Fault-tolerant atomic quantum computation

Abstract

Provided herein are techniques for developing fault-tolerant quantum computation (FTQC) systems by leveraging quantum error correction (QEC) strategies in a neutral atom quantum information processing system. The techniques described herein reuse and replenish qubits in the quantum circuit during operation. To implement such mid-circuit qubit reuse, the quantum information processing system described herein uses a "spin-to-position" qubit conversion method for quantum state measurement. This spin-to-position qubit conversion allows non-destructive, loss-resolved qubit readout.

Description

Attorney Docket No. : H0776.70181 WO00FAULT-TOLERANT ATOMIC QUANTUM COMPUTATIONCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Patent Application No. 63 / 824,621, filed June 16, 2025, and titled “Fault- Tolerant Atomic Quantum Computation,” and to U.S. Provisional Patent Application No. 63 / 675,566, filed July 25, 2024, and titled “Architecture for Deep Quantum Circuits with Reconfigurable Atom Arrays,” each of which is incorporated herein by reference in its entirety.BACKGROUND

[0002] Quantum information processing techniques perform computation by manipulating one or more quantum objects (e.g., objects that can store quantum states and / or manipulate quantum states). These techniques are sometimes referred to as "quantum computing." The field of quantum error correction (QEC) represents one illustrative approach to realizing large-scale quantum computers. In these approaches, one or more qubit systems are coupled to each other inside a vacuum chamber to form a quantum circuit. Qubit systems in a quantum circuit physically and electronically interact to perform quantum computation. QEC focuses on reducing error propagation within quantum circuits in an effort to realize fault-tolerant quantum computation (FTQC) systems.SUMMARY

[0003] The present disclosure generally provides techniques for developing fault-tolerant quantum computation (FTQC) systems by leveraging quantum error correction (QEC) strategies. The quantum information processing system described herein reuses and replenishes qubits in the quantum circuit during operation. Beneficially, mid-circuit qubit reuse and replenishing increases experimental cycle rates while maintaining constant internal entropy. To implement such mid-circuit qubit reuse, the quantum information processing system described herein uses a “spin-to-position” qubit conversion method for quantum state measurement. This spin-to-position qubit conversion allows non-destructive, loss-resolved qubit readout.

[0004] In some embodiments, the techniques described herein relate to a quantum computer, comprising: a computation chamber, comprising a storage zone, an entangling zone, a readout zone, and a reservoir zone; and at least one controller configured to, during qubit- 1 -#14172487v2Attorney Docket No. : H0776.70181 WO00 preparation and manipulation operations of a quantum information processing device, cause: transfer, from the readout zone of a computation chamber to the entangling zone of the computation chamber, a first plurality of qubits, wherein qubits of the first plurality comprise neutral atom qubits; entangle the first plurality of qubits in a spatial direction; transfer, from the storage zone of the computation chamber to the entangling zone, a second plurality of qubits, wherein qubits of the second plurality comprise neutral atom qubits that have previously been entangled with the first plurality of qubits; entangle the first and second plurality of qubits in a time direction; transfer the first plurality of qubits into the storage zone and transfer the second plurality of qubits into the readout zone; measure qubits of the second plurality; and reinitialize states of the qubits of the second plurality.

[0005] In some embodiments, the techniques described herein relate to a quantum computer, wherein during operation of the quantum information processing device: the storage zone is configured to hold qubits for performance of quantum operations between qubits held in the storage zone, the entangling zone is configured to hold qubits for performance of entangling operations between qubits held in the entangling zone, the readout zone is configured to hold qubits to be measured and / or reinitialized, and the reservoir zone is configured to provide a supply of qubits to one or more of the storage zone, the entangling zone, and / or the readout zone.

[0006] In some embodiments, the quantum computer further comprises a spatial light modulator configured to load qubits into arrangements of qubit traps in at least one of the storage zone, the entangling zone, the readout zone, and / or the reservoir zone.

[0007] In some embodiments, the spatial light modulator is configured to load the qubits into arrangements of qubit traps equal to approximately 852-nm.

[0008] In some embodiments, the spatial light modulator is configured to maintain a position of the entangling zone such that the entangling zone is separated from the storage zone by approximately 40-pm and from the readout zone by approximately 40-pm.

[0009] In some embodiments, the quantum computer further comprises a first pair of crossed acousto-optical deflectors configured to move qubits between the arrangements of qubit traps.

[0010] In some embodiments, the first pair of crossed acousto-optical deflectors are configured to arrange qubits of the first plurality and / or the second plurality in blocks of [[7,1,3]] or [[16,6,4]] codes.- 2 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0011] In some embodiments, the quantum computer further comprises a first laser configured to illuminate arrays of qubits in at least one of the storage zone, the entangling zone, the readout zone, and the reservoir zone with a global Raman beam.

[0012] In some embodiments,, the quantum computer further comprises a second pair of cross acousto-optical deflectors configured to form a local Raman beam to perform singlequbit rotations using the global Raman beam.

[0013] In some embodiments, the global Raman beam, while illuminating qubit arrays during operation of the quantum computer, is further configured to cool qubits in the arrangements of qubit traps using Al lambda-enhanced gray molasses cooling.

[0014] In some embodiments, the quantum computer further comprises a second laser configured to generate a laser beam to illuminate qubits in the storage zone.

[0015] In some embodiments, wherein the second laser is further configured to focus the laser beam to an elliptical waist, wherein a semi-minor axis of the elliptical waist is aligned in a vertical direction relative to a center position of the storage zone.

[0016] In some embodiments, the second laser is configured to generate the laser beam having a wavelength of approximately 1529-nm.

[0017] In some embodiments, the quantum computer further comprises a third and fourth laser configured to perform entangling gates between qubits disposed in the entangling zone using two-photon excitation.

[0018] In some embodiments, the third and fourth lasers are configured to emit 420-nm and 1013-nm Rydberg beams, respectively, configured to excite the qubits disposed in the entangling zone to n = 53 Rydberg states.

[0019] In some embodiments, the quantum computer further comprises a plurality of arbitrary waveform generators configured to control one or more lasers of the quantum computer.

[0020] In some embodiments, a first arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to generate entangling gate pulses to entangle qubits disposed in the entangling zone and / or to perform local detunings of the spatial light modulator.

[0021] In some embodiments, the first arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform through-the-lens (TTL) metering.- 3 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0022] In some embodiments, a second arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform in-phase and quadrature control of the qubits and / or pulse-shaping of a global and a local Raman driving.

[0023] In some embodiments , a third arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform real-time rearrangement of qubits.

[0024] In some embodiments, a fourth arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to control positions of the qubits during operation of the quantum computer.

[0025] In some embodiments, a fifth arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to create light grids for local single-qubit control.

[0026] Some embodiments are directed to a method for performing a quantum information processing operation using a quantum computer, the method comprising: transferring, from a readout zone of a computation chamber of the quantum computer to an entangling zone of the computation chamber, a first plurality of qubits, wherein qubits of the first plurality comprise neutral atom qubits; entangling the first plurality of qubits in a spatial direction; transferring, from a storage zone of the computation chamber to the entangling zone, a second plurality of qubits, wherein qubits of the second plurality comprise neutral atom qubits that have previously been entangled with the first plurality of qubits; entangling the first and second plurality of qubits in a time direction; transferring the first plurality of qubits into the storage zone and transferring the second plurality of qubits into the readout zone; measuring the second plurality of qubits; and reinitializing states of the second plurality of qubits.

[0027] In some embodiments, measuring qubits of the second plurality comprises: illuminating the qubits with counterpropagating laser beams to form a one-dimensional qubit lattice; converting qubits having state |1) to stretched dark states; converting qubits having state |0) to stretched bright states; moving the stretched dark states using acousto- optical deflector (AOD) tweezers while maintaining positions of the stretched bright states; and imaging the qubits.

[0028] In some embodiments, imaging the qubits of the second plurality comprises: performing a first cooling step by detuning the two counterpropagating beams emitted by a first laser; performing a second cooling step by blue-detuning the two counterpropagating- 4 -#14172487v2Attorney Docket No. : H0776.70181 WO00 beams emitted by the first laser; reducing a power of one of the two counterpropagating beams emitted by the first laser; and obtaining optical images of the qubits.

[0029] In some embodiments, the method further comprises identifying a spin state and / or a position of the second plurality of qubits using at least one image obtained from imaging the qubits.

[0030] In some embodiments, the method further comprises performing repetitive stabilizer measurements to blocks of qubits of the first plurality by: physically transporting blocks of the first plurality of qubits to interlace the blocks; and applying interleaved CNOT gates and stabilizer measurements to the interlaced blocks.

[0031] In some embodiments, the method further comprises arranging the blocks of the qubits of the first plurality in blocks of [[7,1,3]] or [[16,6,4]] codes.

[0032] In some embodiments, entangling the first plurality of qubits in the spatial direction comprises performing single-qubit operations between qubits of the first plurality of qubits.

[0033] In some embodiments, entangling the first and second plurality of qubits in the time direction comprises performing a transversal entangling gate between the first and second plurality of qubits.

[0034] In some embodiments, entangling the first and second plurality of qubits in the time direction comprises performing lattice surgery on both or either of the first and second plurality of qubits.

[0035] In some embodiments, reinitializing the states of the second plurality of qubits comprises optically pumping the second plurality of qubits.

[0036] In some embodiments, optically pumping coherent pulses comprises using a Raman- assisted optical pumping scheme to apply coherent 7t-pulses to the second plurality of qubits.

[0037] In some embodiments, the techniques described herein relate to an apparatus for reading out quantum states of neutral atom qubits, comprising: a first and second laser configured to generate counter-propagating laser beams configured to form a one dimensional optical lattice of the neutral atom qubits by illuminating the neutral atom qubits; a third laser configured to convert first qubits of the neutral atom qubits having a quantum state |1) to a stretched dark state by optically pumping the first qubits; a fourth laser configured to convert second qubits of the neutral atom qubits having a quantum state |0) to a stretched bright state by causing coherent Raman transfer of the second qubits or by optically pumping the second qubits; acousto-optical deflector (AOD) tweezers configured- 5 -#14172487v2Attomey Docket No. : H0776.70181 WO00 to move the first qubits to spatially separate the first qubits from the second qubits; and a camera configured to image the first and second qubits.

[0038] In some embodiments, the first and second laser comprises titanium: sapphire lasers configured to generate the counter-propagating laser beams with wavelength of 795-nm and a blue detuning between 50 GHz to 200 GHz from a DI line of the neutral atom qubits.

[0039] In some embodiments, the counter-propagating laser beams are circularly polarized.

[0040] In some embodiments, the third laser is configured to generate an optical pump resonant to an F = 2 to F’ = 3 transition of the neutral atom qubits.

[0041] In some embodiments, the third laser is configured to generate the optical pump comprising light that is circularly polarized.

[0042] In some embodiments, the fourth laser is configured to cause coherent Raman transfer to F = 2, mF = +2.

[0043] In some embodiments, the fourth laser is configured to optically pump the second qubits to cause an F = 1 to F' = 2 transition of the neutral atom qubits.

[0044] In some embodiments, the fourth laser is configured to optically pump the second qubits by generating circularly polarized light.

[0045] In some embodiments, the AOD tweezers are configured to move the first qubits approximately 2 pm over a time period of approximately 500 ps.

[0046] In some embodiments, the third laser is configured to produce two counterpropagating beams that propagate in directions that are parallel to an applied external magnetic field.

[0047] In some embodiments, the two counterpropagating beams are configured to be detuned by two times a Zeeman splitting of adjacent mF levels of the neutral atom qubits.

[0048] Some embodiments are directed to a method of reading out quantum states of neutral atom qubits, the method comprising: illuminating qubits, using a first and second laser, with counterpropagating laser beams to form a one-dimensional qubit lattice; converting qubits having state |1) to stretched dark states; converting qubits having state |0) to stretched bright states; moving the stretched dark states using AOD optical tweezers while maintaining positions of the stretched bright states; and imaging the qubits.

[0049] In some embodiments, maintaining positions of the stretched bright states comprises: ramping up the one-dimensional qubit lattice adiabatically, and pinning the stretched bright states in a lattice potential of the one-dimensional qubit lattice.- 6 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0050] In some embodiments, converting the qubits having state 11 ) to stretched dark states comprises using incoherent pumping generated by a third laser and resonant to an F = 2 to F’ = 3 transition of the neutral atom qubits.

[0051] In some embodiments, the incoherent pumping is performed with a G '-polarized 780 nm repumper.

[0052] In some embodiments, converting the qubits having state |0) to stretched bright states comprises using incoherent pumping generated by a fourth laser resonant to an F = 1 to F' = 2 transition of the neutral atom qubits.

[0053] In some embodiments, the incoherent pumping is performed with a o+-polarized 780 nm repumper.

[0054] In some embodiments, converting the qubits having state \F = l; mF= 0) to stretched bright states comprises using a coherent Raman transfer.

[0055] In some embodiments, imaging the qubits comprises obtaining optical images of the qubits using at least one optical camera.

[0056] In some embodiments, obtaining optical images of the qubits comprises: performing a first cooling step by red-detuning two counterpropagating beams emitted by the first laser; performing a second cooling step by blue-detuning the two counterpropagating beams emitted by the first laser; reducing a power of one of the two counterpropagating beams emitted by the first laser; and imaging the qubits.

[0057] In some embodiments, during the first cooling step, the two counterpropagating beams are detuned by two times a Zeeman splitting of adjacent mF levels of the neutral atom qubits.

[0058] In some embodiments, during the first cooling step, the two counterpropagating beams are red-detuned to perform a F = 2 to F' = 3 transition of the neutral atom qubits.

[0059] In some embodiments, the first cooling step comprises one dimensional polarization gradient cooling (PGC).

[0060] In some embodiments, during the second cooling step, the two counterpropagating beams are blue-detuned to perform a F = 2 to F' = 2 transition of the neutral atom qubits.

[0061] In some embodiments, the second cooling step comprises electromagnetically induced transparency (EIT).

[0062] In some embodiments, the method further comprises identifying, using the optical images of the qubits, a spin state and / or a qubit loss of the qubits.- 7 -#14172487v2Attorney Docket No. : H0776.70181 WO00BRIEF DESCRIPTION OF DRAWINGS

[0063] FIG. 1 is an experimental layout illustrating optical tools, similar to Ref.

[0011] with the addition of beams for local cooling, imaging and hiding to enable qubit re-use experiments. The gray panel is a schematic view of control infrastructure for programming quantum circuits. The entire waveform for all AWGs (except for rearrangement) was uploaded at the start of each experimental run. For qubit re-use experiments, the Moving, Raman AOD and Rydberg AWGs loop the same memory segment each layer. The full waveform was programmed for the Raman AWG to ensure phase continuity. For mid-circuit rearrangement, waveforms were calculated on-the-fly using a desktop computer and sent to the Rearrangement AWG operated in first-in first-out (FIFO) mode.

[0064] FIG. 2A shows a processor layout used for qubit re-use experiments and relevant laser beams. Atoms were arranged into storage, entangling and readout zones, with an additional reservoir for refilling lost atoms mid-computation. The 1529-nm hiding beam illuminated the storage zone to preserve coherence of active qubits during imaging in the readout zone. Parallel two-qubit gates were performed in the entangling zone with global Rydberg beams, and local detunings were optionally applied to selected gate sites using an SLM. The readout zone was illuminated with local beams for ID PGC imaging and EIT cooling, as well as two counter-propagating lattice beams to form a spin-dependent potential for readout via spin-to-position conversion. The entire array was addressed with global Raman control for dynamical decoupling. The same Raman light was directed through a pair of crossed AODs for local single-qubit gates. Global imaging and lambda-enhanced gray molasses cooling light were used for the initial loading.

[0065] FIG. 2B is a level diagram of the relevant atomic transitions of87Rb.

[0066] FIG. 3 is a flowchart illustrating a process 300 for performing quantum information processing, in accordance with some embodiments of the technology described herein.

[0067] FIGs. 4A-4B show that spin-to-position conversion for non-destructive, loss- resolved qubit readout was accomplished with a state-selective ID lattice that converted the atom spin state into position. FIG. 4B shows measured Rabi oscillation. This nondestructive readout had 0.46(4)% bit-flip error and 0.24(2)% loss (Methods).

