Modular Rydberg Architecture for Fault-Tolerant Quantum Computing

Abstract

A modular Rydberg architecture for fault-tolerant quantum computing is provided. A first array and a second array of neutral atoms are provided. Each neutral atom has a first state and an excited Rydberg state. Each neutral atom is aligned such that, when in the excited Rydberg state, it imposes a Rydberg blockade on at least its nearest neighbors in the array, thereby implementing a plurality of physical qubits. Each array includes a data qubit and a syndrome qubit. The syndrome qubit is configured to implement a quantum error correction code with respect to the data qubit. Each array includes a subarray of communication qubits having lower dimensionality than the array. Each communication qubit in the first subarray forms a Bell pair with one communication qubit in the second subarray. The first and second arrays of neutral atoms are configured to interact with each other solely through the communication qubits.

Description

[Technical Field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 63 / 357,882, filed July 1, 2022, which is incorporated by reference herein in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under award 1745303 from the National Science Foundation (NSF) and DE-AC02-05CH11231 awarded by the U.S. Department of Energy (DOE) and W911NF2010021 awarded by the U.S. Department of Defense / Defense Advanced Research Projects Agency (DOD / DARPA). The government has certain rights in this invention. [Background technology]

[0003] background Aspects of the present disclosure relate to systems for neutral atom-based quantum computing, and more particularly to a modular Rydberg architecture for fault-tolerant quantum computing. Summary of the Invention

[0004] Quick Overview According to an aspect of the present disclosure, a quantum computing system is provided, the system including a first array and a second array of neutral atoms, each array having a first dimensionality; each neutral atom having a first state and an excited Rydberg state, and each neutral atom aligned to impose a Rydberg blockade on at least its nearest neighbors in the array when in the excited Rydberg state, thereby implementing a plurality of physical qubits; and each array including a plurality of data qubits and a plurality of syndrome qubits, and for each array, the plurality of syndrome qubits are configured to implement a quantum error correction code (e.g., a stabilizer code) on the data qubits. The first array of neutral atoms includes a first subarray of communication qubits, and the second array of neutral atoms includes a second subarray of communication qubits, the first and second subarrays having a second dimensionality lower than the first dimensionality; each communication qubit of the first subarray forms a Bell pair with one communication qubit of the second subarray; and the first and second arrays of neutral atoms are configured to interact with each other only via the communication qubits.

[0005] According to an aspect of the present disclosure, there is provided a method of performing a logical operation between logical qubits, the method comprising: providing a quantum computing system as described above and performing a logical operation between at least one data qubit of a first array and at least one data qubit of a second array.

[0006] According to an embodiment of the present disclosure, there is provided a method of extending a quantum error correcting code (e.g., a stabilizer code) across two non-interacting arrays of particles, the method comprising: providing a quantum computing system as described above; and extending the quantum error correcting code (e.g., a stabilizer code) across the first and second arrays. [Brief explanation of the drawings]

[0007] A brief description of some of the figures in the drawing [Figure 1] FIG. 1 is a schematic diagram of two surface cord patches according to an embodiment of the present disclosure. [Figure 2] FIG. 2 is an exemplary teleported CNOT circuit according to an embodiment of the present disclosure. [Figure 3] FIG. 3 is a graph of logic error rates for an exemplary repetitive and surface code. [Figure 4] FIG. 4 is a graph of CNOT and Bell pair error rate in an exemplary embodiment of the present disclosure. [Figure 5] FIG. 5 is a schematic diagram of a system for quantum computing according to an embodiment of the present disclosure. [Figure 6] FIG. 6 is a schematic diagram of an exemplary cavity configuration, according to an embodiment of the present disclosure. [Figure 7] FIG. 7 is a flowchart illustrating a method of collective measurement according to an embodiment of the present disclosure. [Figure 8] FIG. 8 is an energy level diagram according to an embodiment of the present disclosure. [Figure 9A] 9A-D are schematic diagrams of example cavities illustrating a binary search, according to embodiments of the present disclosure. [Figure 9B] 9A-D are schematic diagrams of example cavities illustrating a binary search, according to embodiments of the present disclosure. [Figure 9C] 9A-D are schematic diagrams of example cavities illustrating a binary search, according to embodiments of the present disclosure. [Figure 9D] 9A-D are schematic diagrams of example cavities illustrating a binary search, according to embodiments of the present disclosure. [Figure 10] FIG. 10 is a schematic diagram of the device of FIG. 5 using free-space entanglement, according to an embodiment of the present disclosure. [Figure 11] FIG. 11 is a schematic illustration of the device of FIG. 5 using cavity entanglement, according to an embodiment of the present disclosure. [Figure 12A] 12A-C are schematic diagrams of surface codes with seams illustrating a limited degree of error, according to embodiments of the present disclosure. [Figure 12B] 12A-C are schematic diagrams of surface codes with seams illustrating a limited degree of error, according to embodiments of the present disclosure. [Figure 12C] 12A-C are schematic diagrams of surface codes with seams illustrating a limited degree of error, according to embodiments of the present disclosure. [Figure 13] FIG. 13 is a graph of the logical error rate for an exemplary noisy syndrome surface and repetition code. [Figure 14] FIG. 14 is a graph of logical error rates for an exemplary code with and without seams, according to an embodiment of the present disclosure. [Figure 15] FIG. 15 is a table of terms used herein. [Figure 16] FIG. 16 is a schematic diagram of an exemplary matching lattice, according to an embodiment of the present disclosure. [Figure 17A] 17A-C are graphs illustrating analytical logical failure bounds according to embodiments of the present disclosure. [Figure 17B] 17A-C are graphs illustrating analytical logic failure boundaries according to embodiments of the present disclosure. [Figure 17C] 17A-C are graphs illustrating analytical logic failure boundaries according to embodiments of the present disclosure. [Figure 17D] 17D-F are graphs illustrating numerical simulations of failure boundaries according to embodiments of the present disclosure. [Figure 17E] 17D-F are graphs illustrating numerical simulations of failure boundaries according to embodiments of the present disclosure. [Figure 17F] 17D-F are graphs illustrating numerical simulations of failure boundaries according to embodiments of the present disclosure. [Figure 18]FIG. 18 is a graph illustrating threshold sag in a configuration having multiple seams, according to an embodiment of the present disclosure. [Figure 19] FIG. 19 is a table of phenomenological bit flip error probabilities according to an embodiment of the present disclosure. [Figure 20] FIG. 20 is a schematic diagram of an apparatus for quantum computing according to an embodiment of the present disclosure. [Figure 21] FIG. 21 illustrates a classical computing node according to an embodiment of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0008] Detailed Description A quantum bit (qubit) is the fundamental building block for quantum computers. By analogy with classical bits (each bit is either a 0 or a 1) used to store information in traditional 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.

[0009] Bits and qubits are each encoded in the state of a real physical system. For example, a classical bit (0 or 1) can be encoded in whether a capacitor is charged or discharged, or whether a switch is "on" or "off." A quantum bit is encoded in a quantum system that has two (or more) distinct quantum states. There are many physical realizations that can be used. 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 distinct quantum states that correspond to different directions of electron spin, nuclear spin, electron orbitals, and molecular rotation / vibration.

[0010] In principle, a qubit can be encoded into any pair of quantum states of an atom / ion / molecule. In practice, an important parameter of a qubit is described by their quantum coherence property. Coherence measures the lifetime of a qubit before its information disappears. This has a close analogy with classical bits: if you prepare a classical bit in the 0 state, it can be randomly flipped to 1 after some time due to environmental noise. Quantum mechanically, the same error can occur: |0> can be randomly flipped to |1> after some characteristic timescale. However, qubits can suffer further errors: for example, they can be flipped into superposition states

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[0011] A quantum computer may generally contain many qubits, each encoded in its own atom, molecule, ion, etc. Besides simply containing qubits, a quantum computer should be able to (1) initialize the qubits, (2) manipulate the state of the qubits in a controlled way, and (3) read out the final state of the qubits. When it comes to qubit manipulation, this is usually broken down into two types: One type of qubit manipulation is the so-called single-qubit gate, which means an operation applied to the qubit individually. For example, this means that the state of a qubit can be flipped from |0> to |1>, or |0> can be flipped into a superposition state

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[0012] In various embodiments of quantum computers, qubits are encoded in energy levels close to the two ground states of an atom, ion, or molecule. An example of this is the hyperfine qubit. Such qubits are encoded in two electrical ground states that differ by the relative orientation of the nuclear spin with respect to the external electron spin. Such state pairs can be selected to be particularly robust / insensitive to environmental perturbations, resulting in long coherence times. These states are separated in energy by the hyperfine interaction energy of the atom / ion / molecule, which is the interaction energy between the nuclear spin and the electron spin. The robustness of a qubit can be understood as the energy division between two particularly stable states. For this reason, such states are referred to as clock states, as stable energy divisions can form excellent frequency references and thus form the basis for atomic clocks. Typical hyperfine splittings between these qubit states are in the 1-13 GHz frequency range.

[0013] To perform single-qubit gates on such hyperfine qubits, it is possible to apply coherent microwave radiation at the precise frequency of the energy division between states. However, this approach has two drawbacks. First, microwaves cannot be applied to just one qubit without affecting neighboring qubits. This is because qubits are encoded in particles that are typically just a few microns apart, and microwaves cannot be focused on such small scales due to their large wavelengths. Second, microwave intensity is quite limited, and therefore the maximum speed of single-qubit gates is correspondingly limited.

[0014] An alternative approach is based on stimulated Raman transitions. In this case, a laser field is applied to an atom / ion / molecule. The laser field is nearly (but not exactly) resonant with an optical transition from one of the ground states to an optically excited state. The laser contains multiple frequency components separated in frequency by an amount exactly equal to the hyperfine splitting of the qubit. The atom / ion / molecule can absorb a photon from one frequency component and coherently emit a different frequency component, and in doing so, it changes its state. This approach benefits from the ability to focus the laser field on individual particles or subsets of particles in quantum computers. The laser field can also be applied at high intensities, allowing for fairly fast gate operations.

[0015] Neutral atom quantum computers encode qubits in individual neutral atoms. Neutral atoms are trapped in a vacuum chamber and levitated by a trapping laser. Most commonly, the trapping laser is an individual optical tweezers, which is an individual tightly focused laser beam that traps an individual atom at its focus. Alternatively, individual atoms can be trapped in an optical lattice formed by a standing wave of laser light, which creates a periodic structure of nodes and antinodes.

[0016] The typical approach for encoding qubits in neutral atoms is the hyperfine qubit approach, in which the two ground states are separated from the qubit by a few GHz. Multi-qubit gates in neutral atom quantum computers are realized using a third atomic state, a highly excited Rydberg state. When one atom is excited into a Rydberg state, neighboring atoms are prevented from being excited into the Rydberg state. This conditional behavior forms the basis for multi-qubit gates such as controlled-NOT gates. The Rydberg state is temporarily used to mediate the multi-qubit gate, and then the atoms are returned from the Rydberg state back to the ground state level to preserve their coherence.

