004] Methods for performing quantum error correction in various embodiments are disclosed. A plurality of data qubits are provided, each of which is placed within a corresponding trap. A plurality of ancilla qubits are provided, each of which is placed within a corresponding trap. The plurality of data qubits and the plurality of ancilla qubits are arranged in a plurality of rows and a plurality of columns, thereby forming a lattice. A plurality of sortings of the plurality of rows and a plurality of columns are performed, each of which sorting places each of the plurality of data qubits within the interaction radius of one of the plurality of ancilla qubits, thereby forming a plurality of proximity pairs. After each of the sortings, a global control pulse is applied to the lattice, thereby applying a gate to each of the plurality of proximity pairs, and thereby encoding a parity check matrix between the plurality of ancilla qubits and the plurality of data qubits.

[0005] In some embodiments, control laser pulses are applied to multiple data qubits before the grid is formed, thereby preparing the multiple data qubits in an initial state.

[0006] In some embodiments, arranging a plurality of data qubits and a plurality of auxiliary qubits in a grid includes moving the plurality of auxiliary qubits in parallel in the grid.

[0007] In some embodiments, performing multiple sorts includes moving one or more rows in a grid in parallel and moving one or more columns in a grid in parallel.

[0008] In some embodiments, the auxiliary qubits include Z stabilizers and X stabilizers.

[0009] In some embodiments, a subset of auxiliary qubits is removed from the grid, and measurements are performed on the subset. In some embodiments, the subset corresponds to a Z-stabilizer. In other embodiments, the subset corresponds to an X-stabilizer.

[0010] In some embodiments, performing multiple sorts involves determining a collision-free path for each of the multiple data qubits and each of the multiple auxiliary qubits for each of the multiple sorts. In some embodiments, determining a collision-free path involves bipartition and recursive sorting of the multiple data qubits and the multiple auxiliary qubits. In some embodiments, the collision-free path is a cubic spline.

[0011] In some embodiments, determining a collision-free path includes decomposing a parity check matrix into a first product graph and a second product graph, determining the routing of multiple rows according to the first product graph, and determining the routing of multiple columns according to the second product graph. In some embodiments, the multiple sorts include row-specific sorts and / or column-specific sorts. In some embodiments, performing the multiple sorts includes applying a pinning beam to at least one of multiple data qubits or multiple auxiliary qubits, thereby maintaining the position of at least one qubit according to the routing of multiple rows or multiple columns.

[0012] In some embodiments, the parity check matrix implements a quantum low-density parity check (qLDPC) code. In some embodiments, the parity check matrix implements a surface code. In some embodiments, the parity check matrix implements a hypergraph product (HGP) code. In some embodiments, the parity check matrix implements a Calderbank-Shor-Steane (CSS) code.

[0013] In some embodiments, a parity check matrix implements the lift product (LP) code. In some embodiments, the LP code is flattened before performing multiple sorts. In some embodiments, flattening the LP code involves iteratively dividing and flattening the LP code.

[0014] In some embodiments, the gate applied to each of the multiple adjacent pairs is a CZ gate.

[0015] In some embodiments, the global control pulse is a laser pulse.

[0016] In some embodiments, the traps corresponding to each of the data qubits and each of the auxiliary qubits are optical traps. In some embodiments, the optical traps corresponding to the data qubits and auxiliary qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), and moving one or more rows and one or more columns includes changing the driving frequency of at least one AOD.

[0017] In some embodiments, one or more rotations are applied during movement. In some embodiments, applying one or more rotations includes applying Raman pulses.

[0018] In various embodiments, a quantum computing system is provided which includes a plurality of data qubits, each located in a corresponding trap, and a plurality of auxiliary qubits, each located in a corresponding trap, and is configured to perform any of the methods described above. [Brief explanation of the drawing]

[0019] [Figure 1] This is a schematic diagram of a quantum information architecture according to the embodiments of this disclosure. [Figure 2] This is an energy level diagram showing the important atomic levels of 87Rb according to embodiments of the present disclosure. [Figure 3] This is a schematic diagram of a quantum processing unit (QPU) according to an embodiment of the present disclosure. [Figure 4] This is a schematic diagram of a qubit grid showing the product structure of a hypergraph product code according to an embodiment of the present disclosure. [Figure 5A] This is a schematic diagram of a qubit grid illustrating the sorting of columns and rows according to embodiments of the present disclosure. [Figure 5B] This is a schematic diagram of a qubit grid illustrating the sorting of columns according to embodiments of the present disclosure. [Figure 5C] This is a schematic diagram of a qubit grid illustrating row sorting according to embodiments of the present disclosure. [Figure 6] This is a schematic diagram of a one-dimensional array of qubits over time, showing a syndrome extraction gate sequence according to an embodiment of the present disclosure. [Figure 7] This is a schematic diagram of a one-dimensional array of qubits over time, illustrating the rearrangement steps according to embodiments of the present disclosure. [Figure 8A] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8B] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8C] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8D] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8E] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8F] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8G] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8H] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8I] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8J] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8K] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8L] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8M] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8N] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8O] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8P] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8Q]This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8R] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8S] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8T] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8U] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8V] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8W] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8X] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8Y] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8Z] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 8AA] This is a schematic diagram of a one-dimensional array of qubits that form a sequence of rearrangement steps for achieving sorting according to an embodiment of the present disclosure. [Figure 9]This is a schematic diagram of a one-dimensional array of qubits over time, illustrating the rearrangement steps according to embodiments of the present disclosure. [Figure 10] This is a schematic diagram of an exemplary two-dimensional graph according to the embodiments of the present disclosure. [Figure 11] This is a schematic diagram of a one-dimensional array of qubits over time, illustrating the use of fixed beams to prevent movement in some of the columns according to embodiments of the present disclosure. [Figure 12] This is a schematic diagram of a qubit grid showing the product structure of a lift product (LP) code according to an embodiment of the present disclosure. [Figure 13] This is pseudocode of an algorithm for an arbitrary 1D atom rearrangement in a logarithmic number of steps according to embodiments of the present disclosure. [Figure 14] This is pseudocode for a product coloration circuit for extracting HGP syndrome according to an embodiment of the present disclosure. [Figure 15] This is pseudocode for a pipelined product coloring circuit for multi-round HGP syndrome extraction according to embodiments of the present disclosure. [Figure 16A] This figure shows an efficient implementation of a quantum LDPC code using an atomic array according to an embodiment of the present disclosure. [Figure 16B] This figure shows an efficient implementation of a quantum LDPC code using an atomic array according to an embodiment of the present disclosure. [Figure 16C] This figure shows an efficient implementation of a quantum LDPC code using an atomic array according to an embodiment of the present disclosure. [Figure 16D] This figure shows an efficient implementation of a quantum LDPC code using an atomic array according to an embodiment of the present disclosure. [Figure 17A] This figure shows the sequence of operations in pipelined syndrome extraction according to the embodiments of this disclosure. [Figure 17B] This figure shows the sequence of operations in pipelined syndrome extraction according to the embodiments of this disclosure. [Figure 18]This is a schematic diagram of an apparatus for quantum computing according to an embodiment of the present disclosure. [Modes for carrying out the invention]

[0020] Quantum low-density parity check (qLDPC) codes can achieve high coding rates and long code distances, offering a promising pathway to low-overhead, fault-tolerant quantum computing. However, the high connectivity required to implement such codes makes their physical realization difficult.

[0021] This disclosure provides a scheme for efficiently implementing constant-rate qLDPC codes using reconfigurable atomic arrays. This scheme leverages the product structure inherent to many qLDPC codes to provide a simple circuit implementation. An efficient parallel rearrangement method for one-dimensional atomic arrays is provided, enabling stabilizer measurements of qLDPC codes with a rearrangement step count that increases logarithmically with respect to the system size when repeated in parallel across different rows or columns. This research enables low-resource-overhead quantum computing using qLDPC codes.

[0022] A qubit is a fundamental building block of a quantum computer. Similar to classical bits (each bit being either 0 or 1) used to store information in conventional computers, a qubit can occupy two distinct states labeled |0> and |1>, or any quantum superposition of two states. In various applications, multiple qubits are entangled to construct multi-qubit quantum gates.

[0023] Bits and qubits are encoded to represent the state of a real-world physical system. For example, a classic bit (0 or 1) might encode whether a capacitor is charged or discharged, or whether a switch is "on" or "off".

[0024] The term "qudit" (quantum digit) refers to a unit of quantum information that can be realized in a suitable d-level quantum system. A collection of N qubits, each capable of measuring a state, can implement an N-level qudit.

[0025] A qubit is encoded in a quantum system with two (or more) different quantum states. There are many physical realizations that can be employed. One example is based on individual particles such as atoms, ions, or molecules isolated in a vacuum. These isolated atoms, ions, and molecules have many different quantum states corresponding to different orientations of electron spin, nuclear spin, electron orbits, and molecular rotation / vibration.

[0026] In principle, a qubit can be encoded into any pair of quantum states of atoms / ions / molecules. In practice, the important parameters of a qubit are described by its quantum coherence property. Coherence measures the lifetime of a qubit until its information is lost. Coherence is very similar to that of a classical bit; if a classical bit is prepared in the state of 0, after some time the 0 state may be randomly flipped to 1 by ambient noise. Quantum mechanically, the same error can occur, where |0> may randomly flip to |1> after a certain characteristic timescale. However, qubits can be subject to further errors; for example, a superposition state (|0>+|1>) / √2 may randomly flip to (|0>-|1>) / √2. In a real quantum computer, qubits must be encoded into quantum states with long coherence properties.

[0027] Quantum computers can generally accommodate many qubits, each encoded with its own unique atoms / molecules / ions, etc. Beyond simply accommodating qubits, a quantum computer should be able to (1) initialize qubits, (2) manipulate the qubit states in a controlled manner, and (3) read out the final states of qubits. Regarding qubit manipulation, this is typically divided into two types: one type of qubit operation is the so-called one-qubit gate, which refers to an operation applied individually to each qubit. This might, for example, invert the state of a qubit from |0> to |1>, or place |0> into a superposition state (|0>+|1>) / √2. A second necessary type of qubit operation is the multi-qubit gate, which acts collectively on two or more qubits, including entangled qubits. Multi-qubit gates are realized through some form of interaction between qubits. Various quantum computing platforms (with different physical codings for qubits) rely on different physical mechanisms for both one-qubit and multi-qubit gates, depending on the physical system that stores the qubits.

[0028] In various embodiments of quantum computers, qubits are encoded into energy levels close to the two ground states of an atom, ion, or molecule. An example of this is a hyperfine qubit. Such a qubit is encoded into two different electronic ground states depending on the relative orientation of the nuclear spin to the outer shell electron spin. Such pairs of states can be selected so that they are particularly robust / unresponsive to environmental perturbations, leading to long coherence times. These states are energy-split 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 a qubit can be understood as the energy splitting between the two states being particularly stable. For this reason, stable energy splitting can form an excellent frequency reference and thus form the basis of an atomic clock, so such states are called clock states. Typical hyperfine splitting between the states of these qubits is in the frequency range of 1–13 GHz.

[0029] To perform a one-qubit gate on such ultrafine qubits, one might apply coherent microwave radiation at the exact frequency of the energy splitting between states. However, this method has two drawbacks. First, microwaves cannot be applied to just one qubit without affecting nearby qubits. This is because qubits are typically encoded into particles that are only a few microns apart from each other, and microwaves cannot be concentrated to such a small scale due to their long wavelengths. Second, the intensity of microwaves is quite limited, and therefore the maximum velocity of a one-qubit gate is correspondingly restricted.

[0030] An alternative approach is based on stimulated Raman transitions. In this case, a laser field, also referred to herein as a Raman pulse, is applied to an atom / ion / molecule. The laser field resonates nearly (but not completely) with the optical transition from one of the ground states to a photoexcited state. The laser field contains multiple frequency components that are frequency-separated by an amount precisely equal to the hyperfine splitting of qubits. The atom / ion / molecule absorbs photons from one frequency component and coherently emits them into a different frequency component, changing its state in the process. This approach benefits from the ability to focus the laser field on individual particles or subsets of particles within a quantum computer. Furthermore, the laser field can be applied at high intensity, enabling faster gate operations.