[0068] FIGs. 5A-5F show spin-to-position conversion. FIG. 5A is a level diagram showing the87Rb hyperfine levels used to engineer a spin- selective one-dimensional optical lattice.- 8 -#14172487v2Attorney Docket No. : H0776.70181 WO00FIG. 5B shows trapping potentials for the bright and dark states. FIG. 5C is a schematic timeline of spin-to-position conversion. The time to transfer the clock qubit to the bright and dark states for readout was typically on the order of roughly 20 ps, but for some of the measurements was several milliseconds due to using a slow global rotation of the magnetic field (FIG. 5E). See Methods text for additional information. FIG. 5D shows transfer of qubit state | 1) to the dark state via resonant optical pumping. FIG. 5E shows transfer of qubit state |0) towards into mF = +1, +2 states. This was achieved with either a coherent Raman transfer to F = 2, mF = +2 or incoherent pumping with c+-polarizcd repumper. Both approaches achieved the same bright state readout fidelity, but in the specific implementation used here the Raman transfer took several milliseconds owing to the rotation of the external magnetic field for driving G transitions (2 and 3). FIG. 5F shows quadratic suppression of readout error due to scattering. The bright state transfer ensured that at least two lattice-induced scattering events were required to cause a readout error, which occurred if the AOD tweezer had not yet moved away as the bright state became unpinned. Scattering further caused diabatic changes in the depth of the lattice potential and may contribute to atom loss.

[0069] FIG. 6 is a flowchart illustrating a process 600 for performing nondestructive qubit readout, in accordance with some embodiments of the technology described herein.

[0070] FIG. 7 is a schematic diagram of an illustrative classical computer 700, which may be used to implement aspects of the technology described herein.

[0071] FIGs. 8A-8D show architectures and mechanisms for fault-tolerant quantum computation. FIG. 8A shows the building blocks of fault-tolerant processing. An architecture based on reconfigurable atom arrays trapped in optical tweezers was used, where the logical processor was segmented into storage, entangling, readout, and reservoir zones. Underlying physical mechanisms were identified and characterized. FIGs. 8B-8D show that stabilizer measurement on a d = 5 surface code was interspersed with global coherent errors injected on the data qubits. Each CZ layer corresponds to one time-step (Methods). Repeated correction reduced error build-up through both the Zeno effect and error tracking (FIG. 8D is at a fixed 0 / 2TT = 0.016). For visual clarity, an acceptance fraction of 50% was used in FIGs. 8C-8D (Methods).

[0072] FIGs. 9A-9G show below-threshold repeated quantum error correction leveraging loss detection. FIG. 9A shows results of repeated rounds of d = 5 surface code using loss- 9 -#14172487v2Attorney Docket No. : H0776.70181 WO00 detection, showing a snapshot of the data block and multiple ancilla blocks (see FIGs. 17A- 17G). FIGs. 9B-9C show that products of stabilizer measurement results between rounds were used to detect qubit errors, which are referred to as ‘detectors’. ‘Bare’ counted loss as state |0), ‘detect loss’ did not make this erroneous assignment, ‘supercheck’ multiplied detectors around lost atoms, and ‘post.’ postselected on all atoms of a detector being present. FIG. 9B shows detector error probability as a function of data qubit loss in each shot, analyzed by partitioning the total dataset. FIG. 9C shows average over all data. FIG. 9D shows logical error per round for a surface code after 4 QEC rounds in both bases, decoded using most- likely error methods (‘bare MLE’), machine learning (‘bare ML’), MLE with loss information, and ML with loss information. ML with loss rendered the error per round for d = 5 as 2. 14(13)x lower than d = 3. No postselection was used. Small points are the four d = 3 quadrants. Results were averaged between |+L) and |0L) initialization bases. See Methods and FIGs. 17A-17G for more details. FIG. 9E shows logical error per round as a function of the mean qubit loss, plotted as a cumulative density function. FIG. 9F shows relative physical error contribution to overall error budget (see Methods). FIG. 9G shows the distribution of detector errors per shot, suggesting the absence of large-scale correlated errors.

[0073] FIGs. 10A-10D explore the interplay of logic gates and entropy removal. FIG. 10A shows atom images illustrating two-qubit logic gates and stabilizer measurement. Lattice surgery was realized using ancillas to measure the logical product Z^Z^ , and transversal gates were realized via atomic motion. FIG. 10B shows quantum circuits for realizing transversal gates and lattice surgery operations. CZ gates were realized as transversal CNOTs and Hadamards. FIG. 10C shows the dependence of error of the logic operation on the ancilla measurement error. Lattice surgery errors rapidly worsened with increasing ancilla measurement errors (injected in post-processing). FIG. 10D shows N repeated logic operations were interspersed with rounds of QEC stabilizer measurement. Transversal CNOT had a lower error than lattice surgery (left), and had an optimum of roughly 3 CNOTs per round, as seen most clearly when modest postselection, characterized by acceptance fractions (AF) was used (right). Error detection was used for lattice surgery in FIG. 10D to compensate for having < d rounds (Methods).

[0074] FIGs. 11A-11D show the synthesis of arbitrary-angle rotations with a universal fault-tolerant gate set. FIG. 11A shows subjecting codes of various dimensionality to global- 10 -#14172487v2Attorney Docket No. : H0776.70181 WO00 rotations. Plateaus are seen at robust angles where the stabilizers also revive. 3D Reed- Muller codes, only with all positive stabilizer signs, have an additional plateau at 45-degrees corresponding to non-Clifford T gates. FIG. 11B shows how programmable angles R(0,were realized by an alternating sequence of H and T gates. Circuit shows an implementation of such a sequence with T implemented via state preparation and H implemented via quantum teleportation. The polar angle plots in FIG. 11C show generated angles using entangled Reed-Muller codes, measured by tomography, for different maximum number of T gates. Experimental results are consistent with theory within statistical error. FIG. 11D provides experimental results showing that the minimum separation between generated angles decreased exponentially with length of sequence. Inset shows a rescaling where e is the angular separation to the target angle. Bloch sphere shows all measured angles. For visual clarity, variable degrees of acceptance fraction were used (Methods).

[0075] FIGs. 12A-12D show the architecture for constant-entropy computation. FIG. 12A shows an illustration of processes for removing entropy generated by computation. Logical teleportation was used to ensure all physical errors were removed (FIG. 13E). FIG. 12B shows rabi oscillations measured using the same atoms for 150 cycles of non-destructive measurement and re-initialization. Each curve shows a single experimental run, averaged over 200 atoms in parallel. 3D cooling methods were used in this subplot as coherence did not need to be preserved. FIG. 12C shows local cooling with ID polarization gradient cooling (PGC) and electromagnetically induced transparency (EIT). The finite magnetic field was compensated by applying a relative detuning between the two circularly polarized counterpropagating beams, lending to a rotating frame where the effective field was zero. The atom loss and temperature was constant as a function of time and recovered to steady state within one cycle after applying a perturbation (turning off cooling on one layer) to the system (dashed line). FIG. 12D shows additional shielding of the data qubits provided by a 1529-nm shielding laser, which rapidly suppressed decoherence induced by the imaging (resonance is at 1529.365 nm).

[0076] FIGs. 13A-13I show deep logical circuits at constant entropy. FIG. 13A provides a schematic of a 2D cluster state created from Steane codes in space and time. The same atomic qubits were re-used every other time layer for up to 27 layers. FIG. 13B shows results of repeated logical state preparation of Steane codes. Stabilizer error was constant as a function of cycle. FIGs. 13C-13D show logical and physical correlations in ID (FIG.- 11 -#14172487v2Attorney Docket No. : H0776.70181 WOOO13C) and 2D (FIG. 13D) cluster states. Logical correlations persisted while stabilizer error correlations rapidly decayed. In FIGs. 13C-13D and 13E-13I, various degree of acceptance fraction were applied (see Methods for tabulation of all details). FIG. 13E provides a circuit diagram illustrating that logical-level evolution was unitary and allowed the logical operator to propagate throughout the algorithm whereas physical-level evolution was dissipative and did not let physical errors propagate. FIG. 13F shows that high-rate [[16,6,4]] codes were entangled in space and time direction. FIG. 13G shows logical evolution and physical error removal with [[16,6,4]] codes (ID cluster state in time direction). Permutation CNOTs within the block (black curve) extended the correlation length in comparison to nonpermuted qubits (Methods). FIG. 13H shows entanglement structure in space and time, with up to 96 d = 4 logical qubits active simultaneously. FIG. 131 shows logical 2D cluster expectation values (averaged across space and the first 9 teleportation layers) as a function of the number of co-propagating logical operators which agreed (i.e., cluster state stabilizers had the same outcome for each logical qubit within the block).

[0077] FIGs. 14A-14F show one-dimensional imaging and cooling in finite field. FIG. 14A shows an atomic level structure for cooling and imaging of87Rb. The one-dimensional (ID) techniques here relied on coupling hyperfine states separated by dm= 2 with counterpropagating <7* beams. In a finite magnetic field Bext, the level degeneracy was lifted by the Zeeman effect; hereis the Bohr magneton and gr = 1 / 2 the Lande factor. FIG. 14B shows a single-shot local image in 8.6 G external magnetic field. Spin-to-position conversion was used in the readout zone and the reservoir was partially imaged by the tails of the imaging beams. FIG. 14C provides an example of a site-averaged imaging histogram in the readout zone. FIG. 14D provides a schematic timeline of mid-circuit measurement and re-initialization used for deep circuit experiments. Due to latency bottlenecks associated with desktop-computer-based processing of images and rearrangement waveforms, speed was not optimized and comfortable parameters were used. The circuit was 13.5-ms long. Including idle times, the spin-to-position conversion took 4-ms, imaging took 10-ms, cooling and re-sorting took 13.3-ms (~7-ms latency for mid-circuit data processing), and reinitialization took 1.1-ms. FIG. 14E shows a ID polarization gradient cooling (PGC) in finite magnetic field. Top left: interference between the two counterpropagating <7* probe beams generated a helix of linear polarization

[0048] . Detuning the two beams rotated the helix such that, in the rotating frame, a fictitious magnetic field appeared that canceled the external- 12 -#14172487v2Attorney Docket No. : H0776.70181 WO00 field and restored the zero-field PGC cooling mechanism. Bottom left: Atom loss from ID PGC light at different relative beam detunings, A, in varying external magnetic field. The atoms were illuminated for 10 ms under comfortable imaging parameters. A cooling resonance was observed when A matched two times the Zeeman splitting (bottom right). Top right: Atom temperature around the cooling resonance in 4.3 G field, obtained from drop-recapture measurements. FIG. 14F shows ID electromagnetically induced transparency (EIT) cooling. Top: A strong <J+pump beam was combined with a weak o’probe beam to drive transitions between quantum harmonic oscillator states of the optical tweezer, cooling the atom. The EIT Fano resonance ensured heating transitions were suppressed

[0049] . Bottom: Ramp-down measurement of atom temperature. The SLM trap depth was adiabatically ramped down to ~ 5 zK and held for 5-ms before being ramped back up; the atom loss probability from this process probed the temperature of all three motional axes

[0076] . Since the EIT cooling resonance was narrow and depended on trap frequency, spatial inhomogeneity in the SLM trap depths translated to inhomogeneous cooling and broadened the temperature distribution. Even so, the coldest sites after ID PGC imaging and EIT cooling reached lower temperatures than conventional 3D PGC, highlighting the utility of these ID techniques.

[0078] FIGs. 15A-15E show atomic physics of hiding beam at 1529-nm. FIG. 15A, Upper panel: The hiding beam was aligned to the storage zone and a knife-edge was used to suppress its gaussian-tail in the readout zone. Lower panel shows the relevant atomic levels in87Rb. The hiding beam coupled the excited- to -excited state transition and imparted a 6 GHz light-shift on the 5P3 / 2 state. At this detuning, the 5Si / i ground state polarizability was approximately 2 x IO-5times smaller

[0050] , thus maintaining coherence in the hyperfine qubit manifold while shifting the transition far off-resonance from the imaging light. FIG. 15B shows additional dephasing on atoms in the storage zone due to local imaging beams in the readout zone, for various drive powers and detunings. 20-ms of illumination and closer probe detuning were used here than in deep circuit experiments. A fit to a simple model was plotted (described in the Example). For deep circuit experiments, tests were operated at 16 GHz red-detuned from the bare 5P3 / 2 — 4D / transition (not shown). FIG. 15C shows that by scanning the detuning of the probe light, an Autler-Townes (AT) splitting of probe resonance was observed when coupled to the 4Ds / 2-level. For this measurement, tests were operated at a low drive power compared to typical values and the drive detuning was 500- 13 -#14172487v2Attorney Docket No. : H0776.70181 WO00MHz red-detuned from resonance, such that both peaks can be experimentally measured in the limited probe detuning range. FIG. 15D shows the measured AT-splitting scales linearly with the Rabi frequency of the drive, or the square root of the drive power. FIG. 15E provides a level diagram illustrating the coupled three-level ladder system which leads to the emergence of the Autler-Townes feature.

[0079] FIGs. 16A-16B show the characterizing effects of loss and leakage in repeated QEC. FIG. 16A shows superchecks. A lost data qubit resulted in anti-commuting stabilizers and a flickering error pattern. Producting stabilizers surrounding the lost qubit recovered commuting superchecks. FIG. 16B shows detection correlation pij matrix for five rounds of QEC on a d = 5 surface code. Data qubit errors appeared as space-like correlations and ancilla measurement errors as time-like correlations between adjacent layers. Leakage resulted in additional persistent correlations between QEC rounds. By selecting shots with the fewest lost data qubits, these correlations were suppressed, indicating that leakage is dominated by loss which can be detected. Similarly, selecting shots with the most loss enhanced the correlations.

[0080] FIGs. 17A-17G provide additional data for repeated QEC characterization circuit. FIG. 17A shows the processor layout for repeated QEC on a d = 5 surface code. The data qubits and one ancilla block are located in the entangling zone (top) and four additional ancilla blocks are in storage (bottom); one ancilla block was unused in the four-round circuit. FIG. 17B shows the processor layout for repeated QEC on a d = 3 code in one of four possible quadrants. FIG. 17C shows a circuit for four rounds of QEC. For the XZZX rotated surface code

[0121] , Y(zr / 2) gates were applied to one data qubit sublattice (A or B) for preparing and measuring in the X or Z basis. The equivalent circuit was obtained for stabilizer measurement of the CSS rotated surface code upon compiling Y(zr / 2) gates. FIG. 17D shows stabilizer gate ordering. The same pattern was used globally in each round. FIG. 17E shows the detector error probability for d = 5. Faint dashed curves correspond to the 24 individual detectors (12 for the first and last rounds), and solid curves are the mean of all deterministic detectors. In FIG. 17E only, to illustrate the supercheck error distribution, the error of each supercheck was assigned to all detectors from which it is composed, and the resulting effective detectors were plotted; the overall mean is the average of these individual detector values. The first round had lower error due to the neighboring transversal state preparation. The best detector over the central three rounds after loss postselection had 27%- 14 -#14172487v2Attorney Docket No. : H0776.70181 WOOO lower error than the overall mean; one atom had anomalously high error in the Z basis. FIG. 17F is a comparison to simulation. Left: detector error probability, converting loss to qubit state 0. Right: detection correlation matrix pq

[0122] . Both metrics showed good agreement between simulation and experiment in the structure and magnitude of the errors. FIG. 17G shows CZ gate fidelity measured via randomized benchmarking

[0036] . Infidelity due to Rydberg P states was removed by leaving sufficient time or distance between gates. Sparse corresponds to 2x larger separation between gate sites.

[0081] FIGs. 18A-18E explore entropy removal during single- and two-logical-qubit operations. FIG. 18A provides additional data for entropy removal via stabilizer measurement. As studied in FIGs. 8C-8D, the final logical error depended on the balance between the injected error rate (corresponding to the injected 0 / 2^ per time-step) and the entropy removal rate (number of QEC rounds in the fixed total time window). For small injected error, there was an optimal number of QEC rounds (here, 3-4) since stabilizer measurement is imperfect and introduces entropy of its own. The plot used MLE decoding and an acceptance fraction of 66% to enhance salient features. FIG. 18B shows the absence of coherent logical error. Using the same error-injection protocol as in (FIG. 18A), the final logical state was measured in both the X and Z basis. With no QEC, the global coherent error resulted in a coherent logical rotation. With one round of QEC, or more, this coherent rotation vanished. The stabilizer measurement was performed immediately after transversal state preparation, before the majority of the error was injected, such that the lack of logical rotation compared to no QEC could be attributed to the non-deterministic X(Z) stabilizer signs randomizing the response of the Z(X) logical operator to coherent error. The ancillas in the MLE decoding were omitted, and instead, a 50% acceptance fraction was used. All curves are the average of both bases; for left plot, the measurement is the same basis as preparation, and for the right plot it is the orthogonal basis. FIGs. 18C-18E show analysis of logical gate performance of two logical qubits undergoing repeated transversal CNOTs and QEC, with the circuit studied in FIGs. 10A-10D. FIG. 18C shows that the measured detector error probability increased linearly as a function of number of CNOTs applied in each round. FIG. 18D shows the logical error probability as a function of number of CNOTs per round. Fits are to a functional form of A ■ (pqec+ N Apdet, where A, pqec, and Apdet are fitted parameters (circles). Using the fitted values of pqecand Apdet in FIG. 18C produced the predicted logical error probability Pzfpdet) (squares). FIG. 18E shows the results of FIG.- 15 -#14172487v2Attorney Docket No. : H0776.70181 WO0018D divided by total number of CNOT gates. Logical gate fidelity Fi pdet) is 1 - PL(pdet) / (3N ), where 3 is the number of rounds and N the number of gates per round.