[0017] Trapped ion quantum computers use atomic species that are ionized, meaning they have a net charge. In most cases, many ions are trapped in one large trapping potential created by electrodes in a vacuum chamber. The ions are pulled to the minimum of the trapping potential, but Coulomb repulsion between the ions causes them to form crystalline structures centered in the middle of the trapping potential. Most commonly, the ions align into linear chains. Other methods for trapping ions are also possible, such as using optical tweezers or trapping ions individually with local electric fields using more complex on-chip electrode structures.

[0018] Qubits can be encoded in trapped ions in several ways. One common approach is to use the ground-state hyperfine level, as described for neutral atoms. Similar to neutral atoms, in trapped ions with hyperfine qubit encoding, single-qubit gates can use microwave irradiation or stimulated Raman transitions.

[0019] Unlike those in neutral atoms, trapped ion hyperfine qubits rely heavily on stimulated Raman transitions to implement multi-qubit gates. Stimulated Raman transitions can be used both to control the ion's hyperfine state and also to change the ion's state of motion (i.e., momentum addition). This can be understood as absorbing a photon moving in one direction and emitting a photon in a different direction, such that the difference in the photons' momentum is absorbed by the ion. Often, many ions are trapped in a collective trapping potential, each mutually repelling one another, so that changing the state of motion of one ion affects the other ions in the system, and this mechanism forms the basis for multi-qubit gates.

[0020] According to various embodiments of quantum computers, individual particles (atoms / ions / molecules) can first be trapped in an array and aligned in a specific configuration. Next, one or more particles are prepared in a desired quantum state. A quantum circuit can then be implemented by a sequence of qubit operations acting on individual qubits (single-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. Readout can be achieved using an observation system that typically includes an electron-multiplying 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.

[0021] Rydberg atomic arrays have favorable scaling properties (a 256-qubit simulator has already been realized), long qubit coherence times (hyperfine qubits with coherence times greater than 1 second have been experimentally demonstrated using dynamical decoupling methods), and gate speeds exceeding 1 MHz. Single-qubit gates can reach 0.02% error rates, and two-qubit gates have reached fidelity >97% for rubidium and 99.1% for strontium. Furthermore, in both cases, detailed error budgets provide clear paths for further improvement, backed up by the theoretical limit of 99.9%. Furthermore, because Rydberg gate fidelity depends only on system size through available laser power, this fidelity is not expected to degrade with system scale. Rapid transport of atoms has also been realized, endowing atomic arrays with reconfigurability on the 100-microsecond timescale, enabling nonlocal gates and qubit transport between designated regions within multifunction quantum processors.

[0022] To address the appearance of errors such as those described above, various quantum error correction codes can be used. Quantum error correction codes restore noisy decohered quantum states to pure quantum states. Stabilizer quantum error correction codes append an ancilla qubit to the protected qubit. Unitary encoding circuits rotate the global state into a subspace of a larger Hilbert space. This highly entangled encoded state corrects local noise errors.

[0023] One class of stabilizer codes known in the art is surface codes. One such surface code is a low-density parity-check (LDPC) code, which has a preferred threshold of about 3%. Data qubits are spread out at the edges (dots) of a square lattice, and measure X and measure Z syndrome qubits check for bit and phase flip errors, respectively, on adjacent data qubits.

[0024] Quantum computers built with noisy components require error correction (QEC) to scale. For a given QEC code to suppress errors, the components must introduce noise below a code-dependent threshold (ideally about 10x less)—errors must be removed faster than they accumulate. Many different QEC codes are currently being investigated, but to date, surface codes have the highest threshold, allowing about 1% errors from circuit-level components. This results in a target of 0.1% total circuit-level errors (including gate and memory errors) per code cycle for quantum hardware.

[0025] A fault-tolerant quantum computer capable of executing Shor's algorithm for 2000 bits would require thousands of logical qubits and tens of trillions of logical operations, and quantum error correction imposes a significant overhead of tens to thousands of physical qubits per logical qubit. Due to engineering constraints, implementing such a large number of logical qubits in a single device is impractical. Trapped ion systems experience severe gate fidelity degradation for system sizes larger than tens of qubits. Superconducting systems are limited to a few thousand qubits by the size and performance of dilution refrigerators. Rydberg arrays are the most scalable, but are not expected to exceed the 10,000-qubit system size limit. In addition to the above limitations, any given platform may have some maximum size beyond which it becomes impossible to control all of the qubits at once in the same module (e.g., only so many ions can fit per chain, and only so many Rydberg atoms can fit in a vacuum chamber).

[0026] To address these challenges, the present disclosure provides a scalable, modular, fault-tolerant architecture for quantum computing based on Rydberg arrays. 4Approximately 50 Rydberg array modules containing qubits can be linked together to realize a quantum computer with 500,000 qubits.

[0027] While alternative modular architectures focus on very small modules containing 2-5 qubits, the architecture provided herein consists of large modules containing thousands of physical qubits, forming surface-coded patches linked together by optical cavity photonic interconnects. While the low fidelity of Bell pairs created by photonic interconnects is a major challenge for alternative modular architectures, the approach provided herein is uniquely immune to Bell pair infidelity because communication errors only occur along one edge of the code.

[0028] Numerical simulations indicate a threshold for communication errors of approximately 10%. Furthermore, because distillation is not required, the local gate requirement remains approximately 1%. These relaxed communication requirements enable fault-tolerant concatenation of currently available Rydberg arrays of many atoms using only moderate-quality Bell pairs. Quantitative performance estimates are provided showing that a single optical cavity of moderate quality can achieve a Bell pair distribution fast enough to achieve a 10 kHz surface code cycle—significantly faster than current coherence times (>1 s)—as well as faster syndrome readout and atom reloading.

[0029] In summary, the Rydberg array demonstrated the coherence time, gate speed, and expected gate fidelity required to continuously manipulate surface code patches (meeting the 0.1% per code cycle benchmark). Once a surface code can be realized on a single Rydberg array, it is ultimately limited by the size of the array. Estimates of the maximum size constraint for the array vary, but laser power, field of view, and acousto-optic deflector bandwidth all point to an upper limit of approximately 10,000 qubits. Regardless of the exact number of atoms that can be locally controlled, scaling will require concatenation of multiple arrays at some point. Scalability is provided in this disclosure by providing unit modules with sufficient quantum input / output (I / O), so the system can be scaled arbitrarily by simply adding more modules. In various embodiments, optical interconnects are used for quantum I / O, where entanglement distributions enable teleported gates for inter-module manipulation.

[0030] The main challenge for all distributed architectures is that each teleported gate uses a non-locally generated Bell pair (typically of lower fidelity due to the additional complexity of communication between separate modules) and several local operations, which, when combined, cause the Bell pair to have lower fidelity than the local operations themselves. One solution is to distill the many Bell pairs into several higher-fidelity Bell pairs. Even more significant than the excessive time and space overhead required for this distillation is the fact that the distillation itself requires ~10 local operations, which means that for the teleported gate itself to achieve code threshold, the local operations then need to be ~10x less than the code threshold. For this reason, in alternative modular architectures, the threshold for local operations is ~10x stricter than that required for a single large module due to the many local operations required in these distillations and the teleported gate's protocol.

[0031] This disclosure shows that using large modules enables fault-tolerant logic gates between modules based on noisy shared Bell pairs, with minimal increase in requirements for local operations. This makes it possible to employ large code patches that operate below threshold and concatenate them with noisy Bell pairs to form larger error-correcting codes without increased requirements for local gates. While fairly high-fidelity local operations appear necessary for small modules where teleported gates are fundamental operations at the code level, for large modules, protocols are provided in which gate teleportation only occurs on lower-dimensional boundaries between modules. This disclosure provides trial-and-error arguments, backed up by numerical simulations, to show that threshold requirements for local gates are largely unaffected even when connecting separate modules using noisy Bell pairs and teleported gates.

[0032] Thus, Rydberg arrays, supplemented by sufficiently fast generation of noisy inter-module Bell pairs, form unitary modules that can be chained together to form truly scalable, fault-tolerant quantum computers.

[0033] Referring to Figure 1, two surface-code patches 101, 102 in separate modules are connected along a lower-dimensional seam 103. A stabilizer check across the seam is performed using teleported gates 104 (where connected dots and crossed circles indicate entangled pairs). Data qubits are shown as open circles, and syndrome qubits are shown as filled circles. The data and syndrome qubits in columns 110 and 120, shown in gray and making up a small fraction of the total qubits, experience elevated noise levels due to the lower fidelity of inter-module operations. X L ,Z L denotes a logical string operator.

[0034] As shown in Figure 1, fault-tolerantly connecting multiple modules requires that code patches be connected along only one edge. To compute fault-tolerantly across modules, code patches are initialized and maintained across seams. The seam interface regions of local code patches have lower dimensionality than the bulk. Lower dimensionality corresponds to lower entropy, which results in a higher threshold.

[0035] Both the seam and the bulk contribute to the logic error as given in Equation 1.

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[0036] Assuming an error model based on the Rydberg error rate, we see a bulk threshold of about 1%. For comparison, a noisy syndrome repetition code has a threshold of 10%. Because only operations across the seam are noisy (one-quarter per plaquette), only the even qubits on the seam are subjected to one of four noisy operations. The advantage of this approach is that only qubits on the seam experience one teleported gate out of four total, making them an additional factor less sensitive to teleported gate errors than the phenomenological model naively indicates.

[0037] As shown in Figure 1, the noisy teleported gate contacts only one column of qubits and syndromes in each code patch.

[0038] Propagating the error about the teleported gate to the qubit proceeds as follows: B Bell pair occurs with probability 1-p B and probability

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[0039] In an exemplary phenomenological model, qubits experience an average level of noise based on how often they are hit with errors originating from the teleported gate and the local gate.

[0040] Referring to FIG. 2, an exemplary circuit is provided in which an X,Z error on a bell pair used in a teleported gate propagates to the two qubits it operates on.

[0041] A similar model is the CNOT gate (p CNOT ) Propagating errors for Bell and local gates from the model shown in Figure 2 yields: From the Bell Pair Error:

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[0042] Summing up the errors, the teleported gate is for X errors: Control:

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[0043] The following analysis is directed to data qubits on smaller scale modules: during each code cycle, the data qubit is twice the target and twice the control, depending on the gate teleported.

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[0044] For syndrome qubits on small modules, consider a plaquette whose syndrome is four times the target of the teleported CNOT, then p M Inaccurate measurements caused by:

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[0045] This is the probability that the syndrome will flip a bit incorrectly. Similar calculations apply for the star operator.

[0046] Below we provide a seam system analysis. Note that so far X-errors have only been simulated along the seam where it is weakest.