[0031] A neutral atom quantum computer encodes qubits into individual neutral atoms. These neutral atoms are trapped in a vacuum chamber and levitated by a capture laser. The capture laser is most commonly an individual optical tweezers, which are individual, highly focused laser beams that capture individual atoms at the focal point. Alternatively, individual atoms may be trapped within an optical lattice formed from standing waves of laser light that generate a periodic structure of nodes / antenotypes.

[0032] A typical method for encoding qubits in neutral atoms is the hyperfine qubit method, where two ground states split into several GHz levels form a qubit. Multi-qubit gates in neutral-atom quantum computers are realized using a third atomic state, which is a highly excited Rydberg state. When an atom is excited to a Rydberg state, neighboring atoms are prevented from being excited to a Rydberg state. This conditional behavior forms the basis of multi-qubit gates such as controlled NOT gates. The Rydberg state is used temporarily to mediate the multi-qubit gate, and then the atoms are returned from the Rydberg state to the ground state level to preserve their coherence.

[0033] Ion-trap quantum computers use ionized atomic species, meaning the atomic species have a net charge. In most cases, many ions are trapped in a single large trapping potential formed by electrodes in a vacuum chamber. The ions are pulled towards the minimum of the trapping potential, but Coulomb repulsion between ions causes them to form a crystalline structure centered on the center of the trapping potential. Ions most commonly align in a linear chain. Other methods of trapping ions are also possible, such as using optical tweezers or trapping ions individually in a local electric field using more complex on-chip electrode structures.

[0034] Qubits are encoded in captured ions in several ways. One common method is to use the hyperfine levels of the ground state, as described for neutral atoms. Similar to neutral atoms, in captured ions that undergo hyperfine qubit encoding, the 1-qubit gate may use microwave emission or stimulated Raman transitions.

[0035] Unlike neutral atoms, the hyperfine qubits of captured ions rely heavily on stimulated Raman transitions to perform multi-qubit gates. Stimulated Raman transitions can be used not only to control the ion's hyperfine state but also to change the ion's kinetic state (i.e., to add momentum). This can be understood as absorbing a photon moving in one direction and emitting a photon in the other, so that the difference in photon momentum is absorbed by the ion. Since many ions are often trapped in a single collective trapping potential and repel each other, changing the kinetic state of one ion affects other ions in the system, and this mechanism forms the basis of multi-qubit gates.

[0036] According to various embodiments of quantum computers, individual particles (atoms / ions / molecules) can first be captured in an array and arranged in a specific configuration. Next, one or more particles are prepared in a desired quantum state. Then, the quantum circuit can be performed by a sequence of qubit operations acting on individual qubits (1-qubit gates) or groups of two or more qubits (multi-qubit gates). Finally, the state of the particles can be read out to observe the results of the quantum circuit. The readout can typically be performed using an observation system that includes an electron-multiplier CCD (EMCCD) camera image to detect the loaded position of the particles and a second camera image to read out the final state of the particles, for example, by detecting fluorescence emitted by the particles in their final state.

[0037] Quantum information platforms rely on qubit interactions, either to execute quantum gates or to perform analog many-body simulations. However, qubits often interact locally, which limits the connectivity of circuits or analog simulations and constrains the possible computations. Some platforms can communicate non-locally by using a shared bus (e.g., trapped ions), but these shared bus techniques are limited to small systems and therefore still require a way to dynamically move qubits in order to truly scale up the platform.

[0038] Neutral atom arrays can be dynamically reconfigured while preserving quantum coherence and entanglement between qubits by storing quantum information in a hyperfine state and shuttle the atoms with optical tweezers. This technique provides a scalable way to realize quantum information systems with a large number of qubits and arbitrary programmability—where any qubit can perform entanglement gates with any other qubit in the array. Various quantum information circuits that leverage the programmability and nonlocal connectivity achievable with these techniques using high-fidelity two-qubit Rydberg gates are described herein. Examples of high-fidelity Rydeberg gates are described in Levine et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett, vol. 123, no. 17, https: / / link.aps.org / doi / 10.1103 / PhysRevLett.123.170503, and Evered et al., High-Fidelity Parallel Entangling Gates on a Neutral Atom Quantum Computer, arXiv:2304.05420 [quant-ph], https: / / arxiv.org / abs / 2304.05420, which are incorporated herein by reference.

[0039] As will be described in more detail below, the methods provided herein enable a variety of computational scenarios. In some scenarios, multiple neutral atoms are moved in parallel between multiple regions in space. For example, an illumination source may be directed to a first region, and atoms are brought into and out of that region during the application of pulses by the illumination source. Similarly, a camera may be directed to an imaging region, and atoms are brought into and out of that imaging region for imaging. Likewise, atoms may be brought into and out of the blockade radius of other atoms, thereby enabling the application of gates to different groups of atoms at different stages of the algorithm or different layers of the quantum circuit.

[0040] Naturally, various stabilizer codes involve the readout of ancilla qubits, and this disclosure allows for the physical relocation of ancilla qubits to an imaging region separated from the data qubits. In this way, ancilla qubits may be readout without destroying the data qubits.

[0041] More broadly, to facilitate both digital gates between different selections of atoms and analog evolution of the entire array, an array of atoms may be moved between multiple arrays. As used herein, an array of atoms or an array of atoms refers to the relative positions of those atoms to one another. It will be understood that a particular array provides connectivity between qubits, enabling analog evolution according to a particular gate or a particular Hamiltonian. One advantage of the methods provided herein is that atoms may be moved to the vicinity of atoms that were not adjacent within the array. Non-proximate atoms are those that are not in a unit cell in a regular lattice or are not the nearest neighbor in a disordered array. For example, in a rectangular lattice, each atom has eight atoms in its unit cell and therefore eight neighboring atoms (neglecting the edges).

[0042] As further defined below, atoms are moved adiabatically to maintain entanglement. In this specification, the term adiabatic movement refers to movement that avoids transitions of the target atom within its trap. For example, a movement is considered adiabatic if the first time derivative of the acceleration of the target atom is not greater than a given value. Typically, adiabatic movement has a jerk < (size of atom) × (trap frequency). 3 This occurs when... In physics, jerk, or jolt, is a term given to the rate at which the acceleration of an object changes with respect to time.

[0043] In addition to adiabatic migration, in some embodiments, dynamic decoupling is applied during migration. As will be further described below, the π pulse during migration cancels out the phase relaxation induced by the differential light shift of the trap. The differential light shift of the trap changes (depending on its acceleration) as the atom moves within the trap and therefore samples different portions of light intensity and thus undergoes different differential light shifts.

[0044] Generally, the more pulses applied, the greater the decoupling from fluctuations. For example, fluctuations may be caused by variations in laser intensity at different atomic displacement positions or by different magnetic fields in space.

[0045] In embodiments where acceleration and deceleration are symmetrical, both alter the differential optical shift in the same way. Therefore, in such embodiments, it is advantageous to apply a π pulse at the midpoint of the motion. In this way, the changes in the differential optical shift induced by acceleration and deceleration cancel each other out.

[0046] Referring to Figure 1, a quantum information architecture enabled by the coherent transport of neutral atoms is shown. Qubits are transported to perform entanglement gates with distant qubits, enabling programmable and nonlocal connectivity. Atomic shuttles are performed using optical tweezers, and high parallelism between two-dimensional and multiple zones enables selective operations. The inset shows the atomic levels used, with |0> and |1> qubit states being: 87 Rb's m F |r> refers to the clock state of 0, and |r> is the Rydberg state used to generate entanglement between qubits, which will be further explained with respect to Figure 2.

[0047] Figure 2 shows the important parts used. 87 This is an energy level diagram showing the atomic levels of Rb. The Rydberg excitation scheme from |1> to |r> consists of two-photon transitions driven by 420 nm and 1013 nm lasers. A DC magnetic field of B = 8.5 G is applied throughout this study.

[0048] As mentioned above, quantum information systems derive their power from controllable interactions that generate quantum entanglement. However, the natural local nature of these interactions limits the connectivity of quantum circuits and simulations. Nonlocal connectivity can be designed via globally shared quantum data buses, but these methods are limited in either control or size.

[0049] According to various embodiments of this disclosure, this long-standing challenge is addressed by a dynamically reconfigurable array of entangled neutral atoms, shuttled by optical tweezers in two spatial dimensions. Hyperfine states are used to store and transfer quantum information between quantum operations, and excitations to Rydberg states are used to generate the entanglement. Highly parallel operations are enabled by selective qubit operations in different zones where qubits are dynamically shuttled among them. Taken together, these elements enable a powerful quantum information architecture that can be used to realize applications including the generation of entangled states, the generation of topological surface states and torus code states, and hybrid analog-digital quantum simulations.

[0050] Within this architecture, programming a specific quantum circuit involves controlling only a few optical degrees of freedom. Arbitrary tweezers positions in space are controlled by computer-generated holograms, hundreds of atoms are dynamically reconfigured in parallel by two waveforms of a 2D acousto-optic deflector (AOD), and qubital operation is achieved by pulsing the light beam. This flexible optical control enables advanced quantum circuits with only a few classical controls. This architecture allows for an inherently scalable approach, where larger codes do not require an increase in the number of classical controls.

[0051] Various quantum circuits, including quantum error-correction (QEC) codes such as surface codes and Steane codes, can be realized using this method, and the fidelity in this disclosure is already comparable to state-of-the-art experiments on other platforms. Furthermore, parallelized nonlocal connectivity is used to generate torus code states on a torus.

[0052] Referring to Figure 3, the quantum processing unit (QPU) according to this disclosure is shown. This design centers on the parallel, efficient classical control of many logic qubits using a light beam. A one-qubit logic gate can be realized transversally, for example, by illuminating all physical qubits within the same logic qubit block with a light beam. A two-qubit logic gate can also be realized transversally by interlacing two logic arrays of qubits and applying a global light pulse to entangle each pair. For such a gate to be transversal, such a gate must only allow corresponding qubits from different logic arrays to interact, such that the first qubit of the first logic array interacts with the first qubit of the second logic array, and so on.

[0053] Neutral atom systems have the potential for practical-scale computing, for example, millions of identical neutral atom qubits can be trapped in a spatial domain on the millimeter scale. A key challenge is the classical control required to assemble these qubits into a large-scale quantum processor. The full programmability of a single physical qubit generally requires classical control techniques that are extremely complex to operate on millions of qubits. In contrast, the architecture provided herein enables the full programmability of a single logical qubit while requiring only a few classical controls per logical qubit. This makes it possible to reach practical scale by encoding logical qubits into blocks that can be efficiently controlled in parallel. Direct capture and manipulation of more than one million qubits is possible using advanced optical microscope systems (such as those used in modern industrial-scale lithography) with high numerical apertures and wide fields of view exceeding several millimeters, as well as appropriately scaled capture laser power. Further scaling is possible by creating 10 to 100 such processing units, each under its own microscope objective lens, and connecting these units together using photonic links and / or optical lattice transport. This allows for sufficient space, resolution, and power density to implement high-fidelity control of 10M qubits or more.

[0054] The QPU300 is segmented into several key zones, namely, the storage zone 311, the entanglement zone 312, the read zone 313, the atomic load zone 304, and the remote entanglement zone 305. The storage zone 311, the entanglement zone 312, and the read zone 313 form the processor core 301, which in some embodiments has a footprint of 0.5 to 5 mm. 4 ~10 6It contains several qubits. Fresh atoms are continuously reloaded from the distant atom loading zone 304, and the distant remote entanglement zone 305 (using optical interconnects and / or lattice transport) delivers remote bell pair entanglement resources.

[0055] In storage zone 311, idle logical qubits are stored for extended periods, utilizing long qubit coherence times and high-fidelity single-qubit gates; therefore, error correction cycles are required only before logical two-qubit gates. Assuming a performance of one-tenth of the threshold for coherence times of 10-100 seconds, approximately 1% of single-qubit phase relaxation errors can be tolerated before a round of d-cycles of error correction. This corresponds to an allowable storage time of approximately 0.1-1 second before correction is required. Due to the all-to-all connectivity provided by the currently described architecture, idle logical qubits can simply be stored in a storage zone safe from additional errors. Logical qubits are thus stored in high-density blocks, shut out when needed by the algorithm, and error correction is performed only before two-qubit gates, significantly reducing error correction overhead. In various exemplary devices, atoms are stored in high-density storage zones at approximately 1 / (2μm) 2 It is stored at a density of approximately 1 / (10μm) in the active zone. 2 It is stored at this density.