[0082] FIGs. 19A-19C show the theoretical characterization of logic al-error-per-round ratio. FIG. 19A shows numerical simulations of rotated surface code initialized in |0L) undergoing repeated QEC rounds with stabilizer measurement gate ordering in Ref.

[0122] . As a simple model, a theory error channel was applied with error probability p = 0.6%, where qubit resets and measurements experienced uniform single-qubit depolarizing noise, CNOT gates were followed by uniform two-qubit depolarizing noise, and data qubits experienced an idling single-qubit uniform depolarizing channel during ancilla qubit resets. Different LEPR ratios were plotted (ra / (a+2) = LEPR(d) / LEPR(d + 2)) using the PyMatching decoder

[0123] , observing a change of at most 17% as the number of rounds was increased from four. The LEPR for a circuit with N logical qubits and n QEC rounds is defined asthe logical error rate of a fully mixed state. FIG. 19B shows numerical simulations of repeated QEC on a single | +L) surface code using the same circuit as the d = 5 surface code experiment in FIGs. 10A-10D. A simplified error model was used where the experimental error model was taken and all loss rates were turned to 0 and the Pauli error rates were doubled for idle errors, reset, measured, and gate errors (single and two-qubit gates) on both data and ancilla qubits. The LEPR ratios were plotted for varying numbers of QEC rounds using an MLE decoder

[0041] , observing changes of at most 9%, indicating that performance remains stable even for deeper circuits. FIG. 19C shows numerical simulations for circuits with 25 QEC rounds on four logical qubits, with interleaved transversal gate layers using the approximate experimental error model described above. Half of the qubits, selected uniformly at random, were initialized |+L) , while the other half are initialized in |0L). Each gate layer comprises random pairing of transversal CNOT gates followed by logical single-qubit Pauli gates selected uniformly at random. After applying the random gate sequence U , the inverse U1' was applied, followed by transversal measurement in the same basis as initialization. By varying the number of gate layers interspersed between rounds, the LEPR ratio averaged over ~ 100 randomly sampled circuits varied by at most 2%.

[0083] FIGs. 20A-20F show universality with 3D codes. FIG. 20A shows codes of various dimensions subject to global rotation. The same data are shown in FIG. 11A, here plotted without any normalization applied to the logical operator or stabilizer expectation values.- 16 -#14172487v2Attorney Docket No. : H0776.70181 WOOOFIG. 20B shows two-copy measurement of a [[15,1,3]] code after a global rotation

[0124] . The additive Bell magic had a plateau at one unit of magic under a global T. The same magic was obtained by analyzing either the underlying physical 15-qubit system or the single logical qubit with error detection. FIG. 20C shows that the encoded logical state was connected via a unitary Clifford decoding circuit to a product state of the encoded state and Pauli inputs on the physical qubits. The Clifford circuit conserves magic and therefore the total physical state and encoded logical state must have the same total magic. This means that, while 15 physical T’s are applied to the system, only 1 physical T is produced, which can only happen with entanglement. FIG. 20D shows an atom image of a register of 2D and 3D color codes. A transversal CNOT may be performed between the face of the [[15,1,3]] code (control) and the [[7,1,3]] (target). FIG. 20E shows code switching. \TL) was prepared transversally on the 3D [[15,1,3]] code and then measurement and in-software feedforward teleported the logical T onto a 2D [[7,1,3]] code (here also with a H) this illustrates the equivalence between code switching protocols and logical teleportation. Error detection was used in both plots. FIG. 20F shows an error-corrected test of Bell’s inequality

[0047] . This test measured .S' = E(X, T ) + E(X, T ) + E(Y, T ) - E(Y, T1' ), where E(A, B) is the expectation value of the Steane and Reed-Muller codes in the A and B bases, respectively, and obtain S = 1.99(3)x 2 with error detection.

[0084] FIGs. 21A-21D show entropy in deep circuits. FIG. 21 A shows general hypercube encoding. The same entangling circuit structure was combined with programmable input physical states to prepare [[7,1,3]], [[15,1,3]] and [[16,6,4]] codes (members of the family of quantum Reed- Muller codes). For the punctured codes, either the top or bottom qubit was removed. FIG. 21B shows turning off entropy removal mechanisms in the 2D [[7,1,3]] cluster state circuit. In the absence of re-cooling or refilling of loss, physical stabilizer correlations persisted in time. Lost atoms were assigned as qubit state 0 for this plot, and only correlations between codes constructed from the same atomic qubits in every other layer are shown. FIGs. 21C-21D show circuit structure and physical error correlations. By replacing CZ gates by CNOT gates in the ID [[7,1,3]] cluster state circuit, physical errors could propagate beyond a single layer, extending the stabilizer correlations. The product of adjacent stabilizers commutes with these propagated errors, recovering the rapid decay in physical correlations.- 17 -#14172487v2Attorney Docket No. : H0776.70181 WO00DETAILED DESCRIPTION

[0085] In classical computers, bits are a unit of information representing a logical state of one of two classical values, 0 or 1. Similarly, a quantum bit (a “qubit” herein) is a unit of information in a quantum computer. Like classical bits, qubits can occupy two distinct states, such as |0) and | 1), or any quantum superposition of the two states. In some cases, qubits are encoded in quantum systems with two or more distinct quantum states. Many physical realizations of qubits may be employed. As an example, qubits may include neutral atoms isolated within a vacuum chamber. These isolated neutral atoms have many distinct quantum states corresponding to the orientation of electron spins, electron orbits, nuclear spins, molecular rotations, and / or the like.

[0086] Quantum computers generally contain many qubits (e.g., tens, hundreds, and / or thousands of qubits) and perform computational operations, including initializing the qubits for computation, manipulating the state and / or position of the qubits, and reading out the state of the qubits at a given time. Manipulation of the qubit state may be performed using quantum logic gates, which perform mathematical operations on qubits. Two such types of quantum logic gates include a single-qubit gate and / or a multi-qubit gate. A single-qubit gate is a quantum logic gate applied to an individual qubit. For example, a single qubit gate may operate on a qubit in state |0) and change (e.g., flip) the qubit state to state | 1). In contrast, a multi-qubit gate operates on at least two qubits. A multi-qubit gate may, as an example, entangle two qubits, wherein entangling describes linking the two qubits such that the state of one influences the state of the other. One common multi-qubit gate is a controlled NOT gate (“CNOT gate”), which can entangle two qubits and conditionally change the state of one or both qubits. For example, a CNOT gate may be configured to flip the state of a second qubit if and only if the state of a first qubit is |0).

[0087] According to various embodiments of a quantum computer, individual particles (e.g., atoms, ions, molecules, etc.) can first be trapped in an array and arranged into particular configurations. Next, one or more of the arranged particles are prepared in a desired quantum state to act as a qubit. Quantum circuits may then be implemented by performing a sequence of qubit operations, which act on individual qubits (“single-qubit gates”) or on groups of two or more qubits (“multi-qubit gates”). Finally, the state of the qubits can be read out in order to observe the result of the quantum circuit. The readout can- 18 -#14172487v2Attorney Docket No. : H0776.70181 WO00 be accomplished using an observation system that typically includes an electron-multiplied CCD (EMCCD) or optical camera image to detect particles’ loaded positions, and a second camera image to read out the qubits’ final states by, for example, detecting fluorescence generated by the particles in their final quantum states.

[0088] The operation of quantum information platforms are based on interactions between qubits. However, qubits often interact locally, which limits the connectivity of the circuit or the analog simulation and constrains the possible computations. While some platforms can communicate in a non-local way through the use of a shared bus, these shared- bus approaches are limited to small systems and thus still require a way to dynamically move qubits around in order to truly scale up the platform.

[0089] In many embodiments of a quantum computer, a qubit may be encoded in two near-ground-state energy levels of an atom, ion, or molecule. An example of this is a hyperfine qubit. In a hyperfine qubit, the two near-ground-states differ by the relative orientation of the nuclear spin with respect to the outer electron spin. The two states are split by the interaction energy between the nuclear spin and electron spin, typically ranging from frequencies of 1-13 GHz. Hyperfine qubits are frequently chosen owing to their resistance to environmental perturbations and long lifetimes.

[0090] Performing single-qubit gates on hyperfine qubits can be done by applying coherent microwave radiation at the frequency of the energy splitting between the first and second hyperfine qubit states. However, due to the physical proximity of the qubits in the quantum computer, in some systems on the scale of a few microns, microwaves cannot be applied to the first qubit without the microwaves affecting the states of qubits proximate to the first qubit.

[0091] Alternatively, some quantum information processing systems may apply a specific type of laser field to the qubits to perform quantum logic gates. This laser field is nearly resonant with an optical transition from one of the ground states to an optically excited state of a particular qubit. By applying the laser field to the qubit (i.e. pumping into the qubit), the qubit absorbs a nearly resonant first wavelength and generates a second wavelength, and in doing so changes its state. This state change is defined as a stimulated Raman transition (SRT) and the laser field defined as a Raman pulse. Beneficially, the Raman pulse can focus on individual qubits and / or subsets of qubits, mitigating the- 19 -#14172487v2Attorney Docket No. : H0776.70181 WO00 unintended state changes faced in microwave state transitions. Additionally, Raman pulses can be applied with high intensity, resulting in faster quantum gate operations.

[0092] Neutral atom quantum computers are a specific type of quantum computer that encode qubits in neutral atoms. These neutral atoms are trapped in a vacuum chamber and levitated by one or more trapping lasers. Commonly, individual atoms are trapped in an optical lattice, which is formed from standing waves of laser light that produces a periodic structure of nodes and anti-nodes. Alternatively or additionally, optical tweezers may be used to trap individual atoms by using tightly focused laser beams.

[0093] Neutral atom qubits can be used as hyperfine qubits, wherein a first and second ground state is split by frequencies of approximately 1-13 GHz. Multi-qubit gates in neutral atom quantum computers are realized using a third state, which is an excited Rydberg state. Beneficially, when an atom is excited to a Rydberg state, proximal atoms are prevented from exciting to the Rydberg state. This conditional behavior forms the basis for multi-qubit gates, including the CNOT gate previously described. The Rydberg state is used to temporarily mediate the multi-qubit gate before Rydberg excited atoms return to ground state, preserving their coherence. Coherence is a measure of the lifetime of the qubit before its information is lost, and is a parameter frequently used to describe qubits.

[0094] Neutral atom arrays can be dynamically reconfigured while preserving quantum coherence and entanglement between qubits by storing quantum information in hyperfine states and shuttling atoms in optical tweezers. This approach offers a scalable way to realize a quantum information system with large numbers of qubits and arbitrary programmability - where any qubit can perform an entangling gate with any other qubit in the array. Using high-fidelity two-qubit Rydberg gates, various quantum information circuits are described herein that leverage the programmability and nonlocal connectivity achievable with these approaches. Examples of high fidelity Rydberg gates are described in Levine, et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett., vol. 123, issue 17, link.aps.org / doi / 10.1103 / PhysRevLett.123.170503, and Evered, et al., High-Fidelity Parallel Entangling Gates on a Neutral Atom Quantum Computer, arXiv:2304.05420 [quant-ph], arxiv.org / abs / 2304.05420, which are hereby incorporated by reference in their entirety.

[0095] In quantum computers, ideal qubits are encoded to have long coherence properties and, consequently, maintain their lifetime before information is lost. Short- 20 -#14172487v2Attorney Docket No. : H0776.70181 WO00 coherence properties result in a higher error rate and increased information loss. One common error in quantum computation is a bit-flip error, wherein a qubit’ s state changes unexpectedly. For example, a quantum qubit encoded in state |0) may change to state |1) after a characteristic time scale, wherein the characteristic time scale defines the qubit’s coherence. As another example, a qubit in a superposition state (|0) + 11)) / 2 may change to state (|0) — 11)) / V2 after a characteristic time scale.

[0096] A challenge of quantum computation is its inherent sensitivity to errors. Whereas classical computers are composed of intrinsically robust digital bits stabilized by dissipation, quantum states are intrinsically analog objects that evolve in a continuous state space with coherent unitary evolution that does not allow such dissipation. Quantum error correction (QEC) provides a method to realize robust quantum computation. “Physical qubits” describe the two-state systems described previously, including neutral atoms in state |0) or state |1). Physical qubits are notoriously prone to error, so QEC involves entangling a plurality of physical qubits into a singular “logical qubit.” Error-corrected logical qubits that can have exponentially low, digitized errors while performing arbitrary, analog-like computations. Combining a plurality of logical qubits forms the basis of a fault-tolerant quantum circuit.

[0097] The inventors have recognized and appreciated that large-scale fault-tolerant quantum computation (FTQC) remains a formidable challenge for the field of quantum computing. To implement FTQC, errors are dissipatively removed from the physical system, while arbitrary coherent manipulation of the encoded logical qubits is simultaneously performed. Such an operation relies on a broad range of components, many hardware considerations, and a wide range of QEC techniques.

[0098] The inventors have further recognized and appreciated that FTQC may be improved by performing repeated QEC operations, which improve the fidelity of the stored quantum information. In the context of a neutral atom computer, the application of repeated QEC operations to a set of qubits relies on the ability of the neutral atom computer to non- destructively readout the states of the neutral atom qubits such that the qubits, after being measured, may have additional QEC operations applied. To permit the non-destructive readout of neutral atom qubits, the inventors have developed systems and methods, described herein, to perform “spin-to-position conversion” of the qubit states of the neutral atom qubits.- 21 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0099] In some embodiments, techniques for performing non-destructive qubit readout include illuminating qubits with counterpropagating laser beams to form a one-dimensional qubit lattice. Qubits having state |1) are converted to stretched dark states using a laser that incoherently pumps the qubits in state 11). Qubits having state |0) are converted to stretched bright states either by using another laser that incoherently pumps the qubits in state |0) or by using a laser configured to perform coherent Raman transfer of the qubits in state |0). The stretched dark states are then physically moved using acousto-optical deflector (AOD) optical tweezers, while the positions of the stretched bright states are maintained by the optical lattice. Finally, a camera collects optical images of the qubits. These optical images may be used to measure the spin state and / or the position of the qubits, and the qubits can be reinitialized and reused in further quantum operations.

[0100] The inventors have further recognized and appreciated that such non-destructive qubit readout techniques may be used to enable the implementation of deep logical circuits in a neutral atom computer. Accordingly, the inventors have developed systems and methods for operating a neutral atom computer at constant, or approximately constant, entropy.

[0101] In some embodiments, the techniques include transferring, from a readout zone of a computation chamber of the quantum computer to an entangling zone of the computation chamber, a first plurality of qubits. The qubits of the first plurality may comprise neutral atom qubits. Then, the first plurality of qubits are entangled in a spatial direction by performing single-qubit operations between qubits of the first plurality. A second plurality of qubits are transferred from a storage zone of the computation chamber to the entangling zone. Qubits of the second plurality comprise neutral atom qubits that have previously been entangled with the first plurality of qubits in a previous operation of the quantum information processor.

[0102] In some embodiments, the first and second plurality of qubits are then further entangled in a time direction by performing transversal entangling gates between the first and second plurality of qubits. The first plurality of qubits are transferred into the storage zone, and the second plurality of qubits are transferred into the readout zone. The second plurality of qubits are measured in the readout zone by collecting images of the second plurality of qubits with a camera. Finally, the states of the second plurality of qubits are reinitialized by optically pumping into the second plurality of qubits.- 22 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0103] Following below are more detailed descriptions of various concepts related to, and embodiments of, techniques for coupling quantum modes using a linear inductive coupler. It should be appreciated that various aspects described herein may be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only. In addition, the various aspects described in the embodiments below may be used alone or in any combinations and are not limited to the combinations explicitly described herein.

[0104] FIG. 1 is an illustration of a quantum information processing system 100, including a control infrastructure 102 for a quantum computer, according to some embodiments of the technology described herein. In some embodiments, and as shown in the example of FIG. 1

[0105] The system 100 includes a vacuum chamber 104 in which a plurality of atoms 105 are trapped. The vacuum chamber 104 may be made of a transparent material (e.g., glass, quartz, sapphire, etc.) such that the atoms 105 can be optically addressed to implement quantum information processing (e.g., the performance of quantum logical gates, error correction processes, etc.). The atoms 105 may be atoms of an atomic species suitable for use as neutral atom qubits, for example alkali metals (e.g., Rubidium or Cesium), alkali earth metals (e.g., Strontium), and / or the like.

[0106] In some embodiments, the system 100 includes a control infrastructure 102, as shown in FIG. 1. The control infrastructure 102 includes circuitry and / or computing devices configured to control one or more components of the system 100 to implement quantum information processing using atoms 105. The control infrastructure 102 may be implemented using any suitable classical computing systems and / or circuitry (e.g., ASICs, FPGAs, etc.) as described in connection with the example of FIG. 7 herein. In some embodiments, the control infrastructure 102 includes one or more of a Rydberg arbitrary waveform generator (AWG) 154, a Raman AWG 156, a rearrangement AWG 158, a moving AWG 160, and / or a Raman AOD AWG 162, the operations of which are described in more detail below.