[0047] In bulk, gates are not teleported:

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[0048] Along the seam, there are three operations and then one teleported gate on the same side with a local CNOT. L or Y L It is important to consider whether the device is operating in the X direction. L When activated in the Z direction, L It is more important to note about X error correction as well as about

[0049] Considering the plaquette, this is the target of three local CNOTs:

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[0050] To summarize, code patches of distance size 3 to 15 with noisy seams are simulated to determine the effect of seams on code threshold, where the probability of error on a seam is p seam and elsewhere the probability of error is given by p bulkis given by

[0051] Referring to Figure 3, the seam and bulk thresholds are compared. The surface code has a threshold of approximately 10%, and the repeat code has a threshold of approximately 50%. The repeat code with L = 3.0, 5.0, and 7.0 is shown by lines 301, 302, and 303, respectively. The surface code with L = 3.0, 5.0, 7.0, 9.0, and 11.0 is shown by lines 304, 305, 306, 307, and 308, respectively.

[0052] As explained above, fault-tolerant communication between surface-coded patches requires maintaining the surface code across two modules. This can be achieved using teleported gates to perform stabilizer checks across the seam, which in turn requires perturbed Bell pairs. Figure 3 shows that for large surface-coded patches, the fidelity of the perturbed Bell pairs can be very low without substantially affecting the quality of the local operations required to reach threshold. Therefore, it is shown that the local gate requirements necessary to run the local-coded patch can be decoupled from the fidelity required to connect the Bell pairs. This contrasts sharply with alternative distributed architectures, which typically rely on distillation to reach target fidelity, increasing the requirements for local operations by a factor of about 10. Therefore, large modules have the significant advantage of tolerating low-fidelity connections as they scale without any cost to local operations. However, until recently, only hardware platforms with natural photonic integration (trapped ion chains, NV centers) were constrained to local operations, which degrade rapidly with small module or system size. In contrast, Rydberg arrays can support many atoms without local gate depression and can naturally support photonic integration.

[0053] Referring to FIG. 4, the thresholds for large modules (using a seam architecture) 401 and small modules (using one data or syndrome qubit per module) 402 are determined by the local CNOT error (p CNOT ) and Bell pair error (p Bell ) are compared with each other. Bell Even for =0, small modules have better p due to the extra local gates required for gate teleportation. CNOT Furthermore, for small modules, Bell >0, p CNOT must be reduced quickly to compensate. In contrast, for large modules, p Bell increases to about 10% until the required p CNOT does not affect.

[0054] Referring to Figure 5, a system for quantum computing is illustrated that includes two modules 501, 502 with sufficient quantum I / O for fault-tolerant communication. Surface-coded patches 503, 504 in each module are realized using arrays of atoms and connected using teleported gates. The Bell pairs 505 required to execute the teleported gates are generated using either optical cavities (506, 507) or highly multiplexed free-space harvesting, such as APD arrays (508, 509). Once the Bell pairs 505 are generated, they are transported to the seam and used to define stabilizer checks across the seam. The cavities can also be used to speed up qubit readout by transporting the syndrome qubit back to the cavity when readout is desired.

[0055] In view of the above discussion of inter-module logic gates that do not require increased fidelity of logic operations, implementation using Rydberg arrays requires the generation of reasonably good quality Rydberg pairs fast enough to perform all of the stabilizer checks across the seam quickly enough. The goal is to run the code cycles fast enough to overcome decoherence, so that errors are detected and corrected faster than they are introduced.

[0056] To be 10x less than the surface code threshold of about 1%, for example, the intrinsic qubit decoherence timescale shown, τ = 1 s, requires a code cycle every 1 ms or less, so the storage error is 10 per qubit within each code cycle. -3 This means that the entire round of stabilizer checks is completed within 1 ms for each surface code patch (approximately d 2 This means that it is needed both within the bulk of the patch (check) and across the seams that join the patches (check about d).

[0057] Since the local Rydberg gate itself is very fast (about 0.1 μs), the time bottleneck to realizing all these stabilizer checks is distributing all Bell pairs and reading out all syndrome qubits.

[0058] To show this all can be done fast enough, the specific choice of parameters is based on Shaw's algorithm (2N logical qubits and ∼10 -12 The proposed method provides a fault-tolerant quantum computer with logic gate fidelity that can perform logic gate errors. The local Rydberg gate operates at 10x below threshold, which is approximately 4x20 2 This can be achieved by a surface code with an edge length of d=20, which contains =1600 physical qubits (1521 exactly).

[0059] A scheme using a single optical cavity with the following parameters is sufficient:

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[0060] From these parameters the following quantities can be derived:

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[0061] With these reasonable optical cavity parameters, the required time per code cycle to generate all necessary Bell pairs and to read out all syndrome qubits can be estimated. From this, an exemplary workflow is described below detailing sequential operation of sufficient quality between modules to enable final fault-tolerant logic gates.

[0062] Entanglement generation Shared entanglement between modules is the basis of quantum communication. To generate shared Bell pairs, a protocol can be used in which photons entangled with appropriately prepared individual atoms are collected. Because each photon is entangled with its atom, Bell state measurement of two such photons projects the two atoms into a Bell state. The resulting atom-atom entanglement is prospective (the collection is imperfect) but is predicted (by successful joint detection). Once a shared Bell pair is created, local operations within the modules and classical communication between modules can be used to teleport quantum information or to define teleported gates. Achieving a sufficiently fast Bell pair creation rate is crucial for any modular architecture.

[0063] In alternative architectures involving modules with a small number of qubits, the fundamental operation at the level of the quantum error correcting code is a teleported gate, which means that the Bell pair generation rate determines the code cycle time (and consequently must be fast enough to keep idling errors below the code threshold). In the architectures presented herein, modules containing surface code patches running on hundreds or thousands of qubits are large, and teleported gates are required to achieve fault-tolerant operation between modules.

[0064] The resulting requirements on the Bell pair creation speed are even more stringent. For example, a transverse CNOT operation between two distance d logic qubits in separate modules requires 4d 2 This involves a teleported CNOT. Furthermore, error correction cannot be performed in mid-CNOT, as executing a subset of CNOT employs a system outside of the code space. Advantageously, other methods of fault-tolerant logic operation are available. This reveals that the minimum communication required to perform fault-tolerant operation between modules is the amount of communication required to simply maintain a code patch across two modules.

[0065] To maintain a code patch across two modules, 2d star and plaquette stabilizer checks must be performed across the seam using teleported gates that require 2d bell pairs. At this point, various methods of logical manipulation, such as braiding or lattice surgery, can be performed with modest (~2x) overhead in the number of qubits. Therefore, to make this architecture work, the bell pair creation rate must be kept within the coherence time (τ c = 1 s), it must be fast enough to generate a 2d bell pair in less than 1 / 1000 of a second.

[0066] Cavity-enhanced entanglement rate The cavities in each module allow for rapid photon extraction, which is used to entangle atoms in each of the two cavities. Atoms in each cavity emit photons, which are collected, routed to a common location using optical fibers, and measured using photodetectors to entangle the atoms. To estimate the rate of entanglement generation achievable using the cavity parameters provided above, we estimate the photon collection efficiency, the corresponding entanglement success probability, and the repetition rate below.

[0067] The single photon emission success probability for emission is given in Equation 25.

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[0068] The probability of Bell pair success is given in Equation 26.

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[0069] Waiting for 4 ring down times to avoid crosstalk is 4τ cav = 0.48 μs iteration time. This gives an average time to success of 0.48 / 0.19 = 2.5 μs per bell pair. Since a minimum of 2d = 40 bell pairs are required per code cycle, for each code cycle the time spent entangling the atom pairs is 40 × 2.5 μs = 100 μs.

[0070] Similar results are observed with runs with seams around all four edges of the patch.

[0071] Free-space entanglement velocity An alternative to using cavities is free-space collection, such as that using the APD array described above. Free-space entanglement generation is performed using a connected unit consisting of lenses that collect light scattered into free space from appropriately tuned atoms, where the light from each atom is focused, coupled to an individual fiber in the fiber array, and detected independently. Bell-state measurements of photons emitted from two separate atoms produce probable but predicted entanglement between the two atoms via entanglement swapping. Local manipulation can then be used to define the teleported gate.

[0072] To estimate the rate of entanglement generation achievable by the free-space method, we first estimate the photon recovery efficiency, then the corresponding entanglement success probability and iteration rate for one atom, and then multiply by the number of atoms we wish to entangle in parallel.

[0073] The single photon recovery efficiency is:

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[0074] The probability of a successful Bell Pair is:

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[0075] The repetition time is 7 μs and is limited by the repumping time, another overall factor of 50% of the duty cycle is assumed to be due to periodic recooling.

[0076] The average time to success is 14 / 0.0079 = 1.8 ms. If we attempt to entangle 500 atoms at once, the average time to create a single Bell pair is 3.6 μs. Since 2d = 40 Bell pairs are required per code cycle, for each code cycle the time required to generate the required entangled pairs is 40 × 3.6 μs = 144 μs, well below the 1 ms target.

[0077] If repumping and cooling times are performed separately, new atoms are continuously put into place, and iteration times can be expected to decrease. Assuming such highly parallelized preparation and transfer are possible, iteration times faster than 1 μs can be achieved, requiring significantly less parallelization.

[0078] Read Rate In addition to generating Bell pairs, 2d Bell pair qubits and 2d 2 A readout of the syndrome qubit is required every code cycle.

[0079] Cavities 506, 507 can be used to very quickly read out the Bell pairs used for teleportation, and also to very quickly read out the syndrome qubits needed for the code cycle, much faster than 2d Bell pairs can be generated. In some embodiments, a separate cavity is provided for this readout.

[0080] A schematic diagram of an exemplary cavity configuration is provided in Figure 6. An array of Rb atoms 601 is placed in cavity 602 and held by optical tweezers 603. The optical cavity 602 is driven by a weak laser beam 604, the presence / absence of which passing through the cavity indicates the presence / absence of atoms coupled to the cavity and can therefore reveal the state of the atoms. Using the parameters previously shown, collecting approximately 10 photons by driving the cavity to transmit / reflect a laser beam 10x less intense than the atomic saturation point is sufficient to achieve an error of 0.001, which can be done in 0.2 μs of continuous driving. The photons are collected by a single photon counting module (SPCM) 605. t per readout event read Scheduling an approximate total of τ = 1.0 μs allows time for cavity ring-up and ring-down (recall τ cav =0.12μs).

[0081] For each code cycle, two types of qubit measurements must be performed. First, each of the 2d teleported gates shown in Figure 2 requires measuring its associated Bell pair qubit for feedback. These qubits are unbiased and have a 50% chance of being found in their |0> or |1> state, and therefore must be read out individually using the cavity. This means that sequentially reading out 2d = 40 Bell pair qubits per code cycle takes 40 × 1.0 μs = 40 μs.

[0082] Second, for each chord cycle, the 2d 2 The syndrome must also be read out. However, these syndrome qubits are only flipped from |0> to |1> when an error occurs, and this is relatively infrequent. This means that these qubits are highly biased when measured, allowing for optimization of the readout. Many syndrome qubits can be placed in the cavity at the same time, and if one or more of them are in |1>, communication through the cavity is blocked. 2d 2By coupling all of the syndrome qubits into the cavity simultaneously, each qubit that is actually in state |1> can be reduced by approximately log2(2d 2 ) steps. With each qubit's chance p in error state |1>, the average number of reads required is (1 + p(2d 2 )log2(2d 2 For each qubit that gets a hit with a 0.001 error local gate and 4 such local gates per code cycle, p = 0.004, requiring ∼(1 + 0.004 × 800 log 2800) = 32 read events, and t read = 1.0 μs takes about 32 μs.