[0056] Active logic qubits are operated in active zone 312. By utilizing qubit transport, all combinations of 2-qubit gates can be executed within a defined region of space. This significantly reduces the complexity of classical control. For example, all 2-qubit gates can be executed using a single global beam of light, which is dramatically simpler than dealing with each individual qubit. This extraordinary degree of parallelism in logic qubit control is a major advantage of this architecture compared to alternative methods, such as those involving individual control of atomic qubits.

[0057] The readout zone 313 allows for the selective reading of a subset of qubits midway through the circuit without disturbing other qubits. This readout occurs in parallel for the global beam and camera and again requires only one set of classical controls.

[0058] Outside the core processor 301, atoms are constantly reloaded from the load zone 304 and transported to the core processor to execute circuits of any length. Remote bell pairs with other processing units are generated using optical links and / or optical lattice transport 305 and shuttled to the core processor 301 to generate remote logic entanglement. This makes it possible to interconnect 10 to 100 single processing units into a single error-corrected, practical-scale quantum computer.

[0059] The architecture given above enables readout in the middle of the circuit. In particular, this architecture may be paired with high-speed imaging in the readout zone and a classical control loop. Furthermore, various methods may be used to suppress crosstalk errors and detect / correct losses. By continuous reloading of atoms and further suppression of crosstalk, circuit depths of arbitrary length may be achieved.

[0060] To connect multiple units, many high-fidelity, long-range bell pairs may be generated in parallel using grid transport and / or photonic links.

[0061] It will be understood that this architecture is suitable for preserving logic states through repeated measurement and correction in the middle of a circuit. Furthermore, surface code logic qubits may be implemented, for example, by moving auxiliary atoms out of the storage zone reservoir, entangling them with data qubits for syndrome extraction, and moving them to the read zone. This enables high-speed readout and feedback in the middle of a circuit while maintaining the coherence of the data qubits. In various embodiments, the data qubits are protected by positioning the imaging zone approximately 50 microns away, thereby suppressing crosstalk from the readout beam and scattered light from the auxiliary atoms.

[0062] In various embodiments, a fast classical control loop uses auxiliary measurements to determine errors in data qubits, detect qubit loss, and correct it. The lost qubit may then be replaced by a reservoir atom. To achieve a surface code distance several times greater than the maximum code produced in alternative systems, local detuning patterns may be utilized for the space-efficient use of the entanglement zone.

[0063] The architecture currently described may be used to execute algorithms using logic qubits. A zoning technique combined with parallel and efficient optical control of many logic qubits enables the construction of large-scale processors. In an exemplary use case, approximately 10 logic qubits are encoded in the active zone and moved to the storage zone. After all logic qubits have been encoded, the algorithm is executed with appropriate logic 1-qubit gates and logic 2-qubit gates. The flexible and localized 1-qubit control required for the logic 1-qubit gates is implemented with Raman light from a 2D AOD illuminating a grid of single-code blocks. The logic 2-qubit gates are implemented transversally in the entanglement zone. Intermediate readouts are used for a non-Clifford gate teleportation sequence and subsequent fast feedback for logic 1-qubit rotation.

[0064] While specific operating parameters are given below as examples, it will be understood that improvements in fidelity in 2-qubit gate errors may be achieved through various further optimizations. For example, increasing the output and detuning of the Rydberg laser reduces laser scattering errors and suppresses other errors by increasing the gate velocity. Cooling the atoms to the moving ground state (thus suppressing Doppler phase relaxation errors) and utilizing a 10-fold higher laser output theoretically yields a gate fidelity of over 99.8%. Further improvements can be made by continuously increasing the laser output, but alternative means such as single-photon excitation to the Rydberg P state or alkaline earth-based systems are also available. The processor speed can be increased to a logical qubit cycle time of approximately 10 microseconds by increasing the collection efficiency, by utilizing cavity-based or ensemble-based readout schemes, or by increasing the travel speed with deeper optical tweezers.

[0065] In order to reach circuits at any depth, the atoms may be continuously reloaded. Thus, some embodiments use loading into a distant magneto-optical trap (MOT) and transporting the atoms within an optical lattice conveyor belt.

[0066] In various embodiments, crosstalk during readout is suppressed by keeping auxiliary atoms away from the data qubits.

[0067] Further scaling of the quantum processor can be achieved by connecting two or more microscope objective lenses either by atomic transport or by an optical communication link. In various embodiments, the first approach utilizes a novel ability of atomic rearrangement combined with the use of an optical lattice conveyor belt for coherently transporting qubits between multiple active optical control regions and distributing entanglement. In various embodiments, the second approach utilizes photon-mediated entanglement between different atomic array nodes with more than 10 4 qubits. A high entanglement rate can be achieved by parallel nanophotonic or bulk optical cavities, and the large size of the atomic arrays can provide further parallelism. This approach also enables modular construction of quantum processor units that can be flexibly rewired and linked together.

[0068] Quantum error correction (QEC) is at the heart of fault-tolerant quantum computing (FTQC). A crucial component of FTQC is the error correction code, which explains how quantum information should be encoded in a redundant manner with the goal of reducing the error rate of computation. Quantum error correction is typically performed by measuring the Pauli operators (called stabilizer generators) of the QEC code (also called stabilizer code) to detect errors. An important subclass of stabilizer codes is the Calderbank-Shor-Steane (CSS) code, in which all non-identical components of the stabilizer generator are either all Pauli X operators or all Pauli Z operators. Quantum error correction in CSS codes works by measuring all stabilizer generators and applying corrections based on the observed results.

[0069] Low-density parity-check (LDPC) codes are a natural class of CSS codes to consider for implementation. LDPC codes are a family of stabilizer codes where every stabilizer generator acts on a fixed number of qubits, and every qubit involves a fixed number of generators. The codewords of parity-check codes are formed by combining blocks of binary information digits with blocks of check digits. Each check digit is the modulo 2 sum of a pre-specified set of information digits (i.e., a sum equal to 1 if the normal sum is odd, and equal to 0 if the normal sum is even). The rules for forming the check digits are represented by a parity check matrix H, where the columns of H represent binary information digits and the rows of H represent the check digits. Low-density parity-check codes are codes specified by a parity check matrix containing mostly 0s and relatively few 1s, such that the columns and rows have relatively small weights (i.e., sums across columns or rows). Therefore, the parity check matrix is ​​a sparse matrix.

[0070] LDPC codes have been very successful in classical contexts because they approach the upper limit of the amount of information that can be reliably transmitted through noisy channels. Many modern technologies, such as WiFi, DVB-T, and 5G, rely on LDPC codes for error correction. Their quantum generalization requires that an additional condition be met: that the X-check and Z-check are commutative. Families of such codes have been constructed. However, conventional methods for achieving quantum error correction, such as surface codes, are generally very costly in terms of resource overhead, requiring millions of qubits to solve the problem of interest.

[0071] Methods based on quantum low-density parity-check codes (qLDPCs) offer a promising avenue for reducing the resources required and potentially enabling quantum computation with constant spatial overhead. However, realizing qLDPC codes with better code parameters (number of qubits encoded, code distance) requires long-distance connectivity between qubits, which makes their physical implementation quite difficult. The required long-distance and multi-layer connectivity has not been demonstrated to date.

[0072] This disclosure demonstrates an implementation of qLDPC codes using a reconfigurable atomic array (RAA), the hardware architecture for quantum computing described above. The product structure present in many qLDPC codes harmonizes naturally with the parallelism provided by the acousto-optic deflector, a core technology of the RAA platform. When combined with a novel algorithm for performing arbitrary 1D qubit rearrangements in log(L) time (where L is the linear dimension of the system), while respecting the hardware constraints of current atomic shuttle technologies, this results in an efficient implementation of qLDPC codes within the scope of the experimental capabilities shown.

[0073] Referring to Figure 4, the product structure of the hypergraph product code is shown. The hypergraph product code is constructed from two classical LDPC codes. The classical codes are shown on the left and above, with circles representing data bits and squares representing check bits. Data qubits are located at each intersection of the two classical data bits (type DD, filled circles with crosses) and at each intersection of the two classical check bits (type CC, filled circles without crosses). Z-stabilizer checks are located at the intersections of horizontal data bits and vertical check bits, while X-stabilizer checks are located at the intersections of horizontal check bits and vertical data bits. Each stabilizer is connected to data bits along the same row and column with the same connectivity as the classical codes, as shown with respect to the Z check in the upper left. Other connections are omitted for ease of visualization.

[0074] Quantum error-correcting codes attempt to encode k logical qubits into a large number of physical qubits n. One particularly convenient way to achieve this is through stabilizer formalism, in which several stabilizer checks are applied to the physical qubits and the eigenvalues ​​of the product of Pauli operators over them are monitored. For simplicity, the following considerations focus on CSS codes, in which each stabilizer generator is either a product of X operators or a product of Z operators, as described above. Logical operators are operators that commute with all stabilizers but do not fall within the span of stabilizers. The minimum weight logical operator (i.e., the logical operator with the fewest non-identical elements) defines the code distance d, which provides a rough characteristic of the number of errors a given code can handle. Together, these parameters [[n, k, d]] provide a useful characterization of QEC codes.

[0075] With respect to quantum codes, all operations are performed using imperfect quantum gates. This is in contrast to the situation in classical communications, where the coding and decoding steps are nearly perfect and errors occur only during the communication itself. Therefore, for syndrome extraction to be fault-tolerant, it is likely necessary that all qubits are involved in a limited number of operations; in other words, a QEC code should be a low-density parity check code in which each stabilizer has a constant weight that does not increase with the code size, and each data qubit is involved in a fixed number of stabilizers.

[0076] Surface codes are an example of qLDPC codes. However, unlike many other such codes, a single surface code patch encodes only a single logical qubit, and therefore many patches—and thus significant overhead—are required to encode many logical qubits. In contrast, a further family of qLDPC codes can achieve a constant encoding rate, i.e., the ratio of logical qubits to physical qubits remains constant as the code size increases. Furthermore, a "asymptotically good" family of such codes is available in which both the number of encoded qubits and the code distance scale linearly with respect to the number of physical qubits, thereby enabling low-overhead quantum computing and significantly reducing resource costs compared to conventional methods.

[0077] Another example of a qLDPC code, which also forms the basis of subsequent codes, is the hypergraph product code (HGP). A hypergraph is a graph in which every edge connects to exactly two vertices, and therefore each edge has a cardinality of 2. Here, we start with two classical LDPC codes and construct a quantum code from the product of the two classical codes that inherits many of the properties of the classical codes. As shown in Figure 4, we can construct a hypergraph product starting with two classical LDPC codes arranged horizontally (401) and vertically (402), respectively. On the associated 2D grid (403), data qubits are placed at every intersection of data bits (filled circles with crosses) (e.g., data qubit 404 at the intersection of data bits 405 and 406) and at every intersection of check bits (filled circles without crosses) (e.g., data qubit 407 at the intersection of check bits 408 and 409). Z-stabilizer checks are placed at every intersection of the horizontal data bits and the vertical check bits (for example, stabilizer qubit 410 at the intersection of data bit 405 and check bit 408), while X-stabilizer checks are placed at every intersection of the horizontal check bits and the vertical data bits (for example, stabilizer qubit 411 at the intersection of check bit 409 and data bit 406). Along each row and column of the quantum code, the qubits are connected in the same way as their corresponding classical codes, as shown with respect to the upper left Z-check 412.

[0078] Logical operators are inherited from the underlying classical codes, and one can choose a basis such that each logical qubit is supported in a single row or column. Mathematically, the parity check matrices of the two underlying classical codes (where each row indicates the bits whose sum should be even if there are no errors)

[0079]

number

[0080] ,

[0081]

number

[0082] When expressed as such, the X and Z stabilizer check matrices of the HGP code can be written as follows:

[0083]

number

[0084] r i Classical [n] with linearly independent checks (i = 1, 2) i , k i d i For a given code, the resulting quantum code has parameters [[n1n2+ r1r2, k1k2, min{d1, d2}]]. Surface codes are a special case of hypergraph product codes where the classical code is a 1D repeating code. However, instead, k i = O(n i ), d i = O(n i By selecting a classical expander code as the underlying classical code, the resulting quantum code (known as a quantum expander code) encodes a linear number of logical qubits k = O(n),

[0085]

number

[0086] It has a scaling distance d, where d is the minimum number of qubits that must be touched to change one logical codeword to another. Distance d addresses the QEC code's ability to correct errors, so a code with distance d can correct any error affecting up to (d - 1) / 2 physical qubits. This is because an error E that is anticommutative with the stabilizer element M changes the codeword's eigenvalue from +1 to -1 with respect to M. Thus, measuring the eigenvalue of the stabilizer's origin yields a binary vector of length n-k called the error syndrome, which can be used to identify which error occurred. Due to the expansion properties of the underlying graph, quantum expander codes support single-shot quantum error correction, i.e., unlike surface codes, repeated rounds of syndrome extraction are not required to achieve fault tolerance.