[0107] In some embodiments, the system 100 includes a spatial light modulator (SLM) 106, a first pair of acousto-optical deflectors (AODs) 108 (also known as “optical tweezers”), a second pair of AODs 116, and a local SLM detuner 118. Light (e.g., laser beams) generated by the SLM 106 and the first pair of AODs 108 passes through a first polarized beam splitter (PBS) 110 and subsequently a first dichroic mirror 112 before passing through a trapping- 23 -#14172487v2Attorney Docket No. : H0776.70181 WO00 objective 114. Light (e.g., laser beams) generated by the second pair of AODs 116 and the local SLM detuner 118 is steered to the trapping objective 114 by dichroic mirrors 112 and 120. Thereafter, light passing through the trapping objective 114 is focused into the vacuum chamber 104.

[0108] During operation of the system 100, in some embodiments, the SLM 106 is configured to load atoms 105 disposed within the vacuum chamber 104 into static magnetooptical traps, referred to herein as static traps. The static traps may be 852-nm static traps, although it should be appreciated that alternative static trap sizes (e.g., as suitable for the atomic size of atoms 105) may be utilized, as aspects of the technology described herein are not limited in this respect.

[0109] In some embodiments, once atoms 105 are disposed in static traps generated by SLM 106, the rearrangement AWG 158 and / or the moving AWG 160 are configured to control the operation of the first pair of AODs 108 to rearrange the atoms 105 into a plurality of arrays using sets of moving traps. Static traps are optical traps that remain in a fixed spatial position during operation of the system 100. Moving traps are optical traps configured to spatially move in the vacuum chamber 104. The first pair of AODs 108 may be configured to generate 852-nm moving traps, although it should be appreciated that alternative moving trap sizes (e.g., as suitable for the atomic size of atoms 105) may be utilized, as aspects of the technology described herein are not limited in this respect.

[0110] In some embodiments, while atoms 105 are being arranged into static and / or moving traps, a camera 124 (e.g., a CMOS camera) may be used to capture optical signals generated by the atoms 105 and exiting the vacuum chamber 104 through an imaging objective 126. The captured optical signals may be passed from the camera 124 to CPU 122, which may be configured to analyze the optical signals to determine the spin-state and / or position of the atoms 105. For example, the CPU 122 may be configured to determine whether any atom losses have occurred, which result in unoccupied spaces in the atom arrangements and may produce undesirable or incorrect computation results.

[0111] In some embodiments, the output from CPU 122 may be provided as feedback to the rearrangement AWG 158. The rearrangement AWG 158 and moving AWG 160 may be connected to a switch 180 coupled to two optical channels 181 A, 18 IB, each of which are coupled to one AOD of the first pair of AODs 108. The optical channels 181 A, 18 IB permit control, by either the rearrangement AWG 158 or the moving AWG 160, of the movement- 24 -#14172487v2Attorney Docket No. : H0776.70181 WO00 of trapped atoms in the x- and y-direction, respectively. Based on the feedback received from CPU 122 (e.g., that there has been a loss of atoms in the moving traps, etc.), the rearrangement AWG 158 may be configured to cause the first pair of AODs 108 to further rearrange the atoms 105 in the moving traps and perform real-time rearrangement of atoms.

[0112] In some embodiments, the atoms 105 may be further optically addressed (e.g., to implement quantum logical gates or to implement other processes) by one or more lasers. A first laser 128 is configured to generate a global Raman beam that passes through a PBS 130 and dichroic mirror 132 before entering the vacuum chamber 104. The global Raman beam generated by the first laser 128 is configured to control global rotations of the quantum states encoded in the atoms 105 and to perform dynamic decoupling throughout the circuit by illuminating the arrays of qubits with the global Raman beam. In some embodiments, the first laser 128 is controlled by the Raman AWG 156 and the acousto-optical modulator (AOM) 174.

[0113] In some embodiments, a second laser 134 generates a hiding beam that passes through a fourth dichroic mirror 136 before subsequently passing through the third dichroic mirror 132 and then entering the vacuum chamber 104. The second laser 134 may be configured to shield qubits in a given lattice from light contamination from other lasers in the system 100. In some embodiments, the second laser 134 is configured to focus the laser beam is configured to focus an elliptical waist, wherein a semi-major axis of the elliptical waist is aligned in a vertical direction relative to a quantum lattice on which it is incident. In some embodiments, the second laser 134 may be configured to have a wavelength of approximately 1529-nm.

[0114] In some embodiments, a third laser 138 is configured to generate a first Rydberg beam that passes through a fifth dichroic mirror 140 before entering the vacuum chamber 104. A fourth laser 142 is configured to generate a second Rydberg beam that passes through the fourth dichroic mirror 136 before subsequently passing through the third dichroic mirror 132 and entering the vacuum chamber 104. As shown in FIG. 1, the third and fourth laser 138, 142 perform single-qubit control using two-photon Raman excitation with the first and second Rydberg beams generated by the two lasers, respectively. This two-photon excitation is the basis for entangling gates performed during operation of the quantum computer. In some embodiments, the first Rydberg beam generated by the third laser 138 is of wavelength 420-nm and the second Rydberg beam generated by the fourth laser 142 is of wavelength- 25 -#14172487v2Attorney Docket No. : H0776.70181 WO001013-nm. In some embodiments, wherein the neutral atom qubits comprise87Rb, the Rydberg beams generated by the third and fourth laser 138, 142 are configured to excite the neutral atoms to n = 53 Rydberg states

[0115] In some embodiments, a fifth laser 144 generates a first counter-propagating light beam of a first polarization that passes through the second PBS 130 before subsequently passing through the third dichroic mirror 132 and entering the vacuum chamber 104. A sixth laser 146 generates a second counter-propagating light beam of a second polarization that passes through a third PBS 148 before subsequently passing through the fifth dichroic mirror 140 and entering the vacuum chamber 104.

[0116] In some embodiments, the first and second polarizations are in opposite directions. In some embodiments, the counterpropagating beams generated by the fifth and sixth laser are configured to perform local cooling with ID polarization gradient cooling (PGC) and electromagnetically induced transparency (EIT).

[0117] In some embodiments, a seventh laser 150 generates a lattice beam of a third polarization that passes through the second PBS 130 before subsequently passing through the third dichroic mirror 132 and entering the vacuum chamber 104. Additionally, an eighth laser 152 generates a lattice beam of the third polarization passes through the third PBS 148 before passing through the fifth dichroic mirror 140 and entering the vacuum chamber 104. In some embodiments, seventh and eighth lasers are configured to generate the counterpropagating laser beams with wavelength of 795-nm and a blue detuning between 50 GHz to 200 GHz from a DI line of the neutral atom qubits. The counterpropagating light beams are configured to form a one-dimensional potential lattice to trap the qubits, forming a onedimensional qubit lattice. In some embodiments, the fifth laser 144 and seventh laser 150 generate their respective beams in parallel. In some embodiments, the sixth laser 146 and the eighth laser 152 generate their beams perpendicular to each other.

[0118] In some embodiments, the same Raman beam generated by the first laser 128 is simultaneously redirected through a local path which focuses on the second pair of AODs 116 direct the Raman beam to individual atoms. Consequently, the Raman beam focused by the second pair of AODs 116 performs local single-qubit rotations. The Raman AOD AWG 162 includes two optical channels 183 A, 183B that control the programmable light grids generated by the local Raman AOD 116 in the x- and y-direction, respectively. The Raman AOD AWG 162 may be configured to create light grids for local single-qubit control.- 26 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0119] In some embodiments, the Rydberg AWG 154 includes a plurality of acousto-optic modulators (AOMs) and at least two generators to perform through-the-lens (TTL) metering. In the Rydberg AWG 154, a first and second AOM 164, 166 controls the third laser 138 and fourth laser 142, respectfully. The Rydberg AWG 154 may be configured to generate entangling gate pulses to entangle qubits and / or to perform local detunings of the local SLM detuner 118. In the embodiment in FIG. 1, the first and second AOM 164, 166 are configured at 420-nm and 1013-nm wavelengths, respectively. A third AOM 168 is configured to control the local SLM detuner 118. A first TTL generator 170 performs TTL metering on the first pair of AODs 108 and a second TTL 172 performs TTL metering on the spatial light modulator 106.

[0120] In some embodiments, a fifth AOM 176 controls the local Raman AOD 116. The Raman AWG 156 includes a laser source 178, which, in the embodiment in FIG. 1, is configured to operate at 6.8 GHz. The Raman AWG 156 may be configured to perform in- phase and quadrature control of the qubits and / or pulse-shaping of a global and a local Raman driving.

[0121] FIG. 2A shows a schematic of the quantum computer processor 200 within the vacuum chamber 104, according to some embodiments. As shown in the example of FIG. 2A, in some embodiments, the plurality of atoms 105 are arranged in a storage zone 202, an entangling zone 204, a readout zone 206, and a reservoir zone 208. A global Raman beam 210 (e.g., a beam generated by the first laser 128) illuminates the qubits in all four zones. A hiding beam 212 (e.g., a beam generated by the second laser 134) illuminates the qubits in the storage zone 202. A first Rydberg tophat beam 218 (e.g., a beam generated by the sixth laser 146) and a second Rydberg tophat beam 220 (e.g., a beam generated by the fourth laser 142) both illuminate the entangling zone 204. A first and second counterpropagated lattice beam 214, 216 (e.g., beams generated by the seventh laser 150 and the eighth laser 152, respectively) illuminate the readout zone 206. Additionally, a first and second counterpropagated imaging beam 222, 224 (e.g., beams generated by the fifth laser 144 and the sixth laser 146, respectively) also illuminates the readout zone 206.

[0122] In some embodiments, the storage zone 202 contains qubits that, during operation of the quantum computer, will entangle with qubits in the entangling zone 204. The hiding beam 212 illuminates the qubits in the storage zone 202 to reduce decoherence in the storage qubits induced by imaging in the readout zone 206. Additionally, the hiding beam 212 is- 27 -#14172487v2Attorney Docket No. : H0776.70181 WO00 configured to reduce error from the Rydberg beams 218, 220. In some embodiments, the storage zone 202 is above the entangling zone 204 by 52-|jm. In some embodiments, each column of storage qubits is separated by 11 -pm.

[0123] In some embodiments, a local SLM beam 225 (e.g., a beam generated by the local SLM detuner 118) may perform local detunings at selected gate sites in the entangling zone 204. The global Raman beam 210 may perform parallel two-qubit gates in the entangling zone. A local Raman beam 226 (e.g., a Raman beam generated by the first laser 128 and directed by the second pair of AODs 116), performs local single-qubit gates. In some embodiments, the entangling zone 204 is separated by 40-pm from the storage zone 202 and readout zone 206 to ensure negligible error on stored qubits from the tails of the Rydberg beams 218, 220.

[0124] In some embodiments, measurement and re-initialization of qubits occurs in the readout zone 206, positioned beneath the entangling zone 204. In some embodiments, the readout zone is 12 rows tall (e.g., approximately 55 pm tall) with two rows of traps per atom for the lattice readout. In some embodiments, six blocks of qubits are interlaced horizontally for storage, corresponding to five 6x6 ancilla blocks and one 5x5 data block. In other embodiments, six blocks of qubits are interlaced horizontally for storage, corresponding to four 6x6 ancilla blocks and two 5x5 data block.

[0125] In some embodiments, and as shown in the example of FIG. 2A, the reservoir zone 208 shown is positioned beneath the readout zone 206 to replenish lost atoms in any of the storage zone 202, entangling zone 204, and / or the readout zone 206.

[0126] FIG. 2B shows a schematic level diagram 250 of the atomic transitions of87Rb, which is used as a neutral atom qubit in some embodiments. The level diagram 250 shows the quantum numbers of an excited87Rb atom, wherein the excitation state of the87Rb atom is controlled by the plurality of lasers in the quantum computer. The87Rb atom qubit in the embodiment is a hyperfine qubit.

[0127] As shown in FIG. 2B, the87Rb qubit is initially encoded into one of two states, a |0) state 252 located at the ground state of87Rb and a |1) state 254 located at a256 of87Rb. The |1) state 254 is 6.8 GHz higher in energy than the |0) state 252. When illuminated by the local Raman beam 226, the |0) state 252 may be excited above the 5P- / 2state 258 to a first excitation state 260, which is detuned above the 5P- / 2state 258. The first and second counterpropagated lattice beams 216 may also be used to excite both the |0)- 28 -#14172487v2Attorney Docket No. : H0776.70181 WO00 state 252 and the |1) state 252 above a 5 / ^ state 258 to a second excitation state 262, which is further detuned about the 5P- / 2state 258 and the first excitation state 260.

[0128] As shown in FIG. 2B, the local SLM beam 225 may excite the 11) state 252 to a third excitation state 266 detuned below a 5P3 / 2state 264. Alternatively or additionally, the local imaging beam 222 may excite the | 1) state 252 to the 5P3 / 2state 264. The first Rydberg tophat beam 218 may excite the | 1) state 252 below a 6P3 / 2state 268 to a fourth excited state 270. Subsequently, the second Rydberg tophat beam 220 may excite the atom from the fourth excited state 270 to a 53SI / 2 state 272, which is equivalent to a Rydberg state 274 (|r)) of87Rb. The double excitation of the |1) state 252 with the first and second Rydberg tophat beams 218, 220 illustrates the two-photon excitation that may be performed in the readout zone 206. Additionally, the hiding beam 212 illuminating the storage zone 202 may excite a87Rb atom from the 5P3 / 2state 264 to a 4£)5 / 2state 276.

[0129] FIG. 3 shows a flowchart illustrating a quantum information process 300 to be performed using a neutral atom quantum computer (e.g., as described in connection with the examples of FIGs. 1-2B herein). The process may be used, in some instances, to perform repeated QEC operations on neutral atom qubits. In some embodiments, the process 300 begins at act 310, in which a first plurality of qubits (e.g., neutral atom qubits) are transferred from a readout zone (e.g., readout zone 206) to an entangling zone (e.g., entangling zone 204). As one example, to implement act 310, the moving AWG 160 may direct the moving AODs 108 to transfer the first plurality of qubits from the readout zone 206 to the entangling zone 204 using moving optical traps.

[0130] After act 310, process 300 may proceed to act 320, in which the first plurality of qubits may be entangled in a spatial direction in some embodiments. To entangle the first plurality of qubits, single-qubit gates may be applied to the qubits of the first plurality. As an example, to implement act 320, the local SLM beam 225 may be used to implement single-qubit gates on the first plurality of qubits to entangle them in a spatial direction. Alternatively or additionally, the local Raman beam 226 may implement single-qubit gates on the first plurality of qubits to entangle them in a spatial direction.

[0131] After act 320, process 300 may proceed to act 330, in which a second plurality of qubits may be moved from a storage zone (e.g., storage zone 202) to the entangling zone in some embodiments. As an example, to implement act 330, the moving AWG 160 may direct- 29 -#14172487v2Attorney Docket No. : H0776.70181 WO00 the moving AODs 108 to transfer the second plurality of qubits from the storage zone 202 to the entangling zone 204 using moving optical traps.

[0132] After act 330, process 300 may proceed to act 340, in which the first and second plurality of qubits may be entangled in a time direction. To entangle the qubits of the first and second plurality, transversal entangling gates may be applied to pairs and / or groups of qubits from the first and second pluralities. As an example, to implement act 340, the first and second Rydberg tophat beams 218, 220 may facilitate two-photon excitation of the qubits to perform transversal entangling gates. Alternatively or additionally, the moving AWG 160 may direct the moving AODs 108 to employ ancilla blocks and perform lattice surgery on data blocks of qubits in the readout zone 206.

[0133] After act 340, process 300 may proceed to act 350, in which the first plurality of qubits may be transferred into the storage zone (e.g., storage zone 202) and the second plurality of qubits may be transferred into the readout zone (e.g., readout zone 206), in some embodiments. As an example, to implement act 350, the moving AWG 160 may direct the moving AODs 108 to transfer the first plurality of qubits into the storage zone 202 via moving optical traps. Additionally, the moving AWG 160 may direct the moving AODs 108 to transfer the second plurality of qubits into the readout zone 206.

[0134] After act 350, process 300 may proceed to act 360, in which qubits of the second plurality may be measured, in some embodiments. As an example, the qubits may be optically imaged (e.g., by an optical camera), and the image analyzed to determine the stored qubit state in each qubit. As one example, to implement act 360, the camera 124 may image the second plurality of qubits in the readout zone 206. The collected images may then be used to determine the spin state and / or position of the second plurality of qubits. Further details on this measurement process are provided herein in connection with FIG. 6.