[0083] Referring to Figure 7, a method for bulk measurement according to an embodiment of the present disclosure is illustrated, and in Figure 8, a corresponding energy level diagram is provided.

[0084] In 701, an N atom 601 is initialized in optical tweezers 603. In 702, a laser 604 pumps the atom 601 to a check state 801. In 703, a Raman π pulse is applied by a laser beam 606 to place the atom 601 in a |0> state 802. In 704, a Raman π pulse is applied by a laser beam 607 to place the atom 601 in a |1> state 803. In 705, an artificial bit-flip error from |1> to |0> is introduced. This step is for testing purposes and is omitted to detect natural errors occurring in operation. In 706, a Raman π pulse is applied by a laser beam 606 to place the atom 601 from the |0> state 802 back into the check state 801. In 707, a 10 μs pulse is applied by a laser 604 to detect any atoms in the check state 801. A binary search is performed to identify any atoms that are in the check state 801 (and therefore any qubits that exhibit an error).

[0085] Referring to Figures 9A-D, a binary search is illustrated. In this example, seven qubits are searched in four steps. In Figure 9A, qubits 1 through 7 are placed in the cavity (shown in dark colors), and qubits 8 through 14 are not placed in the cavity (shown in light colors). A measurement is performed as described above to detect the presence of an error in this group of seven qubits, indicated as "SPCM YES!". In Figure 9B, qubits 4 through 7 (half of the qubits) are removed from the cavity, leaving qubits 1 through 3. A measurement is performed, indicating that qubits 1 through 3 are error-free, indicated as "SPCM NO!". Therefore, the error must be in qubits 4 through 7. In Figure 9C, qubits 1 through 3 are removed from the cavity, and half of the qubits (4 through 5) are reintroduced. A measurement is performed, indicating that qubits 4 through 5 are error-free. In Figure 9D, half of the qubits are removed from the cavity, leaving only qubit 5. A measurement is performed and shows that there is no error in qubit 5. Therefore, the error must be in qubit 4.

[0086] Workflow As described above, Figure 5 shows two unit modules 501, 502, each consisting of an atomic array 503, 504 and an optical cavity 506, 507. The workflow is further illustrated with reference to Figures 10-11, which provide schematic diagrams of free-space entanglement and cavity entanglement embodiments, with arrows indicating atom transport. Modules 1001, 1002, 1101, 1102 include magneto-optical traps (MOTs) 1003, 1004, 1103, 1104, which serve as atom sources. Atoms are transferred to code blocks (Rydberg arrays) 1005, 1006, 1105, 1106 and to free-space entanglement devices 1007, 1008, such as APD arrays or cavity entanglement devices 1107, 1108. Atoms are initialized in the tweezers and loaded into positions suitable for entanglement manipulation via a cavity or Rydberg gate. For each code cycle to run the error-correcting code, a 2d Bell pair for communication between the two modules is generated using photons 1009, 1109 and extracted by optical cavity or free-space entanglement, which takes a total time of ~100 μs.

[0087] After entanglement is predicted by detecting the extracted photon, the Bell pairs are transported to the required positions at the edges of arrays 1005, 1006, 1105, and 1106 in approximately 100 μs, where they are used to define teleported gates for syndrome checking across the seam, effectively merging the two atomic arrays into a single logical code patch. Once the entangled pairs are in place, local Rydberg gates between the data and syndrome qubits perform the parity-check syndrome operation of Figure 5, which takes only a few microseconds.

[0088] Once all of the local Rydberg gates have been performed, the Bell pair qubit and syndrome qubit can be transported back to the cavities 1010, 1011, 1110, 1111 for fast non-destructive readout, requiring approximately 40 μs and 30 μs, respectively, as previously estimated.

[0089] In summary, even if each of the different steps described above are done sequentially, the time for a single error correcting code cycle is still several hundred μs, which is less than the desired target of 1 ms, giving an estimated 1 s coherence time.

[0090] These code cycles are then repeated consecutively, and all d code cycles are sufficient to implement a fault-tolerant logic gate between the two code patches in the two modules.

[0091] With many such modules, and with fiber optic links connecting them all, any logic gate can be performed fault-tolerantly between any of the logical qubits in any of the multiple modules on a timescale of 1-10 ms.

[0092] In Figure 4, we compare the threshold requirements for the large modular architecture described herein (Figure 5) with those of the small modular architecture. We plot the local gate threshold as a function of Bell pair infidelity, showing that our approach enables the execution of surface codes with higher error rates than previously considered possible.

[0093] Referring back to FIG. 1 , two independent surface code patches can each maintain, protect, and error-correct a logically encoded qubit by repeatedly interacting with nearby physical qubits (black and white circles). These interactions between nearby physical qubits in the same code patch are two-qubit gates, e.g., Rydberg gates, and when they occur between physical qubits within the same code patch, they can be referred to as local operations. Error correction within each code patch can function successfully as long as these local operations are performed with a sufficiently low error rate (for surface codes, this is approximately a 1% error rate). To then interact two logical qubits in the two code patches, interactions between qubits in separate code patches, represented by the two-qubit CNOT gate 104 in FIG. 1 , are required, which can be referred to as nonlocal operations. These nonlocal operations are more difficult to achieve with high fidelity than local operations because they occur between physically separated code patches. In the system provided herein, they are realized through the use of Bell pairs distributed between the two code patches, enabling long-distance teleported CNOT gates.

[0094] Conventional analysis estimates that nearby perfectly local operations (<<1%) are necessary to compensate for non-local operations (e.g., ∼10%) that have higher operation errors.

[0095] In contrast, the analysis provided herein shows that two code patches can be made to interact successfully with higher error non-local operations without the need to substantially reduce local operation error. Thus, the present disclosure surprisingly enables interaction of a code patch with a 10% non-local operation error while still using local operations with approximately 1% error.

[0096] Referring to Figures 12A-C, errors resulting from seams are illustrated. Modules 1201 and 1202 each execute a surface code and are connected by seam 1203. Figure 12A shows that only one column of star operators 1204 experiences a higher rate of bit flip and phase flip errors close to seam 1203. No matter how many of these operators experience a phase flip error, it is always detectable and does not result in a logical error. Figure 12B shows that only one column of data qubits 1205 experiences a higher rate of bit flip errors. If a majority of these data qubits experience bit flip errors, a logical error occurs. Figure 12C shows that only one column of plaquette operators 1206 experiences a higher rate of bit flip errors.

[0097] The seam thus forms a quasi-1D system with two columns of qubits that experience a high rate of errors, but only one column corresponding to logical bit flips, as well as a column of plaquette operators and a column of star operators that also experience a higher rate of errors. Thus, imperfect syndrome extraction is integrated into threshold simulations.

[0098] Referring to Figure 13, the results of a numerical simulation using a noisy syndrome are shown. The surface code with the noisy syndrome has a threshold of approximately 3%. The repetition code with the noisy syndrome has a threshold of 10%. Both are plotted here to estimate the error rate for the repetition code, which is 3x larger than for the surface code. Surface codes with L = 3x3, 7x7, and 11x11 are shown by lines 1301, 1302, and 1303, respectively. Repetition codes with L = 3x3, 7x7, and 11x11 are shown by lines 1304, 1305, and 1306, respectively.

[0099] Referring to Figure 14, a comparison between LxL codes with and without seams is shown. These charts include noisy syndrome extraction, and it is estimated that the seam error is 3x worse than the seam error. The prefactor is removed to make the slopes easier to compare. Codes with seams and p=0.0025, 0.005, 0.01, and 0.015 are shown by lines 1301, 1302, 1303, and 1304, respectively. Codes without seams and p=0.0025, 0.005, 0.01, and 0.015 are shown by lines 1305, 1306, 1307, and 1308, respectively.

[0100] Network noise propagation for teleported gates When the interface is realized via distributed entanglement, the entanglement serves as the basis for defining nonlocal teleported gates between qubits in different modules. Figure 2 shows how bit and phase flip noise on a distributed Bell pair propagates to the reference and target qubits in the separate modules where the teleported gate operates. As should be the case since the Bell pair is invariant under the application of XX and ZZ, the propagation is identical to the error occurring in either of the Bell pair qubits. The X and Z noise on the Bell pair (shown by curve 201) used in the teleported gate propagates to the two qubits it operates on. Phase flips propagate only to the reference, and bit flips propagate only to the target.

[0101] boundary By counting walks corresponding to homologically nontrivial error strands, the phenomenological threshold bound for the surface code can be lowered. Surface codes with noise-free syndromes can be decoded by pairing defects on the lattice in 2D and including noisy syndromes, which is extended to the problem of pairing further propagating defects in time as a 3D matching problem. Previous bounds in 2D and 3D, while not narrow, were within about three times the true threshold.

[0102] "Seam" subspace grid (dimension D s <D b ) containing a "bulk" conforming graph lattice (of dimension D b ), we calculate the number of walks and their probabilities, including those that span both the bulk and the seams. b =2, D s = 1 and for the noisy syndrome D b =2+1, D s For σ = 1 + 1), pairing defects on the fitting graph allows decoding of the surface code of the corresponding dimension. If the bulk operates slightly below its own threshold, it can be quantified how strongly the probability of a long "excursion" away from the seam is suppressed. For the maximum objective case (D b =3, D s = 2), and show that it is possible to increase the noise on the seams in exchange for a moderate reduction in noise in the bulk.

[0103] Although these bounds are not narrow, and the combinatorics that track hopping between seams and bulk over-count the walks to produce less impressive results than revealed by exact numerical simulations, the derivatives illustrate many central ideas important for understanding the results and provide insight into more complex situations, such as how code performance is affected by having multiple different seams operating within the same bulk.

[0104] Adaptation graph and MWPM decoder Figure 15 provides an explanation for the following discussion.

[0105] FIG. 16 illustrates an exemplary adaptive grid.

[0106] Since the set of edges in the adaptive lattice {M} corresponds to the data qubits and its vertices correspond to the syndromes, the bound on the set of errors on {M} is

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[0107] During one round of error correction, if enough bits are flipped due to environmental noise combined with the corrections attempted to form several non-trivial chains {γ} across the code, a logic failure occurs. In each round, the errors introduce a random set of bit flips {E}, which occur on both seam edges and bulk edges as solid and hollow Xs, respectively. Minimum weight perfect matches (MWPMs) further bit flip the dashed edges {R} on the seams and bulk, returning the state to code space. The remaining Xs and dashes are then joined together in the set {E+R}. In this example, {E+R} contains a non-trivial chain {γ}.