[0087] While HGPs are not "asymptotically good" in the sense that their distances do not scale linearly with respect to the number of qubits, they form the basis for subsequent structures that achieve linear distances, where additional symmetry reductions are used to reduce the number of qubits required to achieve a given distance. Therefore, understanding the properties of HGPs and being able to implement them also provides a foundation for implementing more complex coding families.

[0088] Numerical simulations of HGP codes yield promising performance. (3,4) Using the hypergraph product between classical expander codes constructed from biregular graphs, several studies have investigated thresholds and logic error rates and found a promising circuit-level threshold of 0.28% and strong evidence that such codes can outperform surface codes as logic memory in moderate system sizes.

[0089] Achieving the desired coding parameters requires a certain number of long-distance connections. In this disclosure, a reconfigurable atomic array (RAA) is employed as a platform to provide this connectivity. In this method, qubits are encoded to long-lived hyperfine degrees of freedom of atoms or nuclear degrees of freedom with a coherence time exceeding 1 second. Raman transitions are used for one-qubit operations, and strongly interacting Rydberg states are used for two-qubit entanglement gates. Due to the blockaded nature of the Rydberg interaction, the gate action is not affected by the exact positions of the atoms and works whenever two atoms are adjacent under global gate laser irradiation.

[0090] By coherently shuttling atoms with optical tweezers, processor connectivity can be reconfigured on the fly, enabling parallel 2-qubit gate operation across the entire system. Importantly, optical tools such as acousto-optic deflectors (AODs) allow for rapid parallel movement of atoms across a grid with only a few classical controls per logical qubit, in contrast to common methods that require several classical controls per physical qubit. Arbitrary connectivity can be achieved with static spatial light modulator (SLM) optical tweezers capable of generating arbitrary trap patterns, and the ability to transfer between AOD traps and SLM traps.

[0091] The RAA platform features efficient parallel control and rearrangement of a large number of qubits, enabling the implementation of long-distance connected quantum processors. Furthermore, as described above, qLDPC codes with improved code parameters (number of qubits encoded, code distance) rely on randomized expander graphs, where the connectivity graph is much more complex. This disclosure shows that, despite these constraints, it is possible to efficiently implement the operations required for HGP codes at a rearrangement depth that scales only logarithmically with respect to the number of qubits involved. The key to this structure is the recognition that the product structure of HGP closely matches the product structure of current AOD hardware. Combined with a novel 1D parallel rearrangement scheme that achieves arbitrary sorting at logarithmic depth without atomic crossing, these techniques enable an immediate implementation of qLDPC codes.

[0092] All logical qubits are |0 L To fault-tolerantly prepare the state of an HGP code, such as the state of an HGP code initialized with |0>, first all physical qubits are prepared with |0>, and then all stabilizers are measured to project onto code space. The repeated measurements of all stabilizers and the performance of Pauli frame tracking enable the preservation of logical quantum information. Transversal readout of all physical qubits in a single basis enables fault-tolerant measurement of the logical state. Thus, most of the important operations are straightforward, with the non-trivial parts being stabilizer measurements.

[0093] Figures 5A to 5C illustrate syndrome extraction by sorting. Figure 5A shows the initial configuration of the qubits (maintaining the legend from Figure 4). Figure 5B shows the result of column sorting, and Figure 5C shows the result of subsequent row sorting. Due to the product structure of the hypergraph product codes, column (row) sorting is sufficient to perform all stabilizer measurements. The same sorting is applied in parallel to all columns (rows).

[0094] Referring to Figure 6, the syndrome extraction gate sequence is shown, where each row of qubits represents the configuration at a given time point, and the gray lines represent the transitions between configurations. As discussed in relation to Figures 5A-5C, each of the two dimensions of the HGP originates from classical coding and has the same connectivity (top). The desired syndrome extraction circuit is implemented by interleaving parallel qubit rearrangements with a global two-qubit gate laser pulse. Time progresses downwards on the page, and the gray lines represent the paths of data qubits and auxiliary qubits from one time step to the next.

[0095] As explained above, the stabilizers of the HGP code are inherited from the corresponding checks of the classical code. As shown in Figure 4, each stabilizer check is connected only to qubits in the same row or column. Horizontally (and vertically as well), the connectivity of the parity checks of the classical code is copied along all rows. In this way, by bringing together two columns that were connected in the original classical code, the necessary gates are performed between all pairs of qubits in those two columns. Thus, syndrome extraction first involves performing a parallel column sort to rearrange atoms into pairs, each pair containing a single data bit and check bit connected in the horizontal classical code. By using the coloration circuit described, the number of sorting layers required is equal to the weight of the largest stabilizer in the classical code. As shown in Figure 6, each rearrangement layer is interleaved with a global Rydberg laser pulse that performs CZ gates between each pair of adjacent atoms. The same procedure is repeated vertically, and rows are sorted to make all the necessary vertical connections, thus completing one round of syndrome extraction.

[0096] To ensure that the correct stabilizers are extracted, the relative order of gates involved in X and Z checks, which involve two shared data qubits, must satisfy the following condition: the X check should interact with both qubits before the Z check, or with both qubits after the Z check. This condition cannot be satisfied when X and Z syndromes of the same cycle are extracted simultaneously, in which case complete syndrome extraction would require two complete cycles of row and column rearrangement. However, syndrome extraction can be shifted so that one round of X syndromes is extracted, and then the Z syndrome of the current round is extracted simultaneously with the X syndrome of the next round. This ensures that the relative order is satisfied and therefore the syndrome extraction schedule is valid.

[0097] Referring to Figure 7, an efficient non-intersecting rearrangement at logarithmic depth is shown. By using a divide-and-conquer algorithm, any 1D rearrangement is performed at a depth logarithmic of the number of qubits. Repeating this across the entire array results in the efficient implementation of the desired rearrangement without requiring intersecting atomic orbitals, which can lead to further loss and decoherence. An initial array is prepared in the workspace to obtain a set of rearrangement steps to move from the initial array 701 to the final configuration 702 (703). The qubits are divided into a left subset and a right subset (704). A recursive sort is performed on the subsets (705). The qubits are then moved to their final positions (706).

[0098] While respecting the hardware constraints of AOD-based atomic shuttles, arbitrary row or column rearrangement can be efficiently performed. One constraint is that different tones of the AOD, which produce different optical tweezers beams, cannot be allowed to intersect during movement. This is due to frequency beating, which occurs when the two tones approach each other and can heat atoms, as well as the possibility of collisions between atoms. Therefore, in order to achieve the rearrangement required for syndrome extraction, it is necessary to develop an efficient non-intersecting atomic rearrangement scheme.

[0099] A divide-and-conquer algorithm is provided herein that decomposes any one-dimensional sort into logarithmic layers, each layer consisting of non-intersecting atom movements that can be executed in parallel. As shown in Figure 7, the goal is to place all L atoms whose final position lies in the right half of the system on the correct side. This is achieved by first moving all atoms to the leftmost available static SLM trap location in the workspace (the first layer in Figure 6), and then moving all atoms that will ultimately end up in the right half to the rightmost L / 2 static traps in the system. The same procedure is applied recursively to the left and right halves of the system, as shown in the middle of Figure 7, until the desired order of atoms is achieved. The final parallel AOD movements transport the atoms to their desired locations for gate operations.

[0100] Since the system size is halved every two layers, the total number of layers required to achieve the desired rearrangement is O(log L). Therefore, any rearrangement in a very large system can be achieved with a small number of layers. This method has some similarities to techniques such as bitonic sort, but different constraints (comparator vs. parallel movement) lead to differences in the algorithm itself.

[0101] With respect to the above-described method, the time required to perform one round of stabilizer measurement using the indicated technology can be estimated. This is based on the following assumption: the transfer time τ between the static SLM trap and the dynamic AOD trap. t , peak atomic transport acceleration a due to cubic spline trajectory p A uniform grid spacing d is used. For simplicity, the number of atoms on the line being rearranged is a power of 2, and L = 2 k It is further assumed that the total number of traps is 3L / 2 to provide sufficient working space for shutting.

[0102] The compactification step of scale s requires a movement of a maximum distance of sd / 2. Moving all target atoms to the right requires a movement of a maximum distance of sd. The two steps can be combined so that atoms are picked up for the next move and atoms are dropped from the previous move at the same time, and thus each layer requires an average of one trap transfer between static and dynamic traps. Using a cubic spline movement trajectory, a movement of distance l takes time

[0103]

number

[0104] It is required.

[0105] Therefore, the total time required for one layer of complete rearrangement is

[0106]

number

[0107] That is the case.

[0108] Recent experiments have shown that τ t = 50 μs,

[0109]

number

[0110] The parameters were given on the order of d = 5 μm. For a medium-sized code consisting of 10,000 qubits (including both data qubits and auxiliary qubits), L ≈ 100. For each gate layer, the total trap transfer time was 0.7 ms and the atom transfer time was 2.3 ms. Assuming a [3,4] biregular graph for the underlying classical expander code, eight rounds of rearrangement are required to measure one complete round of the stabilizer, resulting in a total time overhead of 24 ms, which is only a small fraction of the coherence time of 2 s shown for the neutral atom array. Although this time scale is somewhat longer than the typical readout time scale, the HGP code is single-shot and therefore only one round of stabilizer measurement is required to perform error correction.

[0111] This disclosure provides techniques for the efficient implementation of quantum low-density parity check codes in reconfigurable atomic arrays. These methods leverage the inherent parallelism of existing optical tools and the product structure of many code constructions to enable their efficient implementation on existing hardware.

[0112] These techniques are applicable to other coding families that introduce additional elements on top of hypergraph product structures. Furthermore, these schemes can be readily extended to implementations of lattice surgery gates for qLDPC codes, where the auxiliary code patch can be considered a hypergraph product code between a subsection of the original code and a repeating code. This disclosure enables an immediate implementation of qLDPC codes and significantly reduces the resources required for large-scale fault-tolerant quantum computing.

[0113] Refer to Figures 5A to 8AA, which illustrate an example of a one-dimensional sort. In particular, each set of three consecutive images (e.g., Figures 8A, 8B, and 8C) shows the beginning, midpoint, and end of each step of the sort, with the moving qubits highlighted by vertical dashed lines.

[0114] Referring to Figures 9 to 11, additional algorithms for arbitrary 2D atomic rearrangements are shown. Figure 9 shows 2 k The parallel execution of swap operations between atoms separated by only a few units is shown. Figure 10 shows the graph G×H generated by the product of two subgraphs G and H. Figure 11 shows the use of a fixed beam (square) to prevent movement in some columns. By irradiating with a strong localized light spot, some atoms are prevented from moving under the parallel moving beam, allowing different rows to perform different sorting operations.

[0115] The algorithm includes three key components. Firstly, the algorithm uses the number of extra workspace traps, which is a constant multiple of the total number of traps, to determine the distance 2 k The first method involves a technique for swapping all pairs of qubits separated by a certain distance, where k is an integer. Secondly, the algorithm uses the concept of an extra-fast switchable optical trap as a fixed beam to enable different behavior for different rows or columns. Thirdly, the algorithm uses a routing algorithm that decomposes routing problems on graphs with product structures into three steps on individual components.

[0116] distance 2 k Pairs of qubits separated by only a certain distance are swapped. k By shining tweezers light on half of an atom corresponding to one atom in each pair of qubits separated by a certain distance, one sublattice is picked up and moved to a temporary buffer location. Then, the other half of the atom is picked up and moved to the original position of the first sublattice. Finally, moving the first sublattice from the temporary buffer location to the original position of the second sublattice completes the swap operation.