[0135] After act 360, process 300 may proceed to act 370, in which the quantum states of the second plurality of qubits may be reinitialized in a new state, in some embodiments. Additionally or alternatively, the qubits may be rearranged if, for example, atom loss is detected at act 360. As one example, to implement act 370, the local Raman beam 226 may be used to optically pump the second plurality of qubits to reinitialize them into a new quantum state. Additionally, the moving AWG 160 may direct the moving AODs 108 to replace missing qubits in the readout zone 206 with atoms from the reservoir zone 208 via moving optical traps.- 30 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0136] FIG. 4A shows an illustration of a process 400 of performing spin-to-position conversion, which may be implemented in the readout zone 206 during operation of the quantum computer, in some embodiments. The left column shows steps for the three-step spin-to-position method. The right column shows schematic illustrations that correspond to each of the three steps.

[0137] In some embodiments, at a first step 402, a qubit of an unknown state is trapped in a lattice potential 404. For example, the lattice potential 404 may be formed by the first and second counterpropagated lattice beams 214, 216 as described in connection with FIG. 2 herein. In the example of FIG 4A, the qubit of an unknown state may be in either the |0) spin state 252 or the |1) spin state 254. Optical tweezers generate a first and second spindependent lattice 406, 408, wherein the first and second spin-dependent lattices 406, 408 are configured to separate qubits based on their spin state. The first and second spin-dependent lattices 406, 408 are physically separated 410 to separate qubits in space based on their spin state. In some embodiments, the spin-dependent lattices 406, 408 are physically separated by the physical movement of the spin-dependent lattice 408 (e.g., as controlled by the first pair of AODs 108, as described in connection with the example of FIG. 1 herein).

[0138] In some embodiments, at a second step 412, the qubits in the readout zone 206 are physically separated into one of two position states. The |0) spin state 252 is converted to a stretched bright state 414 and the |1) spin state 254 is converted to a stretched dark state 416. When the spin-dependent lattices 406, 408 are physically separated, the stretched dark state 416 is moved by the movement of the spin-dependent lattice 408. The positions and / or spin states of the qubits can then be measured 418 using an optical camera, according to some embodiments. A measurement 419 may have a value “0” indicating the qubit is of the |0) spin state 252, a value of “1” indicating the qubit is of the 11) spin state 254, or “loss” if the qubit was lost during operation (e.g., by scattering or other means).

[0139] In some embodiments, at a third step 420, the first and second spin-dependent lattice 406, 408 are physically returned to their original locations 422, thereby returning the qubits to their initial position for re-use and reinitialization in the circuit.

[0140] FIG. 4B shows Rabi oscillation plots 430, 440 for qubit measurements under two measurement regimes, a spin-to-loss regime and the spin-to-position described above. A qubit state 432 is plotted against atoms remaining 434 in each of the given measurement regimes. The Rabi oscillation plot for the spin-to-loss regime 430 shows an inverse- 31 -#14172487v2Attorney Docket No. : H0776.70181 WO00 relationship between the qubit state 432 and the atoms remaining 434 in a computation chamber, as measurements performed by this regime are destructive. Conversely, the spin- to-position Rabi oscillation plot 440 shows that atoms remaining 434 goes unchanged as the qubit state 432 changes, as the spin-to-position regime is non-destructive.

[0141] FIG. 5A is level diagram 500 showing the87Rb hyperfine levels used to engineer a spin- selective one-dimensional optical lattice, in accordance with some embodiments. In particular, the level diagram 500 shows hyperfine levels of bright states 252 and dark states 254, where the total angular momentum (F) is shown.

[0142] FIG. 5B shows trapping potential for dark states 520, the trapping potential forbright states 522, and the optical lattice potential 524 to further illustrate the spin-to-position conversion shown in FIG. 4A. A dark state 254 is trapped in a dark state potential 526 with, as an example, an amplitude 528 of approximately ~10MHz. A light state 252 is trapped in a light state potential 530. While the spin state 254 only experiences a light shift from the local SLM beam 225 (i.e., optical tweezers), the state 252 experiences a light shift from boss the local SLM beam 225 and a lattice trapping potential 532. Consequently, the light state potential 530 is a combination of the local SLM beam 225 potential and the potential 532. In the embodiment shown in FIG. 5B, the amplitude of light state potential 530 is approximately 6MHz. The state 252 may localize at the minimum of the light state potential 530 wherein an antinode 536 between the local SLM beam 225 and lattice trapping potential 532 occurs.

[0143] FIG. 5C is a schematic timeline of spin-to-position conversion. The time to transfer the clock qubit to the bright and dark states for readout was typically on the order of roughly 20 ps, but for some of the measurements was several milliseconds due to using a slow global rotation of the magnetic field (FIG. 5E). See Methods text for additional information. Plot 540 shows the state transfer pulse, wherein the state |0) qubits are converted to the bright state and the state 11) qubits are converted to the dark state. The plot 542 shows the duration of the lattice trapping, wherein the bright and dark state qubits are trapped in the optical lattice potential. In the embodiment shown in FIG. 5C, the duration of this trapping is approximately 500ms. The plot 544 shows the applied power of the AOD mediated local SLM beam 225, wherein the bright and dark state qubits are moved during the spin-to- position conversion.- 32 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0144] FIG. 5D shows transfer 550 of qubit state | 1) to the dark state via resonant optical pumping, in accordance with some embodiments described herein.

[0145] FIG. 5E shows transfer of qubit state |0) towards into mF = +1, +2 states. This was achieved with either a coherent Raman transfer 560 to F = 2, mF = +2 or incoherent pumping 570 with G+-polarizcd repumper. The coherent Raman transfer 560 includes a first excitation 562, a de-excitation 564, and a third excitation 566. Both approaches achieved the same bright state readout fidelity, but in the specific implementation used here the Raman transfer took several milliseconds owing to the rotation of the external magnetic field for driving G transitions 564 and 566).

[0146] FIG. 5F shows quadratic suppression 580 of readout error due to scattering. The bright state transfer ensured that at least two lattice-induced scattering events were required to cause a readout error, which occurred if the AOD tweezer had not yet moved away as the bright state became unpinned. Scattering further caused diabatic changes in the depth of the lattice potential and may contribute to atom loss.

[0147] FIG. 6 is a flowchart illustrating a process 600 for performing nondestructive qubit readout, in accordance with some embodiments of the technology described herein. The process 600 may begin at act 610, in which qubits are illuminated by counterpropagating laser beams generated by a first and second laser (e.g., laser 150 / 152 as described in connection with the example of FIG. 1 herein). In some embodiments, the first and second laser may be configured to generate laser beams having a wavelength of approximately 795- nm and operated between 50 and 200 GHz blue-detuned from the DI line. Additionally, the first and second laser may be configured to generate laser beams that are circularly polarized (<j“) such that the laser beams generate a one-dimensional optical lattice. In particular, the circular polarization of the laser beams causes the qubit state \F = 2; mp= +2) to experience approximately a 300 kHz trap frequency in one axis (e.g., because the qubit state experiences a lightshift of approximately 6 MHz). In contrast, the \F = 2; mp= —2) state is a dark state when illuminated by the foregoing laser beams.

[0148] In some embodiments, after act 610, the process 600 may proceed to act 620, in which qubits having state |1) are converted to stretched dark states. Qubits having state |1) are converted to stretched dark states using incoherent pumping generated by a second laser (e.g., laser 138) which is configured to be resonant to an F = 2 to F’ = 3 transition of the neutral atom qubits. A second laser (e.g., laser 142) generates beam configured to optically- 33 -#14172487v2Attorney Docket No. : H0776.70181 WO00 pump qubits in \F = 2; mp= 0) state into the dark state This second laser may have a wavelength of 780-nm and a~ circular polarization.

[0149] In some embodiments, after act 620, the process 600 may proceed to act 630, in which qubits having state |0) are converted to stretched bright states either by optical pumping or coherent Raman transfer. In some embodiments qubits having state \F = 1; mp= 0) encoded as |0) . Optical pumping is performed by incoherent pumping generated by a third laser (e.g., laser 146) resonant to an F = 1 to F' = 2 transition of the neutral atom qubits. The third laser may generate a beam having a wavelength of 780-nm and <J+circular polarization. Alternatively, coherent Raman transfer converts the qubits having state \F = 1; mp= 0) to stretched bright states by using fourth and fifth laser (e.g., lasers 138 and 142) to perform two-photon excitation on individual qubits.

[0150] In some embodiments, after act 630, the process 600 may proceed to act 640, in which the stretched dark states are moved using optical tweezers (e.g., acousto-optical deflector (AOD) tweezers), while the positions of the stretched bright states are maintained (e.g., by pinning to the optical lattice generated in act 610). Lattice is then ramped up adiabatically over approximately 100 ps. Next, the AOD tweezers pick up the dark state atoms and may move them approximately 2 pm over 500 ps while the bright state atoms are pinned in place by a one-dimensional optical lattice (e.g., an optical lattice generated by lasers 150 and 152).

[0151] In some embodiments, after act 640, the process 600 may proceed to act 650, in which the qubits are imaged (e.g., optically). Imaging the qubits comprises obtaining optical images of the qubits using at least one optical camera after a one- or two-step cooling process is performed on the qubits. A first cooling step includes red-detuning two counterpropagating beams generated by a sixth and seventh laser (e.g., lasers 144 and 146). During the first cooling step, the two counterpropagating beams perform a F = 2 to F' = 3 transition of the neutral atom qubits. During the first cooling step, the two counterpropagating beams are detuned by two times a Zeeman splitting of adjacent mp levels of the neutral atom qubits. A second cooling step includes blue-detuning the two counterpropagating beams generated by the sixth and seventh lasers, reducing a power of one of the two counterpropagating beams. During the second cooling step, the two counterpropagating beams perform a F = 2 to F' = 2 transition of the neutral atom qubits.- 34 -#14172487v2Attorney Docket No. : H0776.70181 WO00Subsequently, a camera takes optical images of the qubits. Using the optical images of the qubits, a spin state and / or a qubit loss of the qubits is measured, potentially by a CPU 122.

[0152] An illustrative implementation of a classical computer system 700 that may be used in connection with any of the embodiments of the technology described herein (e.g., such as a controller implementing the method of FIGs. 3 or 6) is shown in FIG. 7. The computer system 700 includes one or more processors 710 and one or more articles of manufacture that comprise non-transitory computer-readable storage media (e.g., memory 720 and one or more non-volatile storage media 730). The processor 710 may control writing data to and reading data from the memory 720 and the non-volatile storage device 730 in any suitable manner, as the aspects of the technology described herein are not limited to any particular techniques for writing or reading data. To perform any of the functionality described herein, the processor 710 may execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., the memory 720), which may serve as non-transitory computer-readable storage media storing processor-executable instructions for execution by the processor 710.

[0153] Computing system 700 may also include a network input / output (VO) interface 740 via which the computing device may communicate with other computing devices (e.g., over a network), and may also include one or more user I / O interfaces 750, via which the computing device may provide output to and receive input from a user. The user VO interfaces may include devices such as a keyboard, a mouse, a microphone, a display device (e.g., a monitor or touch screen), speakers, a camera, and / or various other types of VO devices.

[0154] In this respect, it should be appreciated that one implementation of the embodiments described herein comprises at least one computer-readable storage medium (e.g., RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or other tangible, non-transitory computer- readable storage medium) encoded with a computer program (i.e., a plurality of executable instructions) that, when executed on one or more processors, performs the above-discussed functions of one or more embodiments. The computer-readable medium may be transportable such that the program stored thereon can be loaded onto any computing device to implement aspects of the techniques discussed herein. In addition, it should be appreciated- 35 -#14172487v2Attorney Docket No. : H0776.70181 WOOO that the reference to a computer program which, when executed, performs any of the abovediscussed functions, is not limited to an application program running on a host computer. Rather, the terms computer program and software are used herein in a generic sense to reference any type of computer code (e.g., application software, firmware, microcode, or any other form of computer instruction) that can be employed to program one or more processors to implement aspects of the techniques discussed herein.

[0155] It will be apparent that example aspects, as described above, may be implemented in many different forms of software, firmware, and hardware in the implementations illustrated in the figures. Further, certain portions of the implementations may be implemented as a “module” that performs one or more functions. This module may include hardware, such as a processor, an application-specific integrated circuit (ASIC), or a field-programmable gate array (FPGA), or a combination of hardware and software.

[0156] The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects as described above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present disclosure need not reside on a single computer or processor but may be distributed in a modular fashion among a number of different computers or processors to implement various aspects of the present disclosure.

[0157] Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0158] Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements.- 36 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0159] When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.

[0160] Further, it should be appreciated that a computer may be embodied in any of a number of forms, such as a rack- mounted computer, a desktop computer, a laptop computer, or a tablet computer, as non-limiting examples. Additionally, a computer may be embedded in a device not generally regarded as a computer but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smartphone, a tablet, or any other suitable portable or fixed electronic device.

[0161] Also, a computer may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computer may receive input information through speech recognition or in other audible formats.

[0162] Such computers may be interconnected by one or more networks in any suitable form, including a local area network or a wide area network, such as an enterprise network, and intelligent network (IN) or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks.

[0163] Having thus described several aspects and embodiments of the technology set forth in the disclosure, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the embodiments described herein. Those skilled in the art will recognize or be able to ascertain using no more than routine experimentation many equivalents to the specific embodiments described herein. It is, therefore, to be understood- 37 -#14172487v2Attorney Docket No. : H0776.70181 WO00 that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described. In addition, any combination of two or more features, systems, articles, materials, kits, and / or methods described herein, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the scope of the present disclosure.

[0164] Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0165] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0166] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0167] The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and / or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and / or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and / or B,” when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0168] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This- 38 -#14172487v2Attorney Docket No. : H0776.70181 WO00 definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and / or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

[0169] In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of’ and “consisting essentially of’ shall be closed or semi-closed transitional phrases, respectively.

[0170] The use of “coupled” or “connected” is meant to refer to circuit elements, or signals, which are either directly linked to one another or through intermediate components. Elements that are not “coupled” or “connected” are “decoupled” or “disconnected.”

[0171] The terms “approximately,” “substantially,” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, within ±2% of a target value in some embodiments, and / or within ±1% of a target value in some embodiments. The terms “approximately,” “substantially,” and “about” may include the target value.EXAMPLE

[0172] In this example, reconfigurable arrays of up to 448 neutral atoms were used to implement several elements of a universal, fault-tolerant quantum processing architecture and experimentally explore their underlying working mechanisms. First, surface codes were employed to study how repeated QEC suppresses errors [6, 7], demonstrating 2.14(13)x below-threshold performance in a four-round characterization circuit by leveraging atom loss detection and machine learning decoding [8, 9]. Then, logical entanglement was investigated using transversal gates and lattice surgery [10-12], and extended to universal- 39 -#14172487v2Attorney Docket No. : H0776.70181 WOOO logic through transversal teleportation with 3D [[15,1,3]] codes [13, 14], enabling arbitrary- angle synthesis with logarithmic overhead [5, 15]. Finally, mid-circuit qubit re-use was developed

[0016] , which increased experimental cycle rates by two orders of magnitude and enabled deep-circuit protocols with dozens of logical qubits and hundreds of logical teleportations [17-20] with [[7,1,3]] and high-rate [[16,6,4]] codes while maintaining constant internal entropy. These experiments reveal principles for efficient architecture design, involving the interplay between quantum logic and entropy removal, judiciously using physical entanglement in logic gates & magic state generation, and leveraging teleportations for universality and physical qubit reset. These results established foundations for scalable, universal error-corrected processing and its practical implementation with neutral atom systems.Neutral atom logical processor

[0173] To implement these core building blocks, these experiments utilized a logical processor

[0011] with up to 448 atoms. Qubits were stored in the hyperfine clock states of87Rb atoms trapped in optical tweezers generated by a spatial light modulator (SLM) in storage, entangling, readout, and reservoir zones (FIG. 8A). Quantum circuits were programmed by shuttling qubits in the middle of the computation with a 2D acousto-optic deflector [33-35], high-fidelity entangling operations were realized via fast excitation to Rydberg states

[0036] , and fully programmable single-qubit operations were realized via locally focused Raman beams

[0011] . This system provides control parallelism over logical qubit blocks

[0011] : for logic gates, repeated error correction, and even universal computation, all the physical qubits within the logical block realized identical operations with parallel instructions delivered by optical controls.