[0108] "Edge" refers to the position of a qubit in the adaptation graph. Quantities in brackets {} refer to adaptation graph subsets, and quantities without brackets refer to the size of such subsets. {E} is the set of (possibly cut) edges where a bit-flip error occurs in a given round of error correction, and {R} is the set of edges selected via MWPM that are bit-flipped to attempt to correct {E}. Sum of Error and Recovery Steps

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[0109] The set of errors {E} is

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[0110] Next, MWPM

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[0111] Probability of failure through a particular {γ} The first goal is to ensure that one round of errors, followed by a correction flip from MWPM, S Seam and gamma B The goal is to create a set of bit flips that contains a particular {γ} with a bulk edge, thereby revealing an upper bound on the probability of a logic failure.

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[0112] By definition, MWPM ensures that wt({R})≦wt({E}).

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[0113] Next,

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[0114] By applying the weighting formula 30 definitions, the overall relationship is:

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[0115] The following is Prob(γ SE ,γ BE ) and accurately determine the boundary of γ SE ,γ BE The probability of generating {E} with bit flips overlaps with {γ}. This Prob(γ SE ,γ BE ) is γ S From γ SE and γ B From γ BE is the number of ways to select , which can be bounded,

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[0116] where

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[0117] Since only logical failures are possible if a subsequent round of MWPM correction generates a set of bit flips involving some {γ} that connect opposite sides of the surface code, we then calculate the total logical failure probability P by summing the failure probabilities Equation 35 for all such {γ} that connect two sides of the surface code. fail The boundaries of can be determined:

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[0118] Calculating chains with nontrivial errors P given in Equation 36 failThe upper bound of still depends on all possible {γ}. To obtain an expression that can be more easily evaluated, the following instead sums over a larger set of self-avoiding walks (SAWs) of length l ≥ L. Since all {γ} are such SAWs, then P fail A weaker upper bound on the above can be obtained. In order to assign accurate probabilities, we first discuss below a parameterization of SAWs that accounts for the number of bulk and seam edges they contain.

[0119] In general, the number of SAWs of some fixed length l is defined as n SAW It is shown by n SAW To understand how grows with l, first consider growing a walk on a square lattice by adding edges of some dimension D. A larger D results in faster exponential growth in the number of walks, since it means more directions to choose from all the time edges are added. For a walk of length l, each new edge can be added in any available direction except back towards itself, so one bound is:

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[0120] In this case, γ S We need to count the number of walks as a function of γ and the number of ways to insert bulk displacement segments that bounce off and on the seam. S For walks with seam edges, each length l k To select a location of C along the length of the walk (denoted by k) at which to insert the "displacement"

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[0121] where γ S , C, {l k For a fixed separation of the walk into seam and bulk segments with}, the length of the walk can be traced and the number of directions available for adding edges when extending the walk at a seam or bulk and when encountering a corner can be calculated. If we restrict the edges being added to seams or during bulk displacements,

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[0122] If C=0, there is no SAW with both seam and bulk edges, and the walk is either only in the bulk or in the seam, so S=0 or B=0:

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[0123] Bounding logic failure probabilities Prob SAW (γ s ,{l k},C) be the total probability for generating the error set {E}, then the edge γ s , {l k} and any {γ} with a number C is included in {E+R}. s , {l k} and all {γ} with C are generated by the same probability boundary given by Eq. 35, so from Eq. 35 we can obtain the probability Prob SAW (γ s ,{l k},C) takes a walk n SAW (γ s ,{l k},C). If there are possible poly(L) edges to start a walk on in the compatibility graph:

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[0124] Since forming a homologically non-trivial loop requires an error chain walk with at least L edges extending along the seam in space or time, the number of edges of the combined seam and bulk displacements is large enough to ensure a logic failure probability P fail The boundaries of can be determined:

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[0125] n SAW Incorporating the expressions 38 and 39 into eq. 41 yields terms corresponding to pure bulk chains (S=0) and pure seam chains (B=0), with further summations for walks over seams and bulks where there is at least one displacement C:

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[0126] Here, γ s when < L, all of these geometric sums cannot start with l k = 1. To achieve this, constraints

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[0127] The true upper bound on the sum is determined when the bulk volume is filled by the displacement, so the constraint

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[0128] To handle displacement, we extend the upper bound to k By varying independently up to infinity, the bounds can be simplified and still hold:

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[0129] Then, the value

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[0130] Substituting this into equation 42 gives:

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[0131] Here, identity

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[0132] Where, if there is no displacement C=0:

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[0133] where:

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[0134] A similar argument applies to the crossover term:

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[0135]

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[0136] If we re-express Equation 51, it can be seen that this is equivalent to a small "sag" below the threshold boundary:

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[0137]

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[0138] Equation 54 suggests not only which combination of seam and bulk errors will achieve threshold, but also, important for determining scaling effectiveness, that when the bulk is itself subthreshold, the subthreshold behavior quickly approaches that of the bulk and seam independently. From the above numbers,

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[0139] Referring to Figures 17A-C, analytical logic failure boundaries are shown. b =3 (dotted line) and D s = 2 (dashed line), threshold boundary

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[0140] Referring to Figures 17D-F, numerical simulations are shown. Figure 17D is the same as Figure 17A, but provides a diagram with the exact numerical simulation. Figure 17E is the same as Figure 17B for L=9 and 11, but for the selection p seam =14p bulk ,about

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[0141] Extending the model to two seams By adding multiple seams along the space and / or time directions and calculating paths to jump between different seams, this formalism can be used to understand situations involving repeated crossing gates and code patches that span multiple modules. As shown in Figure 18, this is done by b = 3 Two parallel D s = 2 seams results in an additional threshold sag. In this form, this additional sag can be captured by adding a term that describes the path that jumps between opposite seams, and from this it can be seen that for larger h, the additional sag disappears because longer jumps that traverse further through the bulk are suppressed in probability.

[0142] In this section, these approaches are generalized to understand the case of having these two seams embedded in the same bulk, where error strands can jump between seams in addition to jumping from and to the same seam. We can interpret the results from the previous section in the following way: γ S Increasing by 1 to add more seam edges is:

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[0143] This factor can be interpreted as a sum over different possible ways to add the next seam edge weighted by the "probability" associated with each edge (indeed, the square root of the "probability", due to the above discussion that MWPM can fill in missing edges). The seam edge can then be locally added (μ s Having choices

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[0144] This kind of approach is useful not only for understanding how displacement from a single seam returning to itself behaves, but also for understanding other situations, such as when you have multiple seams in a code. For example, if there are two seams, it makes sense to consider an additional term in the above sum to account for displacements jumping between seams in addition to jumping away from / onto the same seam. In such cases, a term corresponding to all paths leaving one of the seams and reconnecting the opposite seam can be qualitatively added:

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[0145] As done in the previous section, we can collapse the threshold shift for a single seam:

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[0146] As before, we define the whole term in the bracket as

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[0147] Here, the excessive threshold shift disappears for large h; then the long jump through the bulk between seams is highly suppressed and becomes a negligible contributor to the process for adding the edge of the next seam.

[0148] for example

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[0149] Referring to Figure 18, the effect on threshold for two close parallel seams is plotted,

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[0150] Microscopic seam error model The Bell pair and CNOT gates are each modeled as a complete operation followed by all combinations of X, Y, and Z errors on two qubits, each of which is an error

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[0151] Using this error model, we consider a standard surface code patch, where there are no seams and errors arise only from CNOT gate errors and readout errors. If each data qubit is subjected to four parity-check operations per code cycle, the bit-flip probability for a data qubit is

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[0152] Next, consider the situation where two surface code patches are integrated via teleported gates across a seam. Stretching the parity check operator across the seam involves one teleported gate per data / syndrome qubit pair on the seam. Figure 2 shows how the teleported gates propagate bit and phase (X and Z) errors that occur on the bell pair. Bit-flip errors on the bell pair propagate exclusively to the target qubit. Similarly, phase-flip errors on the bell pair propagate exclusively to the control qubit. In total, the bit-flip probability on the control qubit is

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[0153] For a qubit on a seam, the phenomenological weighted error model is as follows: A Plaquette syndrome qubit on a code patch (CP)2 has three local CNOTs (

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[0154] Small module error model For comparison, in the minimum size module configuration, where each module contains a single data or syndrome qubit and just one communication qubit, teleported gates must be used for each two-qubit operation in every check operator, resulting in substantially worse performance.

[0155] In each code cycle, a given data qubit is the target of two teleported CNOTs and the control of two teleported CNOTs, and per code cycle:

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[0156] Similarly, considering the four teleported CNOT target Plackett syndrome qubits, we can see further readout errors.

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[0157] Also, the threshold occurs at p=3% for q=p, so

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[0158] Figure 19 shows the phenomenological bit-flip error probability per code cycle p and q on the data and syndrome qubits. The entries describe how the local operations and bell pairs add to the total phenomenological error probability during a given code cycle. The phase-flip error rate is identical. The "Bulk" and "Seam" columns correspond to the regions shown in Figure 1; for comparison, the "Small Module" column shows the case where all gates in the surface code are made using teleported gates.

[0159] Formation of particle arrays using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a vacuum. Atoms are polarized, and the oscillating electric field of a light beam induces an oscillating electric dipole moment in the atoms. The associated energy shift in the atoms from the induced dipole, averaged over the period of the light oscillation, is called the AC Stark shift. Based on the AC Stark shift induced by light detuned (i.e., offset in wavelength) from the atomic resonance transition, atoms are attracted to light below the resonance frequency and are therefore trapped at a local intensity maximum (for detuned red, i.e., longer wavelength, trapping 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 atom trap. Optical tweezers exploit this principle by focusing a laser into a micron-scale constriction, where individual atoms are trapped at the focus. Two-dimensional (2D) arrays of optical tweezers can be created, for example, by illuminating a spatial light modulator (SLM) that imparts a computer-generated hologram to the wavefront of the laser field. The 2D array of optical tweezers overlaps with a cloud of laser-cooled atoms in a magneto-optical trap (MOT). The tightly focused optical tweezers operate in the "collision blockade" regime, where single atoms are loaded from the MOT, and pairs of atoms are ejected for optically assisted collisions, ensuring that at most one atom is loaded into the tweezers. However, since loading is probabilistic, there is approximately a 50-60% chance that the trap will contain a single atom.

[0160] To prepare deterministic atomic arrays, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into preprogrammed geometries. Atomic rearrangement requires moving atoms in tweezers, which can be facilitated to minimize heating, for example, by using an acousto-optical deflector (AOD) to deflect a laser beam by an adjustable angle controlled by the frequency of an acoustic waveform applied to the AOD crystal. Dynamic tuning of the acoustic frequency translates into smooth movement of the optical tweezers. Multi-frequency acoustic waves generate an array of laser deflections, which, after focusing through a microscope objective, 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 overlaid on top of the SLM tweezers array.

[0161] Exemplary Hardware Optical tweezers arrays constitute a powerful and flexible method for constructing large-scale systems composed of individual particles. Each optical tweezers traps a single particle, including, but not limited to, individual neutral atoms and molecules for applications in quantum technology. Loading individual particles into such tweezers arrays is a stochastic process, in which each tweezers in the system is loaded with a single particle with a finite probability p<1, e.g., p~0.5, over many neutral atom tweezers runs. To compensate for this random loading, real-time feedback can be obtained by measuring which tweezers are loaded and then sorting the loaded particles into programmable geometric structures. This can be done by moving one particle at a time or in parallel.