[0117] This process is shown in Figure 9, and in step 901, 2 1Atoms separated by the position of are swapped. In step 902, 2 2 Atoms separated by that position are swapped.

[0118] As described above, routing algorithms are used that decompose routing problems on product graphs into routing problems on subgraphs. An example of such algorithms known in the art is described in Baumslag, M., Annexstein, F., A unified framework for off-line permutation routing in parallel networks, Math. Systems Theory 24, 233-251 (1991), https: / / doi.org / 10.1007 / BF02090401, which is incorporated herein by reference. However, it will be understood that various alternative algorithms may be used, provided they meet the above-mentioned criteria.

[0119] To perform routing on the product graph G×H (Figure 10), routing is performed in each row based on the routing algorithm on graph G, then in each column based on the routing algorithm on graph H, and finally, another round of row routing based on graph G is performed. A key difference from the more constrained movements of the hypergraph product code is that here, different rows or columns must perform different movements.

[0120] Therefore, to achieve different motions in different columns or rows, localized fixed beams that can be quickly switched on and off are used. As shown in Figure 11, by irradiating selected atoms with a fixed beam (shown as a square, e.g., 1101), it is possible to prevent certain atoms from moving while still applying the same overall motion to all rows or columns. The fixed beams may be implemented, for example, by a digital mirror device (DMD) that can be switched on a very fast timescale.

[0121] Comparing Figure 9 and Figure 11, it can be seen that fixing allows for swapping to be performed only on a selective set of qubit sites.

[0122] Combining all these elements provides a rearrangement algorithm that achieves arbitrary 2D atomic rearrangements in depth O(log N), where N is the total number of qubits. First, the 2D atomic rearrangement is decomposed into row, column, and row sorts using the product routing algorithm described above. Then, the sorts within each row or column are given a distance of 2, where k = 1, 2, ..., log L. k This swap is performed by combining it with a localized fixed beam.

[0123] Any 1D rearrangement can be performed by O(log L) swaps by treating it as a routing problem on a hypercube and recursively applying the product routing algorithm. More specifically, 2 k Routing on a qubit is 2 k Two qubits k-1 It is considered as the product of two graphs, one with 1 qubits and the other with 2 qubits, then routed on the graph with 2 qubits, and then 2 k-1 This can be achieved by routing on a graph of qubits and then, again, on a graph of two qubits. Local fixed beams allow different sorting to be performed on different rows and / or columns. Thus, in some embodiments, the sorting is specific to a row or column rather than universal across all rows or columns.

[0124] The above explanation uses a routing algorithm, cited above and described by Baumslag et al., which decomposes a routing problem on a product graph into routing problems on subgraphs; however, alternative routing algorithms may be used. For example, a bitonic sort-based algorithm, such as the one described by Litinski, D., Nickerson, N., Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections, arXiv:2211.15465 [quant-ph] (2022), https: / / doi.org / 10.48550 / arXiv.2211.15465, which is incorporated herein by reference. This bitonic-based sorting algorithm has a time complexity of O(log 2 It has n).

[0125] Unlike bitonic sort networks, which require bitwise comparisons to determine where each element should go, the algorithms described herein improve this complexity to O(log n) by taking advantage of the fact that the use cases described herein have prior information about the target location.

[0126] In addition to the HGP codes described in other examples, this disclosure may be used to implement lift-product (LP) codes.

[0127] LP codes are a modified version of HGP codes. Quasi-cyclic LP codes are quotient polynomial rings.

[0128]

number

[0129] The two fundamental matrices B1 and B2 above are obtained from two fundamental protographs. The two fundamental matrices are, respectively,

[0130]

number

[0131] and

[0132]

number

[0133] Assume that the two matrices B (on the same polynomial ring) are obtained by taking the hypergraph product between B1 and B2. x and B z You can obtain this.

[0134]

number

[0135] Then there is a process known as lift, B x (B z By replacing each component of ) with its matrix representation using an l×l circulant matrix, the X(Z) check matrix H x (H z ) is obtained. The code size is,

[0136]

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[0137] The number of X tests and Z tests are, respectively,

[0138]

number

[0139] and

[0140]

number

[0141] H x and H z Assuming that is full rank, this results in a [[N, N - M, d]]-LP code.

[0142] The above construction can also be explained using graphs. As an example, Figure 12 shows an LP code using a 3x5 protograph and a lift size of 2. The LP code is constructed by taking the lift over the hypergraph product of two classical protographs. The protographs and their hypergraph product are represented by dashed nodes (e.g., 1201), and the lift is represented by multiple internal nodes (e.g., 1202, 1203) within each dashed node. The internal connectivity between two dashed nodes is given by the matrix representation of the elements of the ring in Equation 3. When the internal nodes are flattened vertically (horizontally), the vertical (horizontal) connectivity between the qubits and checks in each column (row) is the same as that of the lifted classical code left 1204 (up 1205).

[0143] B ij If (B := B1= B2) is non-zero, the i-th dashed check node is connected to the j-th dashed variable node. The protograph lift replaces each dashed node with its two solid internal nodes, and each ring element B ij This is done by defining the connectivity between internal nodes according to the matrix representation of . Equation 3 corresponds to taking the hypergraph product between the protograph and itself, obtaining a grid of dashed nodes. Similar to the hypergraph product sign, the dashed nodes (B x and B z The connectivity between the components of B1 and B2 is inherited from B1 and B2. The qubits and quantum checks are given by the internal nodes after the lift, and their connectivity is inherited from Bx and B z This is given by the matrix representation. An important feature of LP codes is that they still retain some remaining product structure even after the lift. As shown in Figure 12, when the internal nodes are flattened vertically (horizontally), the vertical (horizontal) connectivity between the qubits and checks in each column (row) is the same as that of the lifted classical code on the left (top).

[0144] With respect to matrix components on a polynomial ring, their weights are denoted by the number of terms. For the LP codes constructed in this disclosure, a 3x5-dimensional fundamental matrix with all components of weight 1 is selected, and a family of codes up to 1428 in size can be obtained by increasing the shift-lift size l. The constructed classical parity check is optimized by selecting the fundamental matrix components on the quotient polynomial ring to obtain the best minimum distance for a given shift-lift size l. The selection of fundamental matrix components is also such that the girth (length of the shortest cycle in the Tanner graph) is at least 8, and the minimum distance of the lift-product qLDPC code is the same as the designed classical minimum distance. Note that the selection of the fundamental weight matrix determines the limit of the best possible minimum distance for such code construction. Allowing multiple weights and more general protographs provides greater flexibility in the design of qLDPC codes and improves their minimum distances.

[0145] Referring again to Equation 2, the time overhead of the LP code can be estimated as follows:

[0146] For LP codes, the code layout must first be flattened before implementing the parallel rearrangement scheme. For a given 3x5 protograph, the flattened rectangular array has dimensions of 2n / 8x8. This can be achieved at logarithmic depth using a divide-and-conquer algorithm that flattens the code by half each time. For example, as shown in Figure 12, the code is flattened vertically before establishing vertical connectivity between atoms by rearranging the rows. At that point, the vertical connectivity is the same as for HGP codes, and one can use the efficient 1D rearrangement scheme described above. Thus, the rearrangement time for LP codes is estimated by setting L in Equation 2 to n / 8.

[0147] The rearrangement time in Equation 2 determines the idling error between sequences of entangled gates in the syndrome extraction circuit. Generally, since L is a function of n, t rearrange (n) is a function of the sign size n. Gate error p g When this improves, the idling error p i (n) can also be improved.

[0148]

number

[0149] Assuming this, the coherence time T c and t rearrange The other constant parameters of (n) are listed above.

[0150] Exemplary Method code construction The following sections focus on two families of qLDPC codes, but it will be understood that these results may be asymptotically extended to better codes.

[0151] The first family of codes is the hypergraph product (HGP) code, which is formed from the product of two classical LDPC codes. A geometric outline of the code properties is given above, and the following discussion focuses on alternative algebraic explanations of the codes, providing further details of their code properties. Algebraically, the parity check matrices of the two underlying classical codes (where the rows indicate bits whose sum should be even in the absence of errors)

[0152]

number

[0153] ,

[0154]

number

[0155] Given as follows, the X and Z stabilizer check matrices of the HGP code can be written as follows:

[0156]

number

[0157] r i = n i - k i Classical [n] defined by a set of linearly independent checks (i = 1, 2). i , k i , d i For a linear code, the resulting quantum code has parameters [[n1n2+ r1r2, k1k2, min{d1, d2}]]. Surface codes are a special case of hypergraph product codes where the classical code is a 1D repeating code. However, instead, k i =Θ(n i ), d i =Θ(n iBy selecting a classical code with a good vertex expansion as the underlying classical code, the resulting quantum code (known as a quantum expander code) encodes a linear number of logical qubits k = Θ(n) and distance

[0158]

number

[0159] Such classical expander codes can be asymptotically obtained, for example, from random biholomorphic Tanner graphs and have a high probability of having a sufficient vertex expansion. Logical operators are inherited from the underlying classical codes, and one can choose a basis such that each logical qubit is supported in only a single row or column.

[0160] In this disclosure, HGP codes are constructed by taking the hypergraph product of a classical LDPC code defined by a (3,4) regular Tanner graph, i.e., a bipartite graph having bit nodes of order 3 and check nodes of order 4. By increasing the size of the graph, a constant coding rate can be achieved.

[0161]

number

[0162] A family of HGP codes with the following characteristics is obtained. For each code size, a classical code is selected from randomly generated instances that has the largest distance, the inner diameter of the Tanner graph greater than 6 (the length of the shortest cycle in the Tanner graph obtained by rejection sampling without performing edge swaps), and the largest spectral gap (the gap between the two largest singular values ​​in the check matrix). The hypergraph product of vertex-expanding classical codes yields an HGP code that satisfies the syndrome confinement property and supports single-shot QEC.

[0163] The second family of codes under consideration is the quasi-cyclic lift product (LP) code, which can be considered a hypergraph product code followed by symmetry contraction to reduce the number of qubits required. Algebraically, the quasi-cyclic LP code is a quotient polynomial ring

[0164]

number

[0165] The two fundamental matrices B1 and B2 above are derived from two fundamental protographs (an analogue of classical coding in HGP construction). The two fundamental matrices are, respectively, size

[0166]

number

[0167] and

[0168]

number

[0169] Assume that the two matrices B x and B zHowever, it can be obtained (on the same polynomial ring) by taking the hypergraph product.

[0170]

number

[0171] Then there is a process known as lifting, B x (B z By replacing each component of ) with its matrix representation as an l×l cyclic matrix, the X(Z) check matrix H x (H z ) is obtained. The code size is,

[0172]

number

[0173] The number of X tests and Z tests are, respectively,

[0174]

number

[0175] and

[0176]

number

[0177] The coding rate is

[0178]

number

[0179] The lower limit can be determined by this.

[0180] The above construction can also be illustrated using graphs. For example, Figure 12 shows the elementary matrices.

[0181]

number

[0182] The LP code using the related 3x5 protograph is shown. Protograph checks and bits are indicated by large dashed node. ij If is non-zero, the i-th dashed check node is connected to the j-th dashed bit node. The protograph lift replaces each dashed node with its two solid internal nodes, and each ring element B ij This is done by defining the connectivity between internal nodes according to the matrix representation of . Equation 7 corresponds to taking the hypergraph product between the protograph and itself, obtaining a grid of dashed nodes. Similar to the hypergraph product sign, the dashed nodes (B x and B z The connectivity between the components of is inherited from B. The qubits and quantum checks are given by the internal nodes after the lift, and their connectivity is inherited from B. x and B z This is given by the matrix representation. An important feature of LP codes is that they still retain some remaining product structure even after the lift. As shown in Figure 12, when the internal nodes are flattened vertically (horizontally), the vertical (horizontal) connectivity between the qubits and checks in each column (row) is the same as that of the lifted classical code on the left (top).

[0183] For the LP codes constructed in this disclosure, a 3x5 basis matrix is ​​selected, all components having a single polynomial term, and by increasing the lift size l from 16 to 42, a family of codes with sizes up to 1428 is obtained. Classical parity checking is optimized by selecting the fundamental matrix components on the quotient polynomial ring to obtain the best classical distance for a given lift size l. The selection of fundamental matrix components is such that the inner diameter is at least 8, and the distance of the lifted qLDPC code matches the designed classical distance with high probability. Allowing multiple polynomial terms in each fundamental matrix component and thus allowing more protographs of different sizes provides greater flexibility in the design of qLDPC codes and improves their distances. The classical fundamental matrices used to construct the four LP codes used in this example are given below.