[0174] An improvement was provided by non-destructive, spin-resolved qubit readout using a one-dimensional optical lattice in the readout zone (FIGs. 4A-4B). Whereas conventional readout is realized via spin-to-loss conversion followed by camera readout

[0011] , the state- selective lattice enables splitting the two qubit states into two separate tweezers

[0016] , thereby realizing spin-to-position conversion that enables both loss detection and atom retention. Combined with techniques for mid-circuit re-initialization, this enabled qubit re-use for extended computation as well as a two orders of magnitude increase to the experimental cycle rate, as described below.- 40 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0175] Another addition involved the use of repeated error correction for removing errors produced during quantum algorithms. To probe this function of QEC, a distance-5 surface code logical qubit was used [6, 7] with up to five rounds of stabilizer measurement (FIGs. 8B-8D) In the present example, data qubits were held in static potentials in the entangling zone, and an ancilla block was moved over to perform parallel entangling operations required for measurements of stabilizers. After performing the stabilizer checks, the ancilla block was moved to the storage zone, and a new block was brought in to repeat the cycle. Upon injecting global coherent errors at regular intervals, it was found that repeated correction significantly suppressed logical errors, as coherent errors were projected into incoherent ones (with quadratic suppression) due to the quantum Zeno effect

[0037] and tracked. Although the coherent errors were injected as global, correlated single-qubit rotations, stabilizer measurements converted these into uncorrelated, incoherent physical errors

[0038] , and prevented coherent errors on the logical qubit level (Methods). FIGs. 8C- 8D highlight the function of repeated QEC: the conversion of general errors to correctable bit-flip and phase-flip errors removed entropy by reducing the range of possible errors [5], and detecting the incoherent errors and using them in decoding reduced entropy by keeping track of the state of the system [6].Below-threshold performance

[0176] These tools were utilized to explore below-threshold performance of a surface code qubit under multiple rounds of QEC. Without wishing to be bound by any particular theory, logical errors can be exponentially suppressed as ( / iii)(' / + l , / 2by increasing code distance d if the physical error rate p is below a characteristic threshold pth [5, 7, 39]. FIG. 9A shows the measurement results of the data qubits and multiple sets of ancilla qubits in the configuration is shown in FIG. 8B, obtained with non-destructive, spin-resolved readout (FIGs. 4A-4B), which identified which atoms were lost during the quantum circuit. It was observed that qubit loss events resulted in flickering stabilizer patterns

[0040] (see FIG. 16B and Methods) which created time-correlations between detected error events and led to a sharp rise in detected errors (FIG. 9B).

[0177] To correct for the effect of atom loss, so-called superchecks were constructed (see Ref.

[0040] and Methods) by multiplying stabilizers around the lost atom to recover error information, which resulted in a suppressed detected error probability (FIG. 9B). These- 41 -#14172487v2Attorney Docket No. : H0776.70181 WO00 observations indicated that using the loss information in decoding can greatly improve QEC performance [8, 9]. To take advantage of these features, most-likely error (MLE) [9, 41] and machine learning decoders

[0032] trained on simulated and experimental data were used, incorporating loss information into both (Methods).

[0178] Repeated QEC performance was characterized as a function of the code distance d

[0042] . It was found that d = 5 has a 2.14(13)x lower error per round than d = 3, indicating below-threshold behavior for this four-round characterization circuit (FIG. 9D). In particular, it was found that the loss information and machine learning decoding together improved the QEC performance by a factor of 1.73(13)x compared to conventional methods. While these observations showed below-threshold behavior for repeated QEC rounds, the threshold may worsen by ~ 1.15x in the limit of many repeated rounds and by ~l.lx when incorporating ~ 1 transversal gate per QEC round (see Methods). The error per round is 0.62(3)% for d = 5, and reduced toward 0.1% per round on shots with no qubit loss (FIG. 9E), consistent with a p3scaling and roughly half of these errors being loss.

[0179] These observations are consistent with numerical simulations using a simple empirical error model based on separately characterized error rates (see error budget in FIG. 9F). This observed distribution of errors-per-shot is closely consistent with these simple simulations with uncorrelated Pauli-type and loss-type errors, suggesting the absence of large-scale correlated errors in the system. It was observed that time-correlations are almost fully diminished when postselecting on no qubit loss, indicating that almost all leakage (> 80%, Methods) corresponds to atom loss, while leakage to other hyperfine states appears to be suppressed. Various scattering channels

[0043] may naturally loss-convert, likely due to anti-trapping of metastable states.Stabilizer measurement during logic operations

[0180] Next, repeated QEC was extended to logical operations, by employing two different approaches for realizing quantum logic (FIG. 10A), where the stabilizer measurements played different roles. In the transversal method, a logical gate was realized by physically transporting and interlacing the data qubits of two surface code blocks, and then applying pairwise entangling gates [1, 6, 11]. In the planar setting, logical entanglement was realized via lattice surgery

[0010] , where a joint logical measurement is performed by ancilla qubits added between the two codes (FIGs. 10A-10B). To probe these logical operations, in the- 42 -#14172487v2Attorney Docket No. : H0776.70181 WOOO transversal approach, three rounds of A repeated CNOTs were applied interleaved with two rounds of stabilizer checks. In the lattice surgery approach, a Z / Z^ joint logical measurement was carried out using two rounds of stabilizer measurements

[0010] . The X£X and Z / Z parities of the resulting logical Bell state were measured for both methods.

[0181] To test the role of stabilizer measurements in both cases, additional ancilla measurement errors in postprocessing were introduced (FIG. 10C). A lower logical error for transversal gates was observed, and the lattice surgery was significantly more sensitive to injected measurement errors. To explore the interplay of logic gates and entropy removal, the transversal gate performance as a function of the number of repeated CNOT gates per QEC round was investigated. The optimal performance for several CNOTs per QEC round was found, and using postselection on decoding confidence (see Methods), it was observed that approximately three CNOTs per round was optimal at low error rates (FIG. 10D).

[0182] These results highlight multiple aspects of FTQC. First, the response to ancilla measurement errors highlights a distinction between the two approaches: in the transversal gate setting, the logic was realized directly between the data qubits which store the underlying logical states, and the role of stabilizer measurements is to remove entropy, whereas in lattice surgery, the ancilla measurements directly perform the logic operation and have to be correct. The need for correct stabilizer measurements is the origin of the conventional fault-tolerance assumption of d rounds per QEC cycle [6, 10]. Conversely, with transversal operations, the observed optimum at 3 CNOTs per round corresponds to stabilizer measurements balancing the local entropy generated by the logic gate. This is consistent with theoretical predictions for universal computation with correlated decoding techniques [12, 41, 44]. Second, FIG. 10D clearly shows that the error per logical gate depends on the number of gates applied per QEC round. This highlights a distinction between operations on physical and logical qubits: while physical gate performance can be well-characterized by a single fidelity value F, logical gate performance depends both on the decoding success probability - which reflects the physical entropy of the system, captured by the detector error probability pdet- and on how much that entropy increases per logical gate, quantified by Xpdet. It was found that a simple model for logical gate fidelity FL, where 1 - FLoc [(pdet+ IVA Pdetd + x^2] / with the measured pdetwas consistent with the physical two-qubit gate error, which accurately describes the experimental data (Methods).- 43 -#14172487v2Attorney Docket No. : H0776.70181 WOOOUniversality and synthesizing arbitrary unitaries

[0183] Realization of arbitrary error-corrected unitary operations was investigated. The core ingredient was the Solovay-Kitaev theorem, which states that arbitrary single-qubit rotations can be approximated to exponential precision using only digital gates such as Hadamard H .71 and T = e~ s (a 45-degree rotation around the Z-axis) [5, 15, 24]. While the Eastin-Knill theorem prohibits realizing such a universal gate set with unitary transversal operations

[0045] , it can be circumvented by the introduction of logical measurement which breaks the unitarity constraint

[0013] .

[0184] Non-Clifford T gates and universal rotations were realized with efficient transversal circuits built out of teleportation with 3D codes. FIG. 11A shows experimental measurements where quantum circuits were used (see Methods) to create 2D color codes (Steane codes) [2], and 3D color codes (Reed-Muller codes), and subject them to global phase rotations (p around the Z-axis

[0013] . For comparison, also shown are the results of similar measurements for unentangled physical qubits analyzed as 3D codes, and 3D codes with incorrect values of stabilizers. The various configurations show plateaus in the logical expectation value, and revivals in the stabilizers, for multiples of 90-degree angles (multiples of 180-degree for unentangled qubits). Additionally, 3D codes, prepared in the proper entangled states, also exhibit robustness at multiples of 45-degrees, corresponding to their transversal T gate

[0014] .

[0185] To implement unitary synthesis by the circuit in FIG. 11B, transversal teleportation was utilized. Specifically, multiple Reed-Muller codes were created in the \TL) state, and by applying logical CZ gates followed by X-basis measurements (with feedforwards applied in-software here), the logical information was teleported with an H gate

[0013] . FIG. 11C shows the set of angles generated by the circuit in FIG. 11B using up to three T gates. The resulting logical states span a range of points on the Bloch sphere and matched the expected angles with high precision (consistent with statistical uncertainty). It was observed that the angular spacing between accessible states shrunk exponentially with the number of T gates. This enabled precise synthesis of rotation angles with a logarithmic number of steps (FIG. 11D inset) [5, 24, 46].

[0186] These observations demonstrate that teleportation may be used as a powerful tool for universal processing, allowing precise analog rotations to be built from digital gates. Although the circuit realized here was fully transversal, the measurement, decoding, and - 44 -#14172487v2Attorney Docket No. : H0776.70181 WOOO logical feedforward ensured that the logical information propagated unitarily while physicallevel dissipation served to correct errors. FIGs. 20A-20F explore code- switching and transversal gates between 2D and 3D codes, again finding the underlying source of universality is the measurement of the 3D code. In addition, the role of stabilizers differs fundamentally when generating logical magic. While for Clifford circuits, stabilizer measurements were used to remove entropy, for transversal T gates, correct stabilizer signs (e.g., +1 eigenvalues) were essential for realizing non-Clifford operations (FIG. 11A). This observation revealed that physical entanglement was directly required for logical magic. This may be understood by the fact that, while logical Pauli states such as |+L) are eigenstates of operators XL= X1X2X3..., which was a tensor product of physical operators, states such as \TL) were eigenstates of X1X2X3. . . -4-5^5^ 5^ . . . ), involving a superposition spanning the code that was necessarily entangled. An error-corrected Bell inequality test was performed in FIG. 20F and a CHSH inequality of 1.99(3) x 2 was measured, saturating the quantum bound

[0047] .Deep circuits at constant entropy

[0187] The ability to perform deep-circuit quantum computation on the logical level was explored. The processor was kept at a constant entropy (FIG. 12A) [5], to reduce the accumulation of physical errors. This was challenging because computation inevitably introduced errors in the physical qubit state while, in addition, increasing entropy in the other degrees of freedom, such as the atomic motional state. To ensure that all physical errors were removed and that computation was kept at constant entropy, transversal teleportation was leveraged [13, 18-20]. In this approach (detailed below) the logical information propagated throughout the circuit while physical errors were left behind. Measuring this block then enabled qubit reset, re-cooling, and re-initialization of the physical atoms. After that the block was prepared again in a low entropy state and was then utilized for subsequent teleportation steps.

[0188] To realize constant entropy operation during deep quantum circuits, the atomic internal states, temperature, and atom filling needed to be re-initialized during the computation. In order to achieve this, non-destructive internal state readout (FIGs. 4A-4B) was combined with non-destructive imaging that also re-cooled the atom. While laser cooling is typically achieved using 3D beams in zero magnetic field, a method that enabled - 45 -#14172487v2Attorney Docket No. : H0776.70181 WOOO high-fidelity imaging and cooling operating with focused ID beams in a finite B-field (required for atomic qubit control) was implemented (FIG. 12C) [48, 49]. Coherence of data qubits in the nearby storage zone was protected by applying a 1529-nm shielding beam (FIG. 12D, FIGs. 15A-15E)

[0050] , Moreover, the missing atoms in the array were refilled with atoms from the atomic reservoir (FIG. 12A). With all these methods combined, in FIG. 12C the performance when subjecting the atoms to all the operations (except entangling gates) was measured in a 27-layer circuit, explained below. For instance, by applying a perturbation on the 5th cycle, it was found that the atomic filling and temperature quickly recovered to a steady-state. It was observed that the ID cooling methods nearly reproduced the conventional 3D performance, limited by tweezer-depth in- homogeneity (with potential for improvement, see FIGs. 14A-14F). Repeated operation also allowed for fast cycle rates; as an example, FIG. 12B shows Rabi calibrations with a 4-ms cycle time.

[0189] These tools were used to implement logical algorithms. In the present example, logical blocks were entangled, measured, and teleported in alternating A and B groups (FIG. 13A). Within one layer, a fresh batch of physical qubits (group A) was retrieved from the readout into the entangling zone in order to encode them into error-correcting codes, and then entangled the code blocks with each other (entangling in space direction). Group B was then retrieved (already entangled) from storage, and a transversal entangling gate with the group A block (entangling in time direction) was performed. In the spirit of measurementbased quantum computing

[0019] , group B was moved into the readout zone, group A to the storage zone, and then group B was measured. Group B was then re-initialized and the whole process was repeated in the next layer.

[0190] Repeated state preparation of 32 blocks of [[7,1,3]] (Steane) codes (16 blocks per alternating group) for up to 27 layers was investigated (FIG. 13A), and it was found that the stabilizer expectation value remained constant as a function of the cycle number (FIG. 13B), indicating a steady-state internal entropy. (In this example the fidelity was limited by several sub-optimal choices in the circuit design structure, see Methods for details). As an example algorithm, the [[7,1,3]] codes were entangled in both the time and space directions to realize ID and 2D cluster states and probe the resulting correlations

[0019] . Starting with a ID cluster state in the time direction, FIG. 13C shows the correlations on both the logical-level and the physical-error-level by evaluating the correlator (Z(Z —between coordinates separated in time (Methods). On the logical-qubit level the correlations were observed- 46 -#14172487v2Attorney Docket No. : H0776.70181 WOOO between logical outcomes corresponding to successful algorithm evolution, with a decay with distance corresponding to an effective algorithmic error rate that improved with decreasing entropy (corresponding to lower acceptance fraction). Conversely, for the physical errors, it was observed that the correlations were rapidly suppressed. Similar behavior was found in the 2D cluster state data (FIG. 13D), where the logical cluster state stabilizers were finite across both space and time, but the correlations between stabilizer errors were rapidly suppressed. These properties persisted until the reservoir began to run out of atoms. Leveraging the regular transversal measurements, data qubit loss detection was used as well as a recurrent neural network architecture for decoding these algorithms (Methods).

[0191] These observations may be understood by considering the processes illustrated in FIG. 13E, indicating how the transversal teleportation ensured the removal of physical errors. Whereas the decoding and logical teleportation (with feedforward done in- software) maintained unitary evolution and propagated the logical information, the physical errors remained on the previous block (with errors propagated by entangling gates traveling at most one layer). This structure, where the logical information was natively teleported onto a fresh logical block in the algorithm, thereby ensured that the algorithm proceeded at constant entropy while also performing logic (also leveraged in FIGs. 11A-11D).

[0192] To test this teleportation method for more general encodings, high-rate [[16,6,4]] codes

[0025] were investigated. Such high-rate codes

[0029] have a complex structure that would generally require intricate circuit structuring to ensure leakage removal [7, 9]. In contrast, by directly using the teleportation procedure described above (blue curve in FIG. 13G) it was found that in a temporal ID cluster state the logical information propagated while the physical errors were suppressed with a rate similar to the simpler [[7,1,3]] Steane codes. Such high-rate codes also enabled new opportunities for realizing quantum algorithms. For example, the permutation CNOT operation may entangle logical qubits within the same block simply through re-indexing physical qubits, thereby extending the correlation length (black curve in FIG. 13G).

[0193] 2D entanglement of the [[16,6,4]] blocks (with up to 16 blocks at a time) was also explored, realizing the entangled structure depicted in FIG. 13H. FIG. 131 shows the 2D logical cluster state stabilizers as a function of postselection on shots where the copropagating logical operators agree (i.e., the cluster state stabilizers have the same outcome- 47 -#14172487v2Attorney Docket No. : H0776.70181 WO00 for each logical qubit within the block). It was found that such a procedure further improved algorithm performance. This was because the logical operators - although independent degrees of freedom - were supported on the same physical qubits. While these easily attained in-block correlations were algorithmically useful (as illustrated using [[8,3,2]] codes in Ref.

[0011] ), generating such logical entanglement was only possible because the logical operators overlapped, which was enabled by the underlying physical entanglement (Methods).METHODSSystem Overview

[0194] An overview of the quantum information processing system is provided herein (FIGs. 1, 2A-2B).