[0162] Parallel sorting can be achieved by using two acousto-optical deflectors (AODs) to create multiple tweezers that can pick up particles from an existing particle trapping structure, move them simultaneously, and release them elsewhere. This can involve moving particles around within a single trapping structure (e.g., a tweezer array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezer array and another type of optical / magnetic trap). This sorting is flexible, allowing for the programmed placement of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of the AOD's radio frequency (RF) driving field. Because the AOD's RF driving can be controlled in real time and can include any combination of frequency components, it is possible to create an arbitrary grid of traps (such as a line of arbitrarily positioned traps), move rows or columns of the grid, and add or remove rows and columns of the grid by varying the number, magnitude, and distribution of frequency components in the AOD's RF driving field.

[0163] In an exemplary embodiment, optical tweezers arrays are generated using liquid crystals on a silicon spatial light modulator (SLM), which can programmatically generate flexible arrangements of tweezers. These tweezers are fixed in space for a given experimental sequence and stochastically loaded with individual atoms, so that each tweezers is loaded with a probability of p ∼0.5. Fluorescence images of the loaded atoms are taken to identify in real time which tweezers are loaded and which are empty.

[0164] After detecting which tweezers are loaded, the optical tweezers array and overlapping movable tweezers can dynamically reposition the atoms from their starting positions to fill the target configuration of the trap with near-uniform packing. The movable tweezers are generated using a pair of crossed AODs. These AODs can be used to move one atom at a time to fill the target configuration or to generate a single movable trap that moves many atoms in parallel.

[0165] Referring to FIG. 20 , a schematic diagram of an apparatus 2000 for fault-tolerant quantum computing according to an embodiment of the present disclosure is provided. As shown in FIG. 20 , using a beam generated by a light source 2002 (e.g., a coherent light source; in some exemplary embodiments, a monochromatic light source), an SLM 2004 forms an array of trapping beams (i.e., a tweezer array), which, in the exemplary embodiment shown in FIG. 20 , is imaged onto a trapping plane 2008 within a vacuum chamber 2010 by an optical train including elements 2006 a, 2006 c, 2006 d and a high numerical aperture (NA) objective lens 2006 e. Other suitable optical trains can be used, as will be readily understood by those skilled in the art. Using a beam generated by a light source 2012 (e.g., a coherent light source; in some exemplary embodiments, a monochromatic light source), a pair of AODs 2014 and 2016 with non-parallel (e.g., orthogonal) directions of acoustic wave propagation generate dynamically movable sorting beams. The sorting beam is overlapped with the trapping beam using an optical series such as that shown in Figure 20 (elements 2017, 2006b, 2006c, 2006d, and 2006e). It will be appreciated that other optical series may be used to achieve the same result. For example, sources 2002 and 2012 may be a single source, with the trapping and sorting beams generated by a beam splitter.

[0166] Dynamic motion of the steering beam is achieved using two non-parallel AODs 2014, 2016 arranged in series. In the exemplary embodiment shown in FIG. 20 , one AOD defines the “row” (“horizontal”—‘X’ AOD) direction, and the other defines the “column” (“vertical”—‘Y’ AOD) direction. Each AOD is driven by an arbitrary RF waveform from an arbitrary waveform generator 2020, which is generated in real time by a computer 2022 that processes a feedback routine after analyzing images of the atom loading positions. When each AOD is driven with a single frequency component, a single steering beam (“AOD trap”) is generated in the same plane 2008 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap, and the frequency of the Y AOD drive determines the vertical position; in this way, the single AOD trap can be stepped to overlap any SLM trap.

[0167] In Figure 20, laser 2002 shines a beam of light onto SLM 2004. SLM 2004 can be controlled by computer 2022 to generate a pattern of beams (the "trapping beam" or "tweezer array"). The beam pattern is focused by lens 2006a, passes through mirror 2006b, and is collimated by lens 2006c on mirror 2006d. The reflected light passes through objective lens 2006e to focus the optical tweezers array in vacuum chamber 2010 on trapping plane 2008. The optical tweezers array laser light continues through objective lens 2024a, passes through dichroic mirror 2024b, and is detected by charge-coupled device (CCD) camera 2024c.

[0168] The vacuum chamber 2010 may be illuminated by an additional light source (not shown). Fluorescence from atoms trapped on the trapping surface also passes through the objective lens 2024a but is reflected by the dichroic mirror 2024b onto an electron multiplying charge-coupled device (EMCCD) camera 2024d.

[0169] In this example, a laser 2012 directs a beam of light to AODs 2014, 2016. The AODs 2014, 2016 are driven by an arbitrary wave generator (AWG) 2020, which in turn is controlled by a computer 2022. The cross AODs 2014, 2016 emit one or more beams as described above, which are directed to a focusing lens 2017. The beams then enter the same optical train 2006b...2006e as described above for the optical tweezers array and are focused onto a trapping plane 2008.

[0170] It will be appreciated that alternative optical series components may be used to create an optical tweezers array suitable for use as described herein.

[0171] Excitation of atoms to Rydberg states in an optical tweezers array In separate optical tweezers on the micrometer-length scale, atoms in their ground electronic state have negligible van der Waals interactions. Fortunately, neutral atoms offer a prominent way to switch on strong interactions by coherently exciting the atoms into their Rydberg states.

[0172] The properties of atomic states scale dramatically with the principal quantum number. A Rydberg state is a highly excited electronic state of an atom, in which one of the atom's electrons has a high principal quantum number, n, in the range of 30-100. In the classical picture of the atom, this situation corresponds to one (negatively charged) electron orbiting far away from the (positively charged) ionic core on the atomic length scale, thus forming an oscillating electric dipole. Two atoms excited to the same Rydberg state can exhibit very strong dipole-dipole interactions over distances of tens of microns. Interaction energy

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[0173] Consider an ideal two-level atom with a ground state |g> and a Rydberg state |r>. These two states are laser coupled with a coupling strength set by the angular Rabi frequency Ω, which is an inverse function of the duration of the Rabi cycle and is also called a Rabi flop, which is the periodic absorption and stimulated emission of quanta of energy by the two-level atom in the presence of an oscillatory driving field. The Rabi frequency is proportional to the strength of the coupling between the light and the atomic transition and the amplitude of the electric field of the light. For two such atoms, also referred to herein as Rydberg atoms, the van der Waals interaction energy

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[0174] Several implementations of optical excitation from atomic ground states to target Rydberg states are available. Direct laser excitation using single-photon transitions is the simplest. Wavelengths for such transitions in Rydberg atoms are typically in the ultraviolet. For example, 87The single-photon wavelength for Rb is 297 nm. Ultraviolet lasers pose significant experimental difficulties, for example, due to material decomposition and the unavailability of fiber optics and low-loss optics. Alternatively, two-photon laser excitation can be used to couple the atomic ground state to the target Rydberg state via an intermediate electronic excited state by irradiating the atom from opposite faces with two counterpropagating laser beams.

[0175] Consistent with the above description, the term "blockade" is used herein to refer to the phenomenon in which a laser-stimulated transition of an atom in an interacting atom pair from a first state (e.g., ground state) to an excited state cannot be achieved (is blocked) due to a mismatch between the laser frequency and the shifted energy level of the excited state, where the energy level shift is induced electrically or magnetically. For example, blockade can be achieved by dipole-dipole interaction between two adjacent atoms, where one atom is excited to a Rydberg state.

[0176] Detuning from a resonance with an excited state The coherent evolution of two atoms under laser excitation from the ground state |g> to the Rydberg state |r> is given by the Hamiltonian

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[0177] Furthermore, in two-photon laser excitation schemes, two excitation lasers, typically with one frequency in the blue range of the optical spectrum, e.g., 420 nm, and another frequency in the red or infrared, e.g., 1013 nm, are excited into an intermediate state (

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[0178] It is understood that in various embodiments, the pulse sequences described herein can be generated by computer control of a laser source. Similarly, the detection of conditions described herein can be performed by various techniques known in the art and provided to a computer controller. Thus, it is understood that in various embodiments, computer instructions can be provided to carry out the control and detection steps described herein.

[0179] It will also be appreciated that various methods can be used to read out the state of an array of atoms. For example, quantum gas microscopy can be used to determine whether each atom in the array is in an excited state or a ground state, as described in Browaeys, et al., Many-Body Physics with Individually-Controlled Rydberg Atoms, DOI: 10.1038 / s41567-019-0733-z (available at https: / / arxiv.org / abs / 2002.07413), which is incorporated herein by reference in its entirety.

[0180] Coherent transport of entangled atoms The above-described device can be used to store quantum information in hyperfine states and provide coherent transport of neutral atoms while preventing quantum coherence and entanglement between qubits by shuttling the atoms in optical tweezers. This approach enables the transport of atoms into and out of multiple arrays, cavities, or other modules of an integrated quantum computing system.

[0181] For dynamic reconfiguration, we use mobile traps generated by crossed 2D acousto-optic deflectors (AODs), which allow the transport of atoms to and from static traps, such as those generated by spatial light modulators (SLMs), as well as to and from other modules of the system.

[0182] The transport protocol is optimized to suppress heating and loss by implementing cubic-interpolated atom trajectories, further accompanied by an eight-pulse XY8 robust dynamical decoupling sequence to suppress detuning. Fidelity remains unchanged until the total separation rate is >0.55 μm / μs, corresponding to the onset of atom loss.

[0183] More generally, entanglement is preserved when atoms are transferred adiabatically. The term adiabatic transfer refers to transfer that avoids transitions of the target atom within the trap. For example, a transfer is considered adiabatic if the first time derivative of the target atom's acceleration is not greater than a predetermined value. Typically, adiabatic transfer is considered adiabatic if jerk<(atom size)×(trap frequency) 3 This occurs when:

[0184] Further data on coherent transport is provided in Bluvstein, et al., A quantum processor based on coherent transport of entangled atom arrays, Nature 604, 451-456 (2022) (available at https: / / arxiv.org / abs / 2112.03923), which is incorporated herein by reference.

[0185] 21, a schematic of an example computing node is shown. Computing node 10 is only one example of a suitable computing node and is not intended to suggest any limitation as to the scope of use or functionality of the embodiments described herein. Regardless, computing node 10 may be implemented and / or may perform any of the functions described herein above.

[0186] Among computing nodes 10 are computing systems / servers 12 that are operable with many other general-purpose or special-purpose computing system environments or configurations. Examples of well-known computing systems, environments, and / or configurations that may be suitable for use with computing system / server 12 include, but are not limited to, personal computing systems, server computing systems, thin clients, thick clients, handheld or laptop devices, multiprocessor systems, microprocessor-based systems, set-top boxes, programmable consumer electronics devices, network PCs, minicomputer systems, mainframe computing systems, and distributed cloud computing environments that include any of the above systems or devices.