[0184]

number

[0185] If we denote the lift size l and the classical code distance d after the lift as fundamental matrices, then the fundamental matrices are as follows:

[0186]

number

[0187] The quantum code distance is bounded by the classical code distance of the (lifted) fundamental matrices above. After extensive searching for minimum-weight logical operators using the GAP package, these bounds appear to be strict.

[0188] Atomic rearrangement algorithm Reconfigurable atomic array platforms feature efficient parallel control and rearrangement of multiple qubits, enabling the implementation of long-distance connected quantum processors. As considered herein, optical tools such as acousto-optic deflectors (AODs) can generate a rectangular grid of optical tweezers that can be reconfigured on the fly, enabling the control of large code blocks consisting of thousands of physical qubits with only minimal classical control.

[0189] However, the use of AODs for dynamic rearrangement comes with two significant limitations. First, since the optical spots in the X and Y directions are controlled by separate AODs, the same operation needs to be applied across multiple rows and / or columns. Second, different rows of atoms can be temporarily transferred and stored in static traps such as those based on spatial light modulators (SLMs), but they cannot cross each other due to beating between RF tones and atomic collisions. Therefore, the implementation of qLDPC codes with improved code parameters (number of qubits encoded, code distance), which often rely on pseudorandom expander graphs with complex connectivity graphs, requires the development of efficient atomic rearrangement algorithms.

[0190] Figures 13 to 15 provide a detailed description of the atomic rearrangement algorithms in the form of Algorithms 1 to 3.

[0191] The first component, any 1D atomic rearrangement with logarithmically scaling number of steps, is illustrated in FIGS. 16A to 16D and is described in detail in Algorithm 1, which clearly succeeded for the small example in FIG. 7. Since consecutive layers each halve the system size, the total number of layers required to achieve the desired rearrangement is

[0192]

Number

[0193] Therefore, any rearrangement in a very large system can be achieved with a small number of layers. This method has some similarities to techniques such as bitonic sort, but different constraints (comparator vs. parallel movement) lead to differences in the algorithm itself. The algorithm can also be applied to use parallel qubit swapping over increasing distance as a basic primitive, with the same O(log L) scaling as the system size.

[0194] Figures 16A–16D illustrate an efficient implementation of a quantum LDPC code using an atomic array. Figure 16A is a diagram of an algorithm for performing rearrangement at an arbitrary logarithmic depth. First, all atoms that need to end up in the right half of the system are moved to the right, and then each half is packed into adjacent sites so that there is enough workspace for the subsequent steps. The same procedure can then be repeated recursively for each half of the system by a depth of log(L), where L is the length of the atomic array being rearranged. Figure 16B is a diagram of an HGP code obtained as the product of two classical codes. The lines indicate that parity checks at the syndrome node include the corresponding data node. Figures 16C–16D illustrate the required connectivity, which is implemented by parallel row sorting followed by parallel column sorting.

[0195] The second component is the observation that the product structure of crossed AOD closely matches the product structure present in many qLDPC codes. Based on this observation, Algorithm 2 (Figure 14) provides details of a syndrome extraction circuit for HGP codes, which is called a product coloration circuit because it utilizes a coloration circuit for each of the classical codes of the components. The use of a product coloration circuit is necessary to fully utilize the parallel rearrangement capability across rows and columns, as opposed to alternative coloration circuits or cardinal circuits. Here, the native entanglement gate set of the current atomic array system is diagonal, and therefore, CZ gates and appropriate Hadamard rotations are used to perform syndrome extraction. Under global laser excitation and phase advance, all pairs of qubits within a certain radius of each other (called the blockade radius) undergo a CZ gate, while all individual qubits undergo an identity gate. To analyze these results, the simulation uses CNOT gates as entanglement gates. If the CZ gate is much noisier than the Hadamard gate, this can be physically justified.

[0196] The multiplicative coloration circuit extracts the X and Z syndromes separately, each requiring steps in both the horizontal and vertical directions. Therefore, each coloration of the classical code is Δ C If color is included (with respect to the signs constructed from the (3,4) binormal graphs considered herein, Δ C = 4), the multiplier coloring circuit is 4Δ C It has an entangled layer.

[0197] The product coloring circuit can also be applied to LP codes used herein. As shown in Figure 12, LP codes have the same vertical (horizontal) product connectivity as HGP codes when the internal nodes are flattened vertically (horizontally). Therefore, the same product coloring circuit can be applied to LP codes, provided that there is an additional step of flattening the internal codes during the establishment of the horizontal / vertical connectivity. When a 3×5 basic matrix with all components of weight 1 is used, the product coloring circuit for LP codes has an entangling gate depth of 4×5 = 20.

[0198] To further reduce the depth of the syndrome extraction circuit, algorithm 3 (Figure 15) and Figures 17A-17B provide a modification of the above circuit, which is called a pipelining product coloring circuit. Here, the main challenge is to select the order of gates such that the desired X and Z syndromes are correctly extracted. By performing pipelining and extracting the second round of X syndromes simultaneously with the first round of Z syndromes, the number of entangled layers required to perform d rounds of syndrome measurement is (2d + 2)Δ C While reducing the number of steps, it is possible to ensure that the order of the gates is always valid. This may be particularly relevant to further suppressing the effects of idling errors and improving the execution of logic gates that would otherwise require d rounds of iteration in this scheme.

[0199] Referring to Figures 17A and 17B, the sequence of operations in pipelining syndrome extraction is shown. Figure 17A shows the sequence of entanglement gate steps for the pipelining product coloring circuit described in Algorithm 3, using syndrome extraction with d = 3 rounds. The corner numbers of the X and Z auxiliary qubits indicate the rounds of syndrome extraction they correspond to. Figure 17B is a diagram of the local circuit seen by the data qubit and auxiliary qubit in the same round, with dashed lines indicating moments in different circuits. The syndrome extraction sequence is valid because the X stabilizer interacts with both qubits before the Z stabilizer. A similar analysis can be performed with respect to the commutative relationship with the auxiliary qubit in the next round.

[0200] Dynamic Reconstruction in 2D Tweezers Arrays An exemplary experiment utilizes the apparatus described below. In a vacuum cell, 87 Rb atoms are loaded from a magneto-optical trap into a backbone array of programmable optical tweezers generated by a spatial light modulator (SLM). The atoms are then rearranged in parallel to their defect-free target positions within this SLM backbone by additional optical tweezers generated from a crossed 2D acousto-optic deflector (AOD). After the rearrangement procedure, selected atoms are returned from the static SLM trap to a movable AOD trap, and these movable atoms are then moved to their starting positions within the quantum circuit. Throughout this entire process, the atoms are cooled by polarization gradient cooling. Before running the quantum circuit, camera images of the atoms at their initial starting positions are taken. After the circuit, final camera images are taken to detect the qubit states |0> (presence of atoms) and |1> (loss of atoms after resonant pushout). All data are post-selected by finding the complete rearrangement of AOD and SLM atoms before running the circuit. In some embodiments, each atom remains in a single static trap or a single mobile trap for the entire duration of the quantum circuit.

[0201] The crossed AOD system consists of two independently controlled AODs (AA Opto Electronic DTSX-400) for x and y control of the beam position. Both AODs are driven by independent arbitrary waveforms generated by a dual-channel arbitrary waveform generator (AWG) (Spectrum Instrumentation M4i.6631-x8) and then amplified by independent MW amplifiers (Minicircuits ZHL-5W-1). The time-domain arbitrary waveform consists of multiple frequency tones corresponding to the x and y positions of columns and rows, which are independently modified as a function of time to dynamically steering atoms trapped in the AODs. The complete x and y waveforms are calculated by summing the time-domain profiles of all frequency components, each with a given amplitude and phase. To execute the quantum circuit, the positions of the AOD atoms at each gate location are programmed, and the AOD frequencies are smoothly interpolated as a function of time between gate positions (in a third-order profile). The third-order profile imposes a constant jerk on the atoms, which allows for migration approximately 5 to 10 times faster (without heating and loss) than migration at a constant velocity (first-order profile). The migration protocol applies stretching, compression, and translation of the AOD trap array, meaning that the rows and columns of the AOD never intersect with each other to avoid atomic loss and heating associated with the intersection of two frequency components.

[0202] The AOD tweezers intensity is homogenized throughout the atomic trajectory to minimize phase relaxation induced by the magnitude of the differential optical shift, which changes over time. For this purpose, the intensity of each AOD tweezers is measured at each gate location, and a reference camera in the image plane is used to homogenize them by varying the amplitude of each frequency component, with the amplitude of each individual frequency component interpolated during the movement between the two locations.

[0203] The SLM tweezers beam (830nm) and AOD tweezers beam (828nm) are generated by two separate free-running titanium-sapphire lasers (M Squared, 18-W pump). When projected through a 0.5NA objective lens, the SLM tweezers beam has a waist of approximately 900nm (approximately 1000nm for the AOD beam). When loading atoms, the trap depth is approximately 2π × 16MHz and the radial trap frequency is approximately 2π × 80kHz; when running quantum circuits, the trap depth is approximately 2π × 4MHz and the radial trap frequency is approximately 2π × 40kHz.

[0204] Raman laser system High-speed, high-fidelity 1-qubit operation is a crucial element of the quantum circuit demonstrated in this study. For this purpose, a high-power 795nm Raman laser system was used. F It is used to drive a global 1-qubit rotation between clock states of = 0. This Raman laser system is based on dispersive optics. 795 nm light (Toptica TA pro, 1.8 W) is phase-modulated by an electro-optic modulator (Qubig) driven by a 3.4 GHz microwave (Stanford Research Systems SRS SG384), which is doubled to 6.8 GHz and amplified. The laser phase modulation is converted to amplitude modulation to drive the Raman transition using a chirp Bragg grating (Optigrate). The IQ control of the SG384 is imprinted on the laser amplitude modulation and is therefore used for controlling the frequency and phase of the microwave, giving direct frequency and phase control of the ultrafine qubit drive.

[0205] The Raman laser illuminates the atomic plane from the side with a circularly polarized elliptical beam with waists of 40 μm and 560 μm in the thin axis and tall axis, respectively, with a total average optical power of 150 mW per atom. The large vertical spread ensures less than 1% non-uniformity across atoms, and the shot-to-shot fluctuation of laser intensity is also less than 1%. The Raman laser is operated with blue-detuned intermediate-state detuning at 180 GHz, a two-photon Raman frequency of 1 MHz, and 7 × 10⁻⁶ -5 This results in an estimated scattering error per π pulse (i.e., one scattering event every 15,000 π pulses).

[0206] Qubit coherence and dynamic decoupling In an 830nm trap, the coherence of the ultrafine qubits is T2 * The intervals are characterized by = 4 ms (not plotted here), T2 = 1.5 s (XY16 using a total of 128 π pulses), and T1 = 4 s (including atomic loss). The experiments described herein are performed in a DC magnetic field of 8.5 Gauss. Coherence is further detuned to optical tweezers (with the trap depth kept constant, the difference spectral shift of the tweezers decreases as 1 / Δ, and 1 / T1 is 1 / Δ 3 This can be further improved by using shielding against magnetic field fluctuations (which reduces the amount of atoms lost). For practical QEC operation, it is possible to detect atomic loss in a hardware-efficient manner, and then the atoms can be replaced from the reservoir, which can be reloaded in principle by the MOT to reach any depth of circuit.

[0207] The transport sequence involves a dynamic decoupling sequence. The number of pulses used is a trade-off between maintaining qubit coherence and minimizing pulse errors. In various embodiments, there is an alternation between two types of dynamic decoupling sequences: the XY8 / XY16 sequence composed of individual π pulses with alternating phases that self-correct amplitude errors and detuning errors, and the CPMG-type dynamic decoupling sequence composed of robust BB1 pulses. The CPMG-BB1 sequence is more robust to amplitude errors but incurs additional scattering errors. By selecting from these different sequences and variable decoupling π pulses, the sequence may be empirically optimized for any given experiment to optimize either single qubit coherence (including movement) or the final signal. Typically, the decoupling sequence is composed of a total of 12 to 18 π pulses.