[0195] A cloud containing millions of cold87Rb atoms was loaded in a magneto-optical trap inside of a glass vacuum cell. The Rb atoms were then loaded stochastically into programmable, static arrangements of 852-nm traps generated with a spatial light modulator (SLM, Hamamatsu X13138-02), and then rearranged with a set of 852-nm moving traps generated by a pair of crossed acousto-optic deflectors (AODs, DTSX-400, AA Opto- Electronic) to realize defect-free arrays [62-64]. Al lambda-enhanced gray molasses cooling was used to achieve a loading efficiency of 75%

[0065] . Atoms were imaged with a 0.65-NA objective (Special Optics) onto a CMOS camera (Hamamatsu ORCA-Quest C15550-20UP), chosen for fast electronic readout times. The qubit state was encoded in mF = 0 hyperfine clock states in the87Rb ground-state manifold, with T2 > Is [35, 66], and fast, high-fidelity single-qubit control was executed by two-photon Raman excitation [35, 67]. A global Raman path illuminating the entire array was used for global rotations (Rabi frequency ~ 0.5 MHz, resulting in ~ 5 ps rotations with composite pulse techniques

[0035] ) as well as for dynamical decoupling throughout the entire circuit (typically 1 global 7t pulse per movement). For this work, the micro wave source was upgraded (Rohde and Schwarz, SMW200A) and increased the intermediate- state detuning to 550 GHz (measured scattering error 5 x 10-5per robust SCROFULOUS pulse). Fully programmable local single-qubit rotations were realized with the same Raman light but redirected through a local path which was focused onto targeted atoms by an additional set of 2D AODs. To realize high-fidelity, programmable single-qubit pulses, upgrades were made to the single-qubit entangling rotations by using direct Raman X-type rotations (see Raman gates section). Entangling - 48 -#14172487v2Attorney Docket No. : H0776.70181 WOOO gates (270-ns duration) between clock qubits was performed with fast two-photon excitation using 420-nm and 1013-nm Rydberg beams to n=53 Rydberg states, utilizing a time-optimal two-qubit gate pulse

[0068] detailed in Ref.

[0036] , in this example with the 420-nm laser red- detuned by 4.8 GHz from the intermediate state. During the computation, atoms were rearranged with the AOD traps to enable arbitrary connectivity

[0035] . A technique in the present specification was the ability to perform non-destructive qubit readout, enabling loss detection as well as qubit re-use. This was realized with a one-dimensional optical lattice

[0016] which pinned one of two spin states, used optical tweezers to separate the pinned and unpinned states, and then imaged the atom position. To further enable mid-circuit qubit measurement and re-use on large arrays, methods of low-loss, high-fidelity qubit readout and re-initialization were developed while only needing moderate trap depths (see below). These techniques were implemented for re-using atoms and extending the depth of error- corrected computation.

[0196] The quantum circuits were programmed with a control infrastructure consisting of five arbitrary waveform generators (AWG) (Spectrum Instrumentation), as illustrated in FIG. 1, synchronized to < 10-ns jitter. The 2-channel rearrangement AWG was used for realtime rearrangement, the 2 channels of the Rydberg AWG were used for entangling gate pulses and for local SLM detunings, the 4 channels of the Raman AWG are used for IQ (in- phase and quadrature) control of a 6.8 GHz source [35, 67] (the global phase reference for all qubits) and pulse- shaping of the global and local Raman driving, the 2 channels of the Raman AOD AWG were used for displaying tones that create the programmable light grids for local single-qubit control, and the 2 channels of the Moving AOD AWG were used for controlling the positions of all atoms during the circuit.

[0197] In some embodiments, circuits as long as 1.1 seconds for the experiments in FIGs. 12A-12D and 13A-13I were realized. In order to realize this with the AWGs, a memory segment was generated for one circuit layer for the Moving AWG, Rydberg AWG, and Raman AOD AWG, and then looped these identical memory segments for each layer. This was complicated for the Raman AWG as phase continuity needed to be ensured, and so for simplicity in this work the whole Raman waveform was programmed directly. The entire memory of the Spectrum AWG was filled, and this limited these experiments to 27 layers for some embodiments (an appropriately sized reservoir was selected to have atoms for that many layers).- 49 -#14172487v2Attorney Docket No. : H0776.70181 WO00Details of processor configuration

[0198] This approach to quantum processing was highly programmable, however it was found that each new atomic layout design behaved slightly differently [11, 35, 69]. A close analogy was “design and print a new chip” every time the processor design was changed, but each one required its own specific characterization and calibration. It was observed that- while each configuration created may be slightly different and may have its own specific challenges - with sufficient characterization and optimization ‘nominal’ performance was recovered (i.e., consistent with simple single-qubit and two-qubit error model), and that such a configuration was stable and reproducible once it had been properly set up.

[0199] This example details circuit configurations that required different degrees of characterization in this work. In the repeated QEC rounds on the surface code, the circuit structuring was carefully engineered in a manner where the time would perfectly echo on each qubit. This was greatly facilitated by the symmetric four-gate structure of the stabilizer syndrome extraction circuit. For example, although the local Raman pulses were applied row-by-row, the overall amount of time in superposition - although different for each atom- echoed around a central global n pulse. However, although this enabled the total time echoed, the specific structuring and parity of pulses prevented enabling the overall atomic trajectory echoed

[0011] . As such, homogenizing the AOD trap power over the surface code region was done carefully. Conversely, in order to realize the programmable hypercube codes, the parameters were confined to the general encoding circuit. Even the total time did not echo on each qubit, which thereby greatly impacted performance. This illustrated that each circuit realized was different and, although it was possible to achieve correct ‘nominal’ fidelities, sometimes the layout and circuit design required multiple iterations to find a suitable approach.

[0200] Specific aspects of the processor designs according to some embodiments used in this example are described herein.

[0201] Surface code. For surface code experiments (FIGs. 4A-4B, 8-10), the same static traps were used for mid-circuit storage of ancilla blocks and for readout of all qubits at the end of the computation. The readout zone was 12 rows tall (55 pm) with two rows of traps per atom for the lattice readout, FIG. 9A. Six blocks of qubits were interlaced horizontally for storage, corresponding to five (four) 6x6 ancilla blocks and one (two) 5x5 data block(s) in FIGs. 9A-9G. This interlacing ensured that the dimensions of each qubit block was the- 50 -#14172487v2Attorney Docket No. : H0776.70181 WO00 same in both the readout and entangling zones, preventing heating from AOD intermodulation effects that were observed when compressing or expanding the AOD grid. Two additional columns of traps formed a small reservoir that was used for initial rearrangement, resulting in a total array width of 165 pm.

[0202] The 420-nm and 1013-nm Rydberg tophat beams covered 7 rows of gate sites in the entangling zone and were homogenized to ~ 1 % peak-to-peak variation over a vertical extent of 60 pm. The entangling zone is separated by 40 pm from the storage and readout zone (overlapping in these measurements) to ensure negligible error on stored qubits from the tails of the Rydberg beams.

[0203] Deep circuits. For deep circuit experiments (FIGs. 12A-12D and 13A-13I, and same configuration used in FIGs. 11A-11D), the same 60 pm vertical extent was used for the entangling zone as above. Within this zone, entangling gates were performed simultaneously on up to 256 qubits across 8 rows and 16 columns of gate sites with a horizontal extent of 175 pm. Below the entangling zone was the readout zone, used for measurement and reinitialization of up to 128 atoms arranged in 4 rows. This region was illuminated by counterpropagating imaging and cooling beams (beam waist 50 pm) as well the onedimensional lattice beams (average waist 60 pm) as illustrated in FIG. 2A.

[0204] During mid-circuit imaging, atoms were always held in the storage zone, 50 pm from the entangling zone. To preserve coherence of qubits during the imaging, the storage zone was illuminated by a 1529-nm shielding beam with a beam waist of 35 pm, matching the zone’s vertical extent. These design parameters ensured both that stored atoms did not pick up error from Rydberg beams, as described above, and also that negligible 1529-nm light reached the readout zone and did not cause spurious lightshifts on the imaging and cooling transitions (see “1529-nm shielding beam”). Finally, the reservoir was located directly below the readout zone and contained up to 196 atoms in 6 rows.

[0205] The trap intensities in the entangling and storage zones were set to half of those in the readout and reservoir zones to improve qubit coherence. This was achieved by modifying the target trap intensities in the trap generation algorithm

[0064] . The array was centered on the zeroth diffraction order of the trap SLM to maximize the deflection efficiency.Spin-to-position conversion with a one-dimensional optical lattice- 51 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0206] Non-destructive qubit readout was realized throughout this example through spin- to-position conversion (FIGs. 5A-5F) [16, 70-72]. A one-dimensional optical lattice was formed by two 795-nm counterpropagating local beams, both sourced from the same titanium: sapphire laser (M Squared) and operated at 50-200 GHz blue-detuned of the DI line. Both beams were G” polarized such that F = 2; mF = 1 was a dark state and F = 2; mF = +2 experienced a maximum lightshift of approximately 6 MHz, corresponding to approximately 300 kHz trap frequency in one axis. The close detuning was a balance between minimizing off-resonant coupling to the D2 line for the dark state and reducing scattering and heating from the lattice light. Since the clock state qubit was used for computation, readout was optically pumped F = 2; mF = 0 into the dark state with 780-nm o--polarized light resonant to F = 2 to F' = 3 which was co-propagated with one port of the lattice. To suppress the probability of scattering into the dark state during readout, the F = 1; mF = 0 to F = 2; mF = +2 (bright state) was transferred, which also increased the trap depth. This was achieved via either a coherent Raman transfer or with incoherent G+- polarized 780-nm repumper from F = 1 to F' = 2. The former approach was used in all surface code experiments (FIGs. 8B, 9A-9G, and 10A-10D) and the latter in deep circuit experiments (FIGs. 4A-4B, 11-13), comparable performance was observed from both methods.

[0207] Following these state transfers, the lattice was ramped up adiabatically over approximately 100 ps. AOD tweezers pick up atoms in the dark state moved them by approximately 2 pm over approximately 500 ps; during this, atoms in the bright state were pinned in place by the stronger confinement of the lattice. Finally, the lattice was ramped down and conventional camera-based readout then imaged the position of the atom, which allowed identification of the spin state as well as loss detection. Using the data in FIGs. 4A- 4B, an error probability of 0.87(7)% for the dark state, 0.05(5)% for the bright state, and a 0.24(2)% probability of loss was measured. The asymmetric error arose from trade-offs when simultaneously optimized for loss and readout fidelity and was tuned to be more balanced; typically, due to the pumping fidelity, the dark state error was at least ~ 0.3% higher than the bright state.One-dimensional and finite-field operation for imaging and cooling- 52 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0208] For local cooling and imaging, two counterpropagating 780-nm beams with opposite circular polarization were used [48, 73] (FIGs. 14A-14F). The beams were red-detuned from F = 2 to F' = 3 and had a variable relative detuning; the <J+—polarized beam additionally contained a small repump component. Conventional methods based on polarization-gradient cooling (PGC) require zero magnetic field, however mid-circuit operation requires a finite magnetic field to maintain the quantum state of active qubits. To this end, a scheme for one-dimensional PGC in finite magnetic field was developed. PGC is based on a linear polarization rotating along the beam propagation direction which produces a population imbalance within the hyperfine levels

[0048] ; in finite field, this imbalance is disturbed and the cooling mechanism breaks down

[0074] . By transforming to a frame where the polarization rotates in time, a fictitious field appeared which cancelled the external field and restored the cooling effect. This condition was achieved by detuning the two counterpropagating beams - which were (anti)parallel to the external magnetic field - by two times the Zeeman splitting of adjacent mF levels. As shown in FIG. 14E, this detuning method worked across the full range of magnetic fields investigated (up to 8.6 G). Furthermore, it was broadly applicable to finite-field implementation of any onedimensional technique based on the same polarization configuration used here, for example gray-molasses cooling [65, 75]. While this finite-field PGC was sufficient to image without loss, a second stage of EIT cooling was added to further reduce the atom temperature

[0049] . The scheme, shown in FIG. 14F, used the same beams as the PGC imaging and only required changing to be blue-detuned of F = 2 to F' = 2 (by ~ 80 MHz) and reduced the power in one of the beams. Using both drop-recapture measurements and adiabatic rampdown measurements of the atom temperature

[0076] , the radial and axial atom temperature were probed and were both found to be comparable to three-dimensional techniques (FIG. 12C and FIG. 14F). Furthermore, the steady-state temperature and loss was set only by the EIT cooling fidelity and was independent of the degree of heating introduced from the prior circuit.1529-nm shielding beam

[0209] To preserve the coherence of qubits in the storage zone, they were illuminated with a single beam of 1529-nm light (FIGs. 15A-15E). By coupling the 5P3 / 2state to the 4£)5 / 2state, a strong Stark-shift on the excited 5P3 / 2state was imparted

[0050] . This caused probe light in the readout zone to appear off-resonant to the storage zone atoms while maintaining - 53 -#14172487v2Attorney Docket No. : H0776.70181 WOOO qubit information in the hyperfine manifold of the ground state, see FIG. 15A. The beam was generated by a Connet CoSF-D series 10W fiber laser and was focused down to an elliptical waist of 35 pm x 65 pm. The shorter waist of the beam was aligned vertically to the center of the storage zone. The beam was imaged in a 4f system and a knife-edge was applied in the image plane, approximately 4 beam waists from its center, to suppress its Gaussian-tail. Stray 1529-nm light, even at low powers, degraded the imaging quality in the readout zone. Therefore, it was found that, beam-shaping maintained stable imaging quality and coherence on the storage-zone atoms for the layout of this array.

[0210] Dephasing of the storage-zone qubits was measured as a function of detuning from the bare transition, while readout-zone qubits were illuminated with local probe and repumper light, FIG. 15B. One feature of the spectrum was captured with a simple modelQ probe, Tprobe were the Rabi frequency and scattering rate of the local imaging beams, t was the illumination time, and ALS- was the calculated lightshift of 53 / 2due to the coupling to 4£)5y2and 4£)3 / 2• More complex on-resonance or multi-level features were not captured by this simple model and were particularly sensitive at detunings between the resonances of the 4D-levels

[0077] . During qubit re-use and local imaging, the storage zone at 1529.49 nm with ~ 1.2 W was addressed. This corresponds to an approximate lightshift of 6 GHz on the 5P3 / 2state. The 1529-nm laser was further characterized by varying the detuning of the local imaging light, and a clear Autler-Townes splitting was observed, FIG. 15C. It was found that, the separation of two fitted Lorentzian peaks scales linearly with the square-root of the drive power.Repeated arrangement from reservoir

[0211] The mid-circuit image identified the qubit state as well as which atoms were lost. Before rearrangement, the atoms were recombined into their original tweezers, balancing the trap depth between the AOD and SLM tweezers to minimize loss and used cooling throughout. After this recombination, empty sites were filled using the reservoir.

[0212] In each round of rearrangement, target rows were refilled sequentially with one parallel step per row. All atoms in each step were sourced from a single reservoir row. Efficient horizontal moves were selected and the reservoir-to-target row pairings were- 54 -#14172487v2Attorney Docket No. : H0776.70181 WOOO optimized to minimize travel distance. Finally, since the local imaging beams did not cover the full extent of the reservoir, the reservoir site occupancies were stored from a global image before the circuit began and used reservoir atoms were tracked in software. This led to a slowly growing rearrangement infidelity.Mid-circuit re-initialization

[0213] After qubits were measured and atom loss refilled, the spin state was re-initialized to re-use the qubit. This local state preparation was performed in the readout zone using a Raman-assisted optical pumping scheme [35, 78]. Local Raman was used for the coherent 7t-pulses and the local probe beams were used for resonant depumping of the F = 2 manifold. Due to the close horizontal spacing of traps in the readout zone, the crosstalk between local Raman tweezers was minimized by alternating the applied 7t-pulses between odd and even columns. 24 cycles of pumping per atom was performed over a few hundred microseconds.Local single- qubit gate details

[0214] Single-qubit gates were performed using Raman transitions as previously described in Ref.

[0011] , with several changes to allow X(0) rotations to be directly implemented with high fidelity. One challenge for local X gates was ensuring polarization homogeneity, since the Rabi frequency was sensitive to the degree of circularity. Inhomogeneity was found both across the array, introduced by a sharp dichroic cut-off noted in ref.

[0011] , as well as inhomogeneity within each optical tweezer due to polarization breakdown near the tweezer focus. To reduce the first effect, a second copy of the dichroic was added into the path with a half-waveplate between the pair, such that any angle-dependent phase shifts upon reflection from the dichroics was equally applied to both the s- and p-polarized components and the polarization remained close to circular. Second, polarization breakdown of a circularly-polarized tweezer resulted in an off-axis fictitious field with components both parallel and perpendicular to the external magnetic field (in the plane of the tweezer focus)

[0079] ; the parallel components drove Raman transitions and resulted in dephasing of the clock qubit. Since the magnitude of the maximum off-axis field fell off linearly with tweezer waist, this was mitigated by increasing the waist to 2.5 pm. Finally, to increase the projection of the Rabi frequency drive along the magnetic field axis, the Raman beam was displaced by roughly 1.5 mm within the back aperture of the objective whose size is 5.5 mm, so that the- 55 -#14172487v2Attorney Docket No. : H0776.70181 WO00Raman beam came in at an angle. For all single-qubit gates in this example, robust SCROFULOUS pulses were used

[0080] .AOD intermodulation effects

[0215] Several intermodulation effects were observed from the AODs that resulted in degraded performance for specific AOD moves. First, it was ensured that the frequency tones in a given AOD axis were in an exact frequency comb, as intermodulation can lead to interference and beating near trap frequencies. Second, it was observed that the relative frequencies of the X-frequency-spacing and Y-frequency-spacing also caused an intermodulation effect, and that when beat notes of these were near trap frequencies this also led to heating. As such, the AODs for translations were primarily used, which avoided compressions / expansions of the grid when possible, and incommensurate spacings for X and Y were used to avoid accidental cross-resonances.Analysis of error correlations

[0216] Correlations in errors, in either space or time, may have implications on QEC. This example explains various correlation analyses in this system.