[0187] The computer system / server 12 may be described in the general context of computer system-executable instructions, such as program modules, being executed by a computer system. Generally, program modules may include routines, programs, objects, components, logic, data structures, etc. that perform particular tasks or implement particular abstract data types. The computer system / server 12 may be practiced in a distributed cloud computing environment where tasks are performed by remote processing devices that are linked through a communications network. In a distributed cloud computing environment, program modules may be located in both local and remote computer system storage media, including memory storage devices.

[0188] 21, the computer system / server 12 in the computing node 10 is shown in the form of a general-purpose computing device. Components of the computer system / server 12 may include, but are not limited to, one or more processors or processing units 16, a system memory 28, and a bus 18 coupling various system components such as the system memory 28 to the processor 16.

[0189] Bus 18 represents any one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of a variety of bus architectures, including, by way of example and without limitation, an Industry Standard Architecture (ISA) bus, a MicroChannel Architecture (MCA) bus, an Enhanced ISA (EISA) bus, a Video Electronics Standards Association (VESA) local bus, a Peripheral Component Interconnect (PCI) bus, a Peripheral Component Interconnect Express (PCIe), and an Advanced Microcontroller Bus Architecture (AMBA).

[0190] Computer system / server 12 typically includes a variety of computer system-readable media, which may be any available media that can be accessed by computer system / server 12 and includes both volatile and nonvolatile media, removable and non-removable media.

[0191] System memory 28 may include computer system-readable media in the form of volatile memory, such as random access memory (RAM) 30 and / or cache memory 32. Computer system / server 12 may further include other removable / non-removable, volatile / non-volatile computer system storage media. By way of example only, storage system 34 may be provided for reading from and writing to non-removable, non-volatile magnetic media (not shown, typically referred to as a "hard drive"). Although not shown, a magnetic disk drive may be provided for reading from and writing to a removable, non-volatile magnetic disk (e.g., a "floppy disk"), and an optical disk drive may be provided for reading from or writing to a removable, non-volatile optical disk, such as a CD-ROM, DVD-ROM, or other optical media. In such an example, each may be coupled to bus 18 by one or more data media interfaces. As further shown and described below, memory 28 may include at least one program product having a set (e.g., at least one) program module configured to perform the functions of aspects of the present disclosure.

[0192] A program / utility 40 having a set (at least one) program module 42, as well as, by way of example and not limitation, an operating system, one or more application programs, other program modules, and program data, may be stored in memory 28. Each of the operating system, one or more application programs, other program modules, and program data, or some combination thereof, may comprise the execution of a network environment. The program modules 42 generally perform the functions and / or methodologies of the embodiments described herein.

[0193] The computer system / server 12 may also communicate with one or more external devices 14, such as a keyboard, pointing device, display 24, one or more devices that allow a user to interact with the computer system / server 12, and / or any devices (e.g., network cards, modems, etc.) that allow the computer system / server 12 to communicate with one or more other computing devices. Such communication may occur via an input / output (I / O) interface 22. Furthermore, the computer system / server 12 may communicate with one or more networks, such as a local area network (LAN), a general wide area network (WAN), and / or a public network (e.g., the Internet), via a network adapter 20. As shown, the network adapter 20 communicates with other components of the computer system / server 12 via a bus 18. While not shown, it should be understood that other hardware and / or software components may be used with the computer system / server 12. Examples include, but are not limited to, microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data archival storage systems.

[0194] The present disclosure may be embodied as a system, method, and / or computer program product, which may include computer-readable storage medium(s) having computer-readable program instructions thereon for causing a processor to implement aspects of the present disclosure.

[0195] A computer-readable storage medium may be a tangible device that can hold and store instructions for use by an instruction-execution device. A computer-readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of computer-readable storage media includes the following: portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disc read-only memory (CD-ROM), digital versatile disc (DVD), memory sticks, floppy disks, punch cards, or mechanically encoded devices such as raised structures in grooves with instructions recorded on the grooves, and any suitable combination of the foregoing. As used herein, computer-readable storage media is not understood to be transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission medium (e.g., light pulses through a fiber optic cable), or electrical signals transmitted through wires.

[0196] The computer-readable program instructions described herein may be downloaded from a computer-readable storage medium to each computing / processing device or to an external computer or external storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network may include copper transmission cables, fiber optic transmissions, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions for storage in a computer-readable storage medium within the respective computing / processing device.

[0197] The computer-readable program instructions for carrying out the operations of the present disclosure may be either source code or object code written in any combination of assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state-setting data, or one or more programming languages, such as object-oriented programming languages like Smalltalk, C++, and conventional procedural programming languages like the "C" programming language or similar programming languages. The computer-readable program instructions may execute entirely on the user's computer, partially on the user's computer as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, electronic circuitry, including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA), can execute computer-readable program instructions using state information of the computer-readable program instructions to personalize the electronic circuitry to implement aspects of the present disclosure.

[0198] Aspects of the present disclosure are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer-readable program instructions.

[0199] These computer-readable program instructions may be provided to a processor of a general purpose computer, special purpose computer or other programmable data processing apparatus to manufacture a machine, and the instructions, executing via the processor of the computer or other programmable data processing apparatus, create means for performing the functions / acts identified in the flowchart and / or block diagram block(s). These computer-readable program instructions may also be stored on a computer-readable storage medium that can direct a computer, programmable data processing apparatus and / or other device to function in a particular manner, and a computer-readable storage medium having instructions stored therein includes an article of manufacture containing instructions that perform aspects of the functions / acts identified in the flowchart and / or block diagram block(s).

[0200] The computer-readable program instructions may also be loaded into a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be executed on the computer, other programmable apparatus, or other device to create a computer-implemented process, the instructions executing on the computer, other programmable apparatus, or other device performing the functions / acts identified in the flowchart and / or block diagram block(s).

[0201] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and possible implementation operations of systems, methods, and computer program products according to various aspects of the present disclosure. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of instructions, including one or more executable instructions for performing a specific logical function(s). In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially simultaneously, or the blocks may sometimes be executed in reverse order depending on the functionality involved. It is also noted that each block of the block diagrams and / or flowchart representations, and combinations of blocks in the block diagrams and / or flowchart representations, may be implemented by a special-purpose hardware-based system that performs specific functions or acts or a combination of special-purpose hardware and computer instructions.

[0202] Thus, in a first exemplary embodiment, the invention is a quantum computing system. In a first aspect of the first embodiment, the system includes: a first array and a second array of neutral atoms, each array having one-dimensionality; each neutral atom having a first state and an excited Rydberg state, and each neutral atom is aligned such that when in the excited Rydberg state it imposes Rydberg blockade on at least its nearest neighbors in the array, thereby implementing a plurality of physical qubits; and each array includes a plurality of data qubits and a plurality of syndrome qubits, and for each array, the plurality of syndrome qubits are configured to implement a quantum error correction code with respect to the data qubits. The first array of neutral atoms includes a first subarray of communication qubits, and the second array of neutral atoms includes a second subarray of communication qubits, the first and second subarrays having two-dimensionality less than one-dimensionality; each communication qubit of the first subarray forms a Bell pair with one communication qubit of the second subarray; and the first and second arrays of neutral atoms are configured to interact with each other only through the communication qubits.

[0203] In a second aspect of the first embodiment, the first array of neutral atoms comprises a first edge and the second array of neutral atoms comprises a second edge, wherein: the first subarray of communication qubits is disposed on the first edge and the second subarray of communication qubits is disposed on the second edge. The remainder of the features and example features are as described above with respect to the first aspect of the first embodiment.

[0204] In a third aspect of the first embodiment, for each array of neutral atoms, the plurality of syndrome qubits includes a plurality of Z syndrome qubits and a plurality of X syndrome qubits configured to implement X and Z stabilizers with respect to the data qubits, thereby implementing a quantum error correction code. The remainder of the features and example features are as described above with respect to the first or second aspects of the first embodiment.

[0205] In a fourth aspect of the first embodiment, the system further includes a coupling unit configured to create a Bell pair of the first and second communicating qubits and transport the first communicating qubit to and / or from the first array and the second communicating qubit to and / or from the second array, the remainder of the features and example features being as described above with respect to any of the first through third aspects of the first embodiment.

[0206] In a fifth aspect of the first embodiment, the coupling unit includes first and second resonant optical cavities in optical communication with each other, the first resonant optical cavity configured to receive a first neutral atom, the second resonant optical cavity configured to receive a second neutral atom, and the first and second resonant optical cavities together configured to create a Bell pair from the first and second neutral atoms. The remainder of the features and example features are as described above with respect to any of the first through fourth aspects of the first embodiment.

[0207] In a sixth aspect of the first embodiment, the linking unit includes first and second auxiliary arrays of neutral atoms and first and second avalanche photodiode (APD) arrays in optical communication with the first and second auxiliary arrays of neutral atoms and with each other, the first and second APD arrays together configured to create Bell pairs from the first and second auxiliary arrays of neutral atoms. The remainder of the features and example features are as described above with respect to any of the first through fifth aspects of the first embodiment.

[0208] In a seventh aspect of the first embodiment, each of the first and second arrays of neutral atoms is two-dimensional, the remainder of the features and example features are as described above with respect to any of the first through sixth aspects of the first embodiment.

[0209] In an eighth aspect of the first embodiment, the quantum error correction code is a topological code, a stabilizer code, or a surface code, and the remainder of the features and example features are as described above with respect to any of the first through seventh aspects of the first embodiment.

[0210] In a ninth aspect of the first embodiment, each of the first and second arrays includes: a plurality of data qubits, each of which is a nearest neighbor to two Z syndrome qubits and to two X syndrome qubits; and a plurality of measurement qubits, each of which is a nearest neighbor to four data qubits. The remainder of the features and example features are as described above with respect to any of the first to eighth aspects of the first embodiment.

[0211] In a tenth aspect of the first embodiment, the system further includes at least one confinement system for arranging the neutral atoms in an array, where each neutral atom is disposed at a vertex of a lattice, and where each neutral atom has a Rydberg blockade radius sufficient to block each of at least four nearest-neighbor neutral atoms in the lattice when in an excited Rydberg state. The at least one confinement system includes a laser source aligned to create a plurality of confinement regions; a source of a neutral atom cloud, where the neutral atom cloud is configured to be disposed so as to at least partially overlap with the plurality of confinement regions; and an excitation source for exciting at least some of the neutral atoms from a first state to an excited Rydberg state. The remainder of the features and example features are as described above with respect to any of the first through ninth aspects of the first embodiment.

[0212] In an eleventh aspect of the first embodiment, the grating is a linear grating. The remainder of the features and example features are as described above for any of the first through tenth aspects of the first embodiment.

[0213] In a twelfth aspect of the first embodiment, the neutral atom is 87 Rb atom, 133 Cs atom, 85 Rb atom, 171 Yb atom, 174 Yb atom, 88 Sr atom, 87 Sr atom, 84 Sr atom, 86 Sr atom, 39 K atom, 40 K atom, 41 K atom, 23 Na atom, 6 Li atoms and 7 The remainder of the features and exemplary features are as described above with respect to any of the first through eleventh aspects of the first embodiment.