[0208] Effect of movement on atomic heating and loss Hereinafter, the effect of movement on atomic loss and heating in the harmonic oscillator potential provided by the tweezer trap will be considered. The motion of the trap potential is equivalent to a non-inertial reference frame in which the harmonic oscillator potential is stationary but the atom experiences an apparent force given by F(t) = -ma(t), where m is the mass of the particle and a(t) is the acceleration of the trap as a function of time. The average vibrational quantum number increase ΔN is

[0209]

Eq.

[0210] given by

[0211]

Eq.

[0212] This is the Fourier transform of a(t) evaluated at the trap frequency ω0, and represents the zero-point size of the particle.

[0213]

number

[0214] Therefore, ΔN is the same for all initial levels of the oscillator. Experimentally, an acceleration profile a(t) = jt is applied to an atom from time -T / 2 to +T / 2 to move a distance D with a constant jerk j.

[0215]

number

[0216] Calculating this, simplifying it using ω0T≫1, and assuming a small range of trap frequencies for averaging the oscillation term,

[0217]

number

[0218] It brings about.

[0219] Several important insights can be gleaned from this equation. First, it demonstrates the ability to travel a long distance D with a relatively small increase in time T. Furthermore, in order to maintain a constant ΔN, the travel time T ∝ ω⁰ -3 / 4 Furthermore, in order to perform a large number of movements k for deep circuits, ΔN∝k / T 4 It is possible to estimate that the number of moves can be increased from 5 to 80, for example, by slowing each move from 200 μs to 400 μs. Movement speeds may be further increased by different a(t) profiles, but the quantum speed limit, inevitably imposed by finite resources such as trap depth, ultimately prevents the qubits in the array from moving indefinitely fast.

[0220] Next, equation 11 is compared with experimental observations. Atomic loss is observed with a movement of 55 μm in 200 μs under a constant negative jerk. This rate limit is consistent with the above estimates, ω0 = 2π × 40 kHz and x zpf Using 38 nm, ΔN ≈ 6 is predicted for this migration, corresponding to a clear onset of heating at this migration rate. More quantitatively, a Poisson distribution of mean N and variance N is assumed, and some critical N at which the atom leaves the trap. max Integrate the population exceeding this value. From this analysis, the retention of atoms is

[0221]

number

[0222] It is given by.

[0223] Additional heating and loss in the circuit can also be caused by repeated short drops to execute the 2-qubit gate, where the tweezers are briefly turned off to avoid anti-trapping of the Rydberg state and optical shifts in the ground-Rydberg transition. However, drop-re-trapping measurements suggest that the 500 ns drops used experimentally have a negligible effect up to several hundred drops per atom (corresponding to several hundred CZ gates). Atomic loss and heating as a function of the number of drops are well explained by the diffusion model, and the diffusion model reduces the temperature of the atom by half (thermal motion rate)

[0224]

number

[0225] (reducing by 1 / 2) and drop time t drop Reducing this by half is predicted to increase the number of possible CZ gates per atom to several thousand.

[0226] Implementation of a 2 - qubit CZ gate Two - qubit gates and calibrations may be implemented using the techniques provided herein. In particular, a 2 - qubit CZ gate is implemented by two global Rydberg pulses, each pulse having detuning Δ and length τ, and there is a phase jump ξ between the two pulses. The parameters of the pulses are chosen such that a pair of qubits that are close and under the Rydberg blockade constraint return from the Rydberg state to the ultrafine qubit manifold such that the phase depends on the state of the other qubit. The numerical values of these pulse parameters are as follows. Δ = - 0.377371Ω ξ = - 0.621089×(2π) τ = 0.683201 / [Ω / (2π)]

[0227] Exemplary experiments are performed at a two - photon Rydberg laser frequency of Ω / 2π = 3.6MHz which gives a theoretical τ = 190ns and a theoretical Δ / (2π)= - 1.36MHz. The negative detuning sign is chosen (and made small due to Clebsch - Gordon coefficients) to minimize excitation to the m j = + 1 / 2 Rydberg state (and the coupling to the Rydberg laser is one - third that of the desirable m j = - 1 / 2 state). In this study, a strong blockade between adjacent qubits is provided by making the Rydberg - Rydberg interaction V0 / 2π range from 200MHz to 1GHz.

[0228] Management of spurious phases in the CZ gate A two-qubit gate induces both an intrinsic one-qubit phase and a spurious phase primarily induced by the differential optical shift from the 420nm laser. Under certain configurations, the 420nm-induced differential optical shift of the ultrafine qubit can become extremely large (exceeding 8MHz), resulting in a phase accumulation of approximately 6π in the ultrafine qubit. Therefore, even slight percentage-level variations in the intensity of the 420nm laser can lead to significant qubit phase relaxation.

[0229] This 420-induced phase problem can be addressed by performing an echo sequence, i.e., after the CZ gate, the 1013 nm Rydeberg laser is turned off, a Raman π pulse is applied, and then the 420 nm laser is pulsed again to cancel out the phase induced by the 420 light during the CZ gate. This method echoes out the 420-induced phase but comes at the cost of a twofold increase in 420-induced scattering errors, which are the main cause of errors in 2-qubit CZ gates.

[0230] Echoes between CZ gates. To address these various issues, a Raman π pulse is applied between each CZ gate to echo and cancel out the spurious phase induced by the ultrafine qubit gates. This technique has several advantages. Now, the 420-induced phase is canceled out by the pair of CZ gates without explicitly applying an additional 420 nm pulse to echo each individual CZ gate, thereby reducing the scattering error of the CZ gates in this study by approximately half. This echo technique, which reduces the scattering error induced in each gate, almost compensates for the increased scattering rate induced by spreading the optical power over a wider 2D space, thereby giving a gate fidelity comparable to that of a 2-qubit CZ gate with a fidelity of over 97.4(2)%. Furthermore, the echo between CZ gates also cancels out the intrinsic 1-qubit phase of the CZ gates, eliminating calibration errors for this parameter and canceling out spurious 1-qubit phases induced by all other gates, such as the approximately 0.01 rad phase induced by pulse-off the trap for 2-qubit gates for 500 ns. If the number of CZ gates is odd, an echo is performed for the last CZ gate.

[0231] Sign of detuning in the intermediate state. To further suppress the effects of spurious 420 induced phase, the 420 nm laser uses 6P. 3 / 2 The device is operated to red-detune from the transition (2 GHz). With respect to red-detune, the optical shifts in the |0> and |1> states have the same sign, minimizing the differential optical shift, while with respect to blue-detune below 6.8 GHz, the optical shifts in the |0> and |1> states have opposite signs, amplifying the differential optical shift.

[0232] Sensitivity to axial trap vibrations On the timescale of typical Rydberg excitations using optical tweezers, axial trap oscillation frequencies of several kHz are negligible. However, in circuits that run for long periods of 1.2 ms using Rydberg pulses throughout, axial trap oscillations can have a significant impact. In particular, axial oscillations cause atoms to vibrate inward / outward from the Rydberg beam, and at an estimated axial temperature of approximately 25 μK and an axial frequency of 6 kHz, axial spread occurs.

[0233]

number

[0234] It is estimated that, with respect to a 20-micron waist beam, the effect of this positional broadening is relatively small with respect to the CZ gate pulse parameters, but can be significant for the highly sensitive 420 induced phase, which should be canceled out by echoing the phase induced by the CZ gate separated by approximately 200 μs. When using a 20-micron waist beam and a 420 nm laser with 2.5 GHz blue-detuning, phase relaxation due to axial trap oscillations is significant. To mitigate this adverse effect, the beam waist of the 420 nm laser is broadened to 35 microns (while maintaining a constant intensity), and the laser frequency is changed to 2 GHz red-detuning, which together result in a significant reduction in phase relaxation associated with improper echoing of the 420 nm pulse.

[0235] Shaping and uniformity of Rydberg beams The Rydberg beam is shaped into a variable-size top hat by wavefront control using the phase profile of a spatial light modulator (SLM). This capability allows the beam profile height to be matched to the size of the experimental zone of any given experiment, thereby maximizing the intensity and CZ gate fidelity of the 1013 nm light. The uniformity of the Rydberg beam is optimized until the peak-to-peak inhomogenity is less than 1%. To this end, all aberrations are corrected up to the vacuum chamber window, which results in a few percent of atomic inhomogenity attributable to a defect in the last window. To further optimize uniformity, aberration corrections are adjusted in the top hat by Zernike polynomial correction to the phase profile in the SLM plane (Fourier plane). This procedure reduces the peak-to-peak inhomogenity to less than 1% over the 40-50 μm range in the atomic plane.

[0236] Coherent mapping protocol A coherent mapping protocol is provided to map a common many-body state of the {|1>, |r>} basis to a long-lived, non-interacting {|0>, |1>} basis. To achieve this mapping, a Raman π pulse is applied immediately after Rydberg dynamics to map |1>→|0>, and then a subsequent Rydberg π pulse is applied to map |r>→|1>.

[0237] Even with perfect Raman and Rydberg π pulses (for isolated atoms), there are three key causes of infidelity associated with this mapping process. (1) All populations in a state that breaks the blockade (i.e., two adjacent atoms, both in |r>) are strongly shifted off-resonance with respect to the last Rydberg π pulse. Thus, this atomic population remains in the Rydberg state and is lost. (2) For example, a long-range interaction from the next-nearest-neighbor detunes the last Rydberg π pulse from resonance and thus reduces the fidelity of the pulse. Since the long-range interaction is not the same for all many-body microscopic states, this effect cannot be mitigated by a simple shift in detunement. (3) Phase relaxation of the state occurs over the entire duration of the Raman π pulse, primarily from the Doppler shift between the ground state |0>, |1> and the Rydberg state |r>. While these random on-site detunings also exist among many-body dynamics, turning off the Rydberg drive Ω allows the system to accumulate phase freely, making us particularly sensitive to phase relaxation errors.

[0238] The mechanism of the above error is mitigated as follows: To minimize the error from (1), many-body dynamics are used.

[0239]

number

[0240] This is executed. This minimizes the probability of an atom breaking the blockade to the order of 1%. To help minimize errors from (2),

[0241]

number

[0242] (V NNNThe amplitude of the 420nm laser is doubled for the last π pulse so that (this is the interaction with the next nearest neighbor), reducing the pulse error due to long-range interactions to the order of 1%. Finally, to reduce errors from (3), a fast Raman π pulse is executed with only a 150ns gap between the end of the many-body Rydberg dynamics and the start of the Rydberg π pulse. The 150ns gap is T2 of the {|g>, |r>} basis. * This is relatively short compared to approximately 3-4 μs, resulting in the accumulation of random phase on the order of about 0.02 × 2π rad per particle, but it is further complicated by the fact that the entangled state of N particles in one copy has an entangled state of N particles in the second copy, which accumulates random phase.

[0243] A global Raman beam induces a phase shift in a Raman π pulse, induced by an optical shift of approximately π relative to |r> and |0> and |1>. Similarly, a global 420 nm laser also induces a phase shift in a Rydberg π pulse, induced by an optical shift of approximately π between |0> and |1>. The measurements performed here are coherent (in other words, the singlet state being measured is invariant under global rotation) and are therefore unaffected by these global phase shifts, although these phase shifts can be measured and revealed as appropriate.

[0244] Formation of particle arrays using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a 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 within the atom from the induced dipole, averaged over the oscillation period of the light, is called the AC Stark shift. Based on the AC Stark shift induced by light detuned (i.e., with an offset wavelength) from the atom's resonant transition, the atom is attracted to light below its resonant frequency and thus trapped at the local intensity maximal (for red-detuned, i.e., longer wavelength trap light). The AC Stark shift is proportional to the intensity of the light. Therefore, the shape of the intensity field is the shape of the associated atomic trap. Optical tweezers utilize this principle by focusing a laser to a waist on a micron scale, trapping individual atoms at the focal point. A two-dimensional (2D) array of optical tweezers is generated, for example, by illuminating a spatial light modulator (SLM) that imprints a computer-generated hologram onto the wavefront of the laser field. The 2D array of optical tweezers is superimposed on a cloud of laser-cooled atoms within a magneto-optical trap (MOT). The strongly focused optical tweezers operate in a "collision blockade" regime, where single atoms are loaded from the MOT while pairs of atoms are ejected by light-assisted collisions, ensuring that the tweezers are at best loaded with single atoms, although the loading is probabilistic, so that the trap is loaded with a single atom with a probability of approximately 50-60%.