[0217] De-correlation of global coherent errors by projective measurement. Parallel control enabled, for example, the realization of a transversal entangling gate with a single global pulse of the entangling laser

[0011] . Additionally, error correction natively de-correlates globally correlated errors.

[0218] Consider a code block of qubits with X and Z stabilizers. Applying a global 9 will map each of the X operators to — (X + 10F) = X ■ (1 — 0Z). Consequently, measuring the X-basis component of this qubit probabilistically led to a Pauli Z error on this site with probability 92. For global rotation 9, the logical operator XL= XXXX...maps to X + i0Y)(X + z0T)(X + z0T)(X + z'6»T)... = XXXX... + idY XXX... + (i0)dYYYY... As such, for small 9, logical rotations were exponentially suppressed with the code distance d. As such, even though all the physical qubits received a global rotation 9, the logical qubit state did not receive that same rotation, and after syndrome measurements these errors were converted into incoherent-type errors and were corrected. This is the basis behind the observed suppression in FIGs. 4A-4B and 8A-8D. FIG. 18B shows that the error correction prevented an unintended logical rotation. The logical rotation was even further suppressed by the random stabilizer signs (below).- 56 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0219] Role of stabilizer signs under coherent errors. Stabilizer signs affected the logical qubit’s response to global coherent rotations. For transversal non-Clifford gates, deterministic stabilizer eigenvalues (e.g. = +1) were necessary to correctly implement the logical gate (FIGs. 11A-11D). In contrast, Clifford circuits allowed the eigenvalues to be either +1 or -1, as the signs were tracked through the circuit, giving freedom to engineer how coherent errors interfered. For example, choosing negative signs generated decoherence-free subspaces

[0081] and gave greater robustness of the logical operator against coherent errors. The same principle also suppressed logical coherent errors during computation

[0038] . In particular, stabilizer measurement projected the logical state onto a specific stabilizer configuration with random ± 1 values, which corresponds to a random configuration of physical X and Z flips which did not commute with the coherent rotation. On top of the exponential suppression of logical coherent errors, this further resulted such that the specific rotation angle of scale ~ 9dwas random on each shot, effectively turning these again into incoherent errors on the logical level.

[0220] Decay to Rydberg P states. In Ref.

[0036] FIGs. 18A-18E, the presence of weak correlations was analyzed between CZ gate errors seen in a repeated randomized benchmarking sequence; the origin of these may be due to decay to atomic Rydberg P states. Concretely, during Rydberg gates, roughly 0.07% of the error budget was decay of Rydberg atoms to adjacent Rydberg P states. These states had a strong, long-ranged interaction with the Rydberg 5 states that were used for the gates, and thereby affected gates occurring in a different site, and moreover had lifetimes of over 100 ps. During repeated benchmarking sequences, like the ones reported in Ref.

[0036] , there were only 4 ps between gates, and consequently Rydberg P atoms survived for many layers of gates and corrupted gates in distant sites.

[0221] In FIG. 17G, the CZ gate fidelity was plotted in a repeated benchmarking sequence as a function of the duration between the gates, and it was found that the gate fidelity in this array increased from 99.3% to 99.5% by increasing duration between gates to 100 ps, as the Rydberg atoms decayed or ejected during that time. This also implied that the reported gate fidelity in Ref.

[0036] may have been impacted by this effect and thereby underestimated the maximum gate fidelity. Analogously, it was observed that this gate fidelity reduction was removed by reducing atom density. In quantum circuits based on atom motion, the duration between gates was sufficiently long (e.g., 400 ps for surface code repeated stabilizer- 57 -#14172487v2Attorney Docket No. : H0776.70181 WOOO measurements) for these Rydberg P states to decay or eject

[0082] , which natively fixed this issue, and consequently such effects were not observed during quantum circuits.

[0222] Surface code measurements. In the repeated quantum error correction on the surface code, unexpected error correlations were investigated by plotting the distribution of detector errors in a shot. It was found that these are closely consistent with the distribution as seen by Clifford simulations that assume uncorrelated one- and two-qubit errors. Although this data was only composed of approximately 105detector rounds (14855 shots, 96 detectors per shot across 5 rounds), it was nevertheless indicative of the absence of such events.

[0223] Logical teleportations and deep-circuit measurements. In a physical system, diverse physical errors and imperfections may cause complex correlations. For instance, a leakage event can lead to complex correlations that - without its knowledge - can greatly affect QEC performance. As discussed in the next section, incorporating logical teleportations in an architecture ensured such errors were removed. FIGs. 13A-13I verified that such teleportations rapidly removed errors and ensured that errors were not correlated in either time or space. FIGs. 21C-21D plot the correlations when cooling and sorting were turned off and it was found that correlations in such a case did not rapidly decay.Loss detection for improved QEC

[0224] Leakage types with neutral atoms. Leakage errors, which took the qubit out of the two-level computational subspace, are typically accounted for for in error correction. The three dominant leakage errors with neutral Rubidium (or other alkali) atoms were:• Loss events. Loss events were when the atom was physically lost from the optical trap. Due to the blockade nature of the gate, doing a gate with a lost atom simply turned off the gate while still applying gate error (it is identical to the atom being in state 0 which is also dark to the Rydberg laser).• Leakage to other hyperfine states in the ground-state manifold. In the limit of a large magnetic field, these states were off-resonant and behaved the same as a lost atom (turning off CZ gates). However, they were not detected through loss detection. Moreover, in the practical operating conditions of 8.6 GHz, the level spacings of 6 MHz (compared to Rabi frequency of 4.6 MHz) showed that adjacent hyperfine states could still off-resonantly couple to the Rydberg state and lead to repeated errors.- 58 -#14172487v2Attorney Docket No. : H0776.70181 WO00• Leakage to Rydberg states. Population left in the Rydberg manifold affected subsequent gates, and led to large error correlations. For example, many-body Rydberg evolution in dense systems observed so-called avalanche errors where a macroscopic fraction of the system had an error

[0083] . In this approach with a low atomic density and several hundred microseconds between gates, the Rydberg atoms either decayed to the ground state or were expelled from the tweezer. In this way, such Rydberg leakage either converted into an error within the computational subspace, a leakage into adjacent hyperfine states, or a loss event.

[0225] It was observed that with several hundred microseconds between gates, the effects of Rydberg leakage were not apparent, and during the repeated QEC data, leakage was at least 80% loss (FIG. 16B).

[0226] Effect of loss during repeated QEC. Although losses simply turned off subsequent gates, these led to distinct signatures to account for in the QEC design. Ancilla loss was detected in the projective measurement, and the loss of a data qubit corresponded to unknown loss of a degree of freedom from the system [40, 84]. Without adjusting the stabilizer measurement pattern to account for such a loss, the ancilla qubits functioned as measuring operators which anti-commuted with each other and thereby led to a ‘flickering’ pattern around the lost data atom. This flickering pattern was akin to the behavior in a subsystem code

[0085] and revealed that the flickering may continue for arbitrarily long times. Without accounting for the loss, this then appeared as strong time correlations which were observed in FIG. 16B.

[0227] Erasure information and superchecks. It was useful to detect atom loss for two reasons. First, knowing about the lost atom greatly enhanced the decoding performance. While bit-flip and phase-flip errors were inferred by stabilizers, direct detection of qubit errors - or so-called erasures - showed that one already had direct information about where the errors were. Such erasure information thereby greatly improved decoding performance [8, 9, 20, 31, 86, 87]. For example, while only (d-l) / 2 Pauli-type errors were corrected, up to (d-1) erasure-type errors could be corrected. Erasures were not detected as soon as they occurred, instead they were detected at the final qubit measurement, constituting delayed- erasure information.

[0228] Second, although lost atoms led to anti-commuting stabilizer measurements and a flickering error pattern, these were accounted for with the use of so-called superchecks, - 59 -#14172487v2Attorney Docket No. : H0776.70181 WO00 illustrated in FIG. 16A [40, 84]. While individual stabilizer checks around a lost atom were anti-commuting, taking products of multiple checks created superchecks which again commute with each other. FIG. 9B shows that such superchecks were able to remove the sharp rise in detected error that occurred with increasing data loss.Decoding

[0229] MLE and error-model tuning. To decode the surface code experiments in FIGs. 4A- 4B, 9A-9G, and 10A-10D, the delayed-erasure MLE decoder described in Ref. [9] was used, augmenting the MLE decoder developed in Ref.

[0041] to leverage loss information. In particular, the MLE decoder took as input the stabilizer measurements and the probabilities of the physical error sources in the circuit, and output the most likely combination of errors consistent with the syndrome. The circuit error model was constructed using Stim

[0088] to initially contain information about the Pauli error sources in the circuit, then it was updated for each shot to reflect the detected atom losses. In particular, after an atom was lost, all subsequent gates were canceled, generating different potential errors depending on when the loss occurred. Therefore, all potential locations a qubit loss could have originated from were considered (e.g., initialization, gates, movement, or idling prior to measurement), then each of the resulting error patterns and their probabilities were added to the error model for that shot. Errors producing the same syndrome were combined into a composite error mechanism and their probabilities were correspondingly reweighted, as in Ref.

[0088] . This process explicitly accounted for both propagated Pauli errors from the gate cancellations and the invalidation of stabilizers, which were handled using superchecks.

[0230] To optimize the performance of the MLE decoder, the probabilities of different error sources in the circuit error model were fine-tuned. In particular, each physical operation was associated with both a Pauli and loss error rate. The error probabilities in these channels were then treated as variables which were optimized using the covariance matrix adaptation evolution strategy

[0089] in order to minimize the logical error rate on a dataset of approximately 10000 shots (different shots from the final dataset were used for evaluating the fidelity).

[0231] To quantify the benefit of using loss information, in FIGs. 9A-9G the ‘bare MLE’ decoder did not update the circuit error model based on the loss information, and assigned each loss event to a 0 measurement. It was found that the loss information improved the measured d = 3 / d = 5 error ratio from 1.24(5) to 1.69(8).- 60 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0232] Finally, the confidence of the MLE correction was quantified for each shot by comparing the probability of the most likely error pQand the probability of the most likely error which provided the correction to the logical Pauli observable pi [90-93]. The more similar these two error probabilities were, the less confident the decoder was in its correction. Therefore, postselection on increasing pQ / (p0+ px) improved the accuracy of the results, which were use in FIG. 10D when repeated logical gates were investigated.

[0233] ML decoder - surface code. A fully connected neural network was employed to decode measurement outcomes from the surface code experiment using machine learning (ML) shown in FIGs. 9A-9G [32, 94- 96]. The decoding task was formulated as a supervised binary classification problem: the input features were measurement outcomes from the experiment, and the output was a label indicating whether the initial state was |0L) or |1L). The ML architecture was a fully connected feedforward network comprising four linear layers, each followed by batch normalization and a Gaussian Error Linear Unit (GELU) activation, as illustrated below: decoder = nn . Sequential ( nn . Linear ( input_size , 1024 ) , nn . BatchNormld ( 1024 ) , nn ,GELU( ) , nn . Linear ( 1024 , 512 ) , nn . BatchNormld ( 512 ) , nn ,GELU( ) , nn . Linear ( 512 , 256 ) , nn . BatchNormld ( 256 ) , nn ,GELU( ) , nn . Linear ( 256 , 1 ) , nn . Sigmoid ())

[0234] Training proceeded in three stages: raw training, ensembling, and fine-tuning.

[0235] Raw training — The decoder was trained on simulated data generated through circuit-level simulations that incorporated both Pauli and loss errors. Measurement outcomes showed values of 0, 1, or 2, corresponding to the qubit being in the |0) state, the - 61 -#14172487v2Attorney Docket No. : H0776.70181 WO0011) state, or being lost, respectively. These were one -hot encoded, so the feature vector of a given shot was 3 x (# of measurements). To create balanced training data, random software flips were applied with probability 1 / 2 along the relevant logical operator, yielding ensembles of |0L) , | 1L) for the Z memory and |+L) , |—L) for the X memory. In addition to the raw {0, 1, 2} measurement values, the neural network was provided with calculated detector outcomes and logical operator values. These additional features helped the model learn from structured correlations in the data. Detector values were computed as binary parities (0 or 1) over specified stabilizer regions; if a measurement showed a loss (2), it was assigned a value of 0 when computing detector parities. Logical operator values were calculated along each row or column, depending on the basis. A hidden layer size of 1024, the Adam optimizer with an initial learning rate of 10-3, and a weight decay of 10-2were used. The learning rate was decreased by a factor of 0.3 if the validation loss did not improve for 10 epochs. Training was performed independently for 10 total experimental configurations: two with code distance d = 5 (in the Z and X bases) and eight with d = 3 (covering four spatial quadrants in both bases). In the pre-training phase, each model was trained on 200 million simulated shots and validated on 20 million simulated shots. It was observed that decoder performance was largely robust to small perturbations in the error model, and thus precise tuning of simulation parameters was not necessary. For a batch size of ~ 104shots, the inference time per shot was 0.33 ps on a GPU (NVIDIA-A100).

[0236] Ensembling — To account for training variability and enhance robustness, the full training procedure as repeated with 10 different random seeds, resulting in 10 independently trained models per experiment. These were ensembled together by computing the geometric mean of their output probabilities. The resulting ensembled ML decoder achieved a logical error per round (LEPR) of 0.78(4)% for d = 5 and 1.37(3)% for d = 3.

[0237] Fine-tuning — To improve decoding performance, each pre-trained decoder was fine-tuned on experimental data taken from designated training sets (independent of the final dataset). The d = 5 decoders were fine-tuned on approximately 37000 shots per basis. For the d = 3 decoders, approximately 2500 shots per basis, per quadrant was used. The neural network architecture remained unchanged, and fine-tuning was performed using the Adam optimizer with a learning rate of 10-3and a weight decay of 8 x 10-2. The resulting ensemble of fine-tuned ML decoder achieved a LEPR of 0.71(4)% for d = 5 and 1.33(4)% for d = 3.- 62 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0238] Hybrid — When comparing the MLE and ML decoders, it was found that they did not predict the same logical state on all shots and, in particular, differed on shots where one of the decoders had low confidence in its prediction. To further enhance performance, a hybrid decoder was constructed that combined the output confidences of the ensembled ML decoder with those from the delayed-erasure MLE decoder, where the MLE confidence was derived from comparing the probabilities of the most-likely error and the most-likely error which provided the opposite logical outcome. The final prediction was given by the weighted geometric mean of the two confidence values with weights of 0.4 and 1 for the MLE and ML, respectively. This resulted in a final value for the reported LEPR of 0.62(3)% for d = 5 and 1.33(4)% for d = 3, which corresponds to the ML with loss decoder reported in FIGs. 9A-9G.

[0239] MLE decoder - lattice surgery. In the lattice surgery experiment (FIGs. 10C-10D), a joint ZZ measurement was performed using additional stabilizer checks along the common vertical edge between the two surface codes (“seam”). Both codes were prepared in |+L) and performed two rounds of stabilizer checks on the effective d ld surface code lattice, measuring both the codes’ stabilizer checks and the new seam checks in each round. To measure the ZZ parity of the resulting logical Bell state, with the delayed-erasure MLE decoder [9, 41], two decoding procedures were used. Eirst, only the ancilla measurements were used to obtain the result of the lattice surgery Z Zf measurement given by the product of the seam Z stabilizers. Second, Z Zf was obtained directly from the data qubit measurements, using the prior ancilla measurements in decoding. This measured the ZZ parity of the logical Bell state was obtained using lattice surgery. The seam checks from the final data qubit measurement were not included. A shot is counted as an error, Z^Z ; = — 1, if these two decoding procedures disagreed.

[0240] To obtain the XX Bell state parity, all data qubits in the X basis were measured and the joint X X operator was decoded spanning both codes. The final logical error probability was given by the mean of the XX and ZZ parities.

[0241] ML decoder - deep circuits. To decode the ID and 2D cluster states of logical [[7,1,3]] and [[16,6,4]] codes in FIGs. 13A-13I, a convolutional neural network (CNN) was employed. Since error correlations in the cluster state did not propagate beyond two CZ gates (see FIG. 13E), a convolutional window of size 3 was sufficient to capture the relevant correlations. The decoder architecture comprised three components: an encoder, a- 63 -#14172487v2Attorney Docket No. : H0776.70181 WOOO convolutional block, and a