[0214] In a thirteenth aspect of the first embodiment, the thirteen plurality of data qubits have a CNOT error (p CNOT The remainder of the features and example features are as described above with respect to any of the first through twelfth aspects of the first embodiment. In a fourteenth aspect of the first embodiment, the Bell pair has an error (p Bell The remainder of the features and example features are as described above with respect to any of the first through thirteenth aspects of the first embodiment.

[0215] In a second exemplary embodiment, the invention is a method of performing a logical operation between logical qubits. In a first aspect of the second embodiment, the method includes: providing a quantum computing system as described above with respect to any of the first to fourteenth aspects of the first embodiment; and performing a logical operation between at least one data qubit of a first array and at least one data qubit of a second array.

[0216] In a third exemplary embodiment, the invention is a method of extending a quantum error correcting code across two non-interacting arrays of particles. In a first aspect of the third embodiment, the method comprises extending a quantum error correcting code across the first and second arrays: as described above with respect to any of the first to fourteenth aspects of the first embodiment.

[0217] In a second aspect of the second or third embodiment, the method further includes creating a Bell pair of the third and fourth communicating qubits; and transporting the third communicating qubit to the first array and the fourth communicating qubit to and / or from the second array. The remainder of the features and example features are as described above with respect to the first aspect of the second or third embodiment.

[0218] In a third aspect of the second or third aspects, executing the quantum error correction code includes dividing the plurality of syndrome qubits into a plurality of subsets; and for each of the plurality of subsets, simultaneously measuring the syndrome qubits therein. The remainder of the features and example features are as described above with respect to the first or second aspects of the second or third aspects.

[0219] In a fourth aspect of the second or third aspect, executing the quantum error correction code further comprises sequentially transferring each of a plurality of subsets of syndrome qubits into the optical cavity for said measurement, the remainder of the features and example features being as described above with respect to any of the first through third aspects of the second or third aspect.

[0220] In a fifth aspect of the second or third aspect, measuring a syndrome qubit in each of the plurality of subsets includes placing a syndrome qubit that is not in the subset being measured in a shelf state prior to measurement, with the remainder of the features and example features being as described above with respect to any of the first through fourth aspects of the second or third aspect.

[0221] In a sixth aspect of the second or third aspect, executing the quantum error correction code includes identifying one or more syndrome qubits in an error state by incrementally measuring and dividing a plurality of syndrome qubits into the subsets, the remainder of the features and example features being as described above with respect to any of the first through fifth aspects of the second or third aspect.

[0222] In a seventh aspect of the second or third embodiment, the plurality of data qubits has a CNOT error (p CNOT The remainder of the features and example features are as described above with respect to any of the first through sixth aspects of the second or third embodiment.

[0223] In an eighth aspect of the second or third embodiment, the bell pair has an error (p Bell The remainder of the features and example features are as described above with respect to any of the first through seventh aspects of the second or third embodiment.

[0224] In a fourth exemplary embodiment, the present invention provides a method for generating a plurality of Bell pairs of neutral atoms, each of which has a first state and an excited Rydberg state, each of the plurality of Bell pairs including a first communication qubit and a second communication qubit; transporting each of the first communication qubits of the plurality of Bell pairs to a first array of neutral atoms including a first plurality of syndrome qubits and a first plurality of data qubits; transporting each of the second communication qubits of the plurality of Bell pairs to a first array of neutral atoms including a second plurality of syndrome qubits and a second plurality of data qubits. performing at least one Rydberg gate between the first or second plurality of syndrome qubits and the first or second plurality of data qubits; transporting the first and / or second plurality of syndrome qubits into an optical cavity; driving the optical cavity with a coherent light source; measuring transmission through the optical cavity to thereby detect the presence or absence of errors in the first and / or second plurality of syndrome qubits; and returning the first and / or second plurality of syndrome qubits to their respective arrays of neutral atoms.

[0225] The description of various embodiments of the present disclosure is presented for illustrative purposes and is not intended to be exhaustive or limited 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 selected to best explain the principles of the embodiments, practical applications, or technical improvements to technology found in the market, or to enable others skilled in the art to understand the embodiments disclosed herein.

Claims

1. A quantum computer computing system comprising a first array and a second array of neutral atoms, each array having one dimension, Each neutral atom has a first state and an excited Rydberg state. Each neutral atom, when in the excited Rydberg state, is aligned to impose a Rydberg blockade on at least its nearest neighbor in the array, thereby executing multiple physical qubits. Each array includes a plurality of data qubits and a plurality of syndrome qubits, and for each array, the plurality of syndrome qubits are configured to execute a quantum error correction code with respect to the plurality of data qubits. moreover, The first array of neutral atoms includes a first subarray of communication qubits, the second array of neutral atoms includes a second subarray of communication qubits, and the first and second subarrays have a two-dimensionality lower than the one-dimensionality. Each communication qubit of the first subarray forms a bell pair using one communication qubit of the second subarray. A quantum computing system in which the first and second arrays of the neutral atoms are configured to interact with each other only through the communication qubits.

2. The first array of neutral atoms includes a first edge, and the second array of neutral atoms includes a second edge. The first subarray of the communication qubit is located at the first edge. The system according to claim 1, wherein the second subarray of communication qubits is located at the second edge.

3. The system according to claim 1, wherein for each array of neutral atoms, the plurality of syndrome qubits include a plurality of Z syndrome qubits and a plurality of X syndrome qubits configured to perform X and Z stabilizers with respect to the data qubits of the plurality of data qubits, thereby executing the quantum error correction code.

4. The system according to claim 1, further comprising a coupling unit configured to create a bell pair of first and second communication qubits, and to transport the first communication qubit to and / or from the first array, and the second communication qubit to and / or from the second array.

5. The system according to claim 4, wherein the connecting unit includes first and second resonant optical cavities that communicate optically with each other, the first resonant optical cavity configured to accept a first neutral atom, the second resonant optical cavity configured to accept a second neutral atom, and the first and second resonant optical cavities together configured to form the bell pair from the first and second neutral atoms.

6. The system according to claim 4, wherein the coupling unit comprises first and second auxiliary arrays of neutral atoms, and first and second avalanche photodiode (APD) arrays that optically communicate with the first and second auxiliary arrays of neutral atoms and with each other, the first and second APD arrays together being configured to create the bell pair from the first and second auxiliary arrays of neutral atoms.

7. The system according to claim 1, wherein each of the first and second arrays of neutral atoms is two-dimensional.

8. The system according to claim 1, wherein the quantum error correction code is a topological code, a stabilizer code, or a surface code.

9. Each of the data qubits in the plurality of data qubits of the first and second arrays is the nearest neighbor to two Z syndrome qubits and two X syndrome qubits, The system according to claim 8, wherein each of the first and second arrays further includes a plurality of measurement qubits aligned such that each of the plurality of syndrome qubits in the first and second arrays is the nearest neighbor to four data qubits.

10. A system further comprising at least one confinement system for aligning neutral atoms in an array, wherein each neutral atom is positioned at a vertex of a lattice and each neutral atom has a Rydberg blockade radius sufficient to block each of at least four nearest neighbor neutral atoms in the lattice when it is in the excited Rydberg state, The at least one confinement system is Laser sources aligned to create multiple confinement regions, A source of a neutral atom cloud, wherein the neutral atom cloud is configured to be positioned so as to at least partially overlap with the plurality of confinement regions, and An excitation source for exciting at least some of the neutral atoms from the first state to the excited Rydberg state. The system according to claim 1, including the following:

11. A method for performing a logical operation between logical qubits, wherein the method is - A step of providing a quantum computing system comprising a first array and a second array of neutral atoms, wherein each array is one-dimensional, Each neutral atom has a first state and an excited Rydberg state. Each neutral atom, when in the excited Rydberg state, is aligned to impose a Rydberg blockade on at least its nearest neighbor in the array, thereby executing multiple physical qubits. Each array includes a plurality of data qubits and a plurality of syndrome qubits, and for each array, the plurality of syndrome qubits are configured to execute a quantum error correction code with respect to the plurality of data qubits. Furthermore, the first array of neutral atoms includes a first subarray of communication qubits, and the second array of neutral atoms includes a second subarray of communication qubits, and the first and second subarrays have a two-dimensionality lower than the one-dimensionality. Each communication qubit of the first subarray forms a bell pair with one communication qubit of the second subarray, thereby extending the quantum error correction code across the first and second arrays. The first and second arrays of neutral atoms are configured to interact with each other only through the communication qubits, step, - A step of performing a logical operation between at least one data qubit of the first array and at least one data qubit of the second array. Methods that include...

12. The first array of neutral atoms includes a first edge, and the second array of neutral atoms includes a second edge, The first subarray of the communication qubits is located at the first edge, The second subarray of the communication qubit is located at the second edge. The method according to claim 11.

13. The method according to claim 11, wherein for each array of neutral atoms, the plurality of syndrome qubits include a plurality of Z syndrome qubits and a plurality of X syndrome qubits configured to perform X and Z stabilizers with respect to the data qubits of the plurality of data qubits, thereby executing the quantum error correction code.

14. The steps include creating bell pairs for the third and fourth communication qubits, The steps include transporting the third communication qubit to the first array and the fourth communication qubit to and / or from the second array. The method according to claim 11, further comprising:

15. Executing the quantum error correction code Dividing the aforementioned multiple syndrome qubits into multiple subsets, For each of the aforementioned subsets, the syndrome qubit is measured. The method according to claim 11, including the method described in claim 11.

16. The method according to claim 15, wherein the syndrome qubit is measured simultaneously for each of the plurality of subsets.

17. Executing the quantum error correction code The method according to claim 15, further comprising moving each of the plurality of subsets of syndrome qubits sequentially into the optical cavity for the measurement.

18. Measuring the syndrome qubit in each of the plurality of subsets, The method according to claim 15, further comprising, before measurement, arranging the syndrome qubits that are not in the subset being measured in a shelf-like state.

19. The method according to claim 15, wherein executing the quantum error correction code includes identifying one or more syndrome qubits of an error state by incrementally measuring the plurality of syndrome qubits and dividing them into subsets.

20. A step of forming a plurality of bell pairs of neutral atoms, each neutral atom having a first state and an excited Rydberg state, and each of the plurality of bell pairs including a first communication qubit and a second communication qubit, A step of transporting each of the first communication qubits of the plurality of bell pairs to a first array of neutral atoms, wherein the first array of neutral atoms includes a first plurality of syndrome qubits and a first plurality of data qubits. A step of transporting each of the second communication qubits of the plurality of bell pairs to a second array of neutral atoms, wherein the second array of neutral atoms includes a second plurality of syndrome qubits and a second plurality of data qubits. The steps include performing at least one rude verifier between the first or second plurality of syndrome qubits and the first or second plurality of data qubits, The steps include transporting the first and / or second plurality of syndrome qubits into an optical cavity, The steps include driving the optical cavity using a coherent light source, The steps include measuring the transmittance through the optical cavity to detect the presence or absence of errors in the first and / or second plurality of syndrome qubits, The steps include returning the first and / or second plurality of syndrome qubits to the first and / or second arrays of their respective neutral atoms. A method for executing quantum error correction code, including the implementation of quantum error correction code.

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