[0245] To prepare a deterministic atomic array, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into a pre-programmed shape. Atomic rearrangement requires moving atoms within tweezers that can be smoothly steered to minimize heating, for example, by deflecting a laser beam by an adjustable angle controlled by the frequency of an acoustic waveform applied to an AOD crystal using an acousto-optic deflector (AOD). Dynamic adjustment of the acoustic frequency translates to smooth movement of the optical tweezers. Multi-frequency acoustic waves create an array of laser deflections, which, after focusing through a microscope objective lens, form an array of optical tweezers with adjustable position and amplitude, both controlled by the acoustic waveform. Atoms are rearranged using an additional set of dynamically moving tweezers superimposed on the SLM tweezers array.

[0246] Exemplary Hardware Optical tweezers arrays form a powerful and flexible method for constructing large-scale systems composed of individual particles. Each optical tweezers captures a single particle, including but not limited to individual neutral atoms and molecules, for applications in quantum technology. Loading individual particles into such a tweezers array is a probabilistic process, with each tweezers in the system being filled with a single particle with a finite probability p < 1, for example, p ≈ 0.5 in the case of an implementation of many neutral atom tweezers. To compensate for this random loading, real-time feedback may be obtained by measuring which tweezers have been loaded and then sorting the loaded particles into a programmable shape. This may be done by moving one particle at a time or by moving them in parallel.

[0247] Parallel sorting may be achieved by using two acousto-optic deflectors (AODs) to generate multiple tweezers that can pick up particles from existing particle trapping structures, move those particles simultaneously, and release them in another location. This may involve moving particles within a single trapping structure (e.g., a tweezers array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezers array and another type of optical / magnetic trap). This sorting is flexible and allows for programmed positioning of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of the radio frequency (RF) drive field for the AOD. Since the RF drive of the AOD can be controlled in real time and can include any combination of frequency components, by changing the number, magnitude, and distribution of frequency components in the RF drive field of the AOD, it is possible to generate any grid of traps (e.g., lines of arbitrarily positioned traps), move rows or columns of the grid, and add or remove rows and columns of the grid.

[0248] In an exemplary embodiment, the optical tweezers array is fabricated using a liquid crystal on silicon spatial light modulator (SLM) that can programmatically create a flexible arrangement of tweezers. These tweezers are fixed in space for a given experimental sequence, and individual atoms are probabilistically loaded such that each tweezers is loaded with a probability p ≈ 0.5. Fluorescence images of the loaded atoms are taken to identify in real time which tweezers are loaded and which are empty.

[0249] After detecting which tweezers are loaded, the movable tweezers, overlapping the optical tweezers array, can dynamically reposition atoms from their starting positions to nearly-unity fill the target array of traps. The movable tweezers are made up of pairs of crossed AODs. These AODs can be used to move one atom at a time to create a single movable trap that fills the target array, or to move many atoms in parallel.

[0250] Referring to Figure 18, a schematic diagram of the apparatus 1800 for quantum computing according to embodiments of the present disclosure is provided. As shown in Figure 18, using a beam generated by a light source 1802 (e.g., a coherent light source in some exemplary embodiments -- a monochromatic light source), the SLM 1804 forms an array of capture beams (i.e., a tweezers array) which, in the exemplary embodiment shown in Figure 18, is imaged onto a capture plane 1808 in a vacuum chamber 1810 by an optical train including elements 1806a, 1806c, 1806d and a high numerical aperture (NA) objective lens 1806e. Other suitable optical trains may be used, as will be readily apparent to those skilled in the art. Using a beam generated by a light source 1812 (e.g., a coherent light source, in some exemplary embodiments -- a monochromatic light source), a pair of AODs 1814 and 1816 having non-parallel directions (e.g., orthogonal directions) of acoustic wave propagation generate a dynamically moving sort beam. By using an optical train such as the one shown in Figure 18 (elements 1817, 1806b, 1806c, 1806d, and 1806e), the sort beam is superimposed with the capture beam. Naturally, other optical trains can be used to achieve the same result. For example, light sources 1802 and 1812 could be a single light source, and the capture beam and sort beam could be generated by a beam splitter.

[0251] The dynamic movement of the steering beam is accomplished by using two non-parallel AODs 1814, 1816 arranged in series. In the exemplary embodiment shown in Figure 18, one AOD defines the “row” direction (“horizontal” -- “X” AOD) and the other AOD defines the “column” direction (“vertical” -- “Y” AOD). Each AOD is driven by an arbitrary RF waveform from an arbitrary waveform generator 1820, which is generated in real time by a computer 1822 that processes a feedback routine after analyzing an image of where atoms are loaded. When each AOD is driven by a single frequency component, a single steering beam (“AOD trap”) is generated in the same plane 1808 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, so that a single AOD trap can be steered to overlap with any SLM trap.

[0252] In Figure 18, laser 1802 projects a beam of light onto SLM 1804. SLM 1804 can be controlled by computer 1822 to generate a beam pattern ("capture beam" or "tweezers array"). The beam pattern is focused by lens 1806a, passes through mirror 1806b, and collimated on mirror 1806d by lens 1806c. The reflected light passes through objective lens 1806e to focus the optical tweezers array in the vacuum chamber 1810 onto the capture plane 1808. The laser light from the optical tweezers array continues through objective lens 1824a, passes through dichroic mirror 1824b, and is detected by charge-coupled device (CCD) camera 1824c.

[0253] The vacuum chamber 1810 may be illuminated by an additional light source (not shown). Fluorescence from atoms trapped in the trapping plane also passes through the objective lens 1824a but is reflected by the dichroic mirror 1824b and then reflected by the electron multiplier CCD (EMCCD) camera 1824d. In this example, the laser 1812 directs a beam of light towards AODs 1814 and 1816. The AODs 1814 and 1816 are driven by an arbitrary waveform generator (AWG) 1820, which is further controlled by a computer 1822. The intersecting AODs 1814 and 1816 emit one or more beams as described above, and these beams are directed towards the focusing lens 1817. The beams then enter the same optical trains 1806b...1806e as described above with respect to the optical tweezers array and focus onto the trapping plane 1808.

[0254] It will be understood that alternative optical trains may be used to generate optical tweezers arrays suitable for the uses described herein.

[0255] The descriptions of the various embodiments of this disclosure are presented for illustrative purposes only and are not intended to be exhaustive or limitful to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used herein have been chosen to best describe the principles, practical applications, or technical improvements to the technologies found in the market of the embodiments, or to enable those skilled in the art to understand the embodiments disclosed herein. [Explanation of Symbols]

[0256] 300 QPUs 301 processor cores 304 Atomic Load Zone 305 Remote entanglement zones, optical links and / or optical lattice transport 311 Storage Zones 312 Tangled Zone, Active Zone 313 Read Zones 403 2D Grid 404 Data Qubits 405 data bits 406 data bits 407 Data Qubits 408 test bits 409 test bits 410 Stabilizer Cubit 411 Stabilizer Cubit 412 Z-check 701 Initial Sequence 702 Final Configuration 1101 Fixed beam 1201 nodes 1202, 1203 Internal Nodes 1800 equipment 1802 Light source, laser 1804 SLM 1806a element, lens 1806b element, mirror 1806c element, lens 1806d element, mirror 1806e objective lens 1808 Capture Plane 1810 Vacuum Chamber 1812 Light source, laser 1814 AOD 1816 AOD 1817 element, focusing lens 1820 Arbitrary Waveform Generator, AWG 1822 Computer 1824a Objective lens 1824b Dichroic Mirror 1824c Charge-Coupled Device (CCD) Camera 1824d Electronic Multiplier CCD (EMCCD) Camera

Claims

1. A method for performing quantum error correction, A step of providing a plurality of data qubits, wherein each of the plurality of data qubits is placed within a corresponding trap, A step of providing a plurality of auxiliary qubits, wherein each of the plurality of auxiliary qubits is placed in a corresponding trap, The steps include arranging the plurality of data qubits and the plurality of auxiliary qubits in a plurality of rows and a plurality of columns, thereby forming a grid, A step of performing multiple sorts of the plurality of rows and the plurality of columns, wherein each of the plurality of sorts places each of the plurality of data qubits within the interaction radius of one of the plurality of auxiliary qubits, thereby forming a plurality of proximity pairs. A method comprising the steps of applying a global control pulse to the grid after each of the plurality of sortings, thereby applying a gate to each of the plurality of proximity pairs, and thereby encoding a parity check matrix between the plurality of auxiliary qubits and the plurality of data qubits.

2. Before forming the aforementioned grid, The method according to claim 1, further comprising the step of applying control laser pulses to the plurality of data qubits to prepare the plurality of data qubits in an initial state.

3. The step of arranging the plurality of data qubits and the plurality of auxiliary qubits in the grid is, The method according to claim 1 or 2, comprising moving the plurality of auxiliary qubits in parallel on the grid.

4. The step of performing the aforementioned multiple sorting is, The method according to claim 1, comprising moving one or more rows in the grid in parallel and moving one or more columns in the grid in parallel.

5. The method according to claim 1, wherein the plurality of auxiliary qubits include a Z stabilizer and an X stabilizer.

6. The method according to claim 1 or 5, further comprising the step of removing a subset of the plurality of auxiliary qubits from the grid and performing measurements on the subset.

7. The method according to claim 6, wherein the subset corresponds to a Z stabilizer.

8. The method according to claim 6, wherein the subset corresponds to an X stabilizer.

9. The step of performing the aforementioned multiple sorting is, The method according to claim 1, comprising determining a collision-free path for each of the plurality of data qubits and each of the plurality of auxiliary qubits for each of the plurality of sortings.

10. Determining the aforementioned collision-free path The method according to claim 9, comprising bipartite and recursive sort of the plurality of data qubits and the plurality of auxiliary qubits.

11. The method according to claim 9, wherein the collision-free path is a tertiary spline.

12. Determining the aforementioned collision-free path Decomposing the parity check matrix into a first product graph and a second product graph, Determining the routing of the plurality of rows according to the first product graph, The method according to claim 9, comprising determining the routing of the plurality of columns according to the second product graph.

13. The method according to claim 12, wherein the plurality of sorts include sorts specific to rows and / or sorts specific to columns.

14. The step of performing the aforementioned multiple sorting is, The method according to claim 13, comprising applying a fixed beam to at least one of the plurality of data qubits or the plurality of auxiliary qubits, thereby maintaining the position of the at least one qubit in accordance with the routing of the plurality of rows or the plurality of columns.

15. The method according to any one of claims 1 to 14, wherein the parity check matrix implements a quantum low-density parity check (qLDPC) code.

16. The method according to any one of claims 1 to 14, wherein the parity check matrix implements surface codes.

17. The method according to any one of claims 1 to 14, wherein the parity check matrix implements a hypergraph product (HGP) code.

18. The method according to any one of claims 1 to 14, wherein the parity check matrix implements a Calderbank-Shor-Steane (CSS) code.

19. The method according to any one of claims 1 to 14, wherein the parity check matrix implements a lift product (LP) code.

20. The method according to claim 19, further comprising the step of flattening the LP code before performing the plurality of sortings.

21. The method according to claim 20, wherein the step of flattening the LP code includes iteratively dividing and flattening the LP code.

22. The method according to any one of claims 1 to 21, wherein the gate applied to each of the plurality of adjacent pairs is a CZ gate.

23. The method according to any one of claims 1 to 22, wherein the global control pulse is a laser pulse.

24. The method according to any one of claims 1 to 23, wherein the traps corresponding to each of the plurality of data qubits and each of the plurality of auxiliary qubits are optical traps.

25. The optical traps corresponding to the plurality of data qubits and the plurality of auxiliary qubits are generated by directing a beam of light to at least one acousto-optic deflector (AOD), The method according to claim 24, wherein moving one or more rows and moving one or more columns is a change in the drive frequency of the at least one AOD.

26. The method according to claim 25, further comprising the step of applying one or more rotations during the movement.

27. The method according to claim 26, wherein the step of applying the one or more rotations includes applying a Raman pulse.

28. A quantum computing system comprising a plurality of data qubits, each located in a corresponding trap, and a plurality of auxiliary qubits, each located in a corresponding trap, configured to perform the method according to any one of claims 1 to 27.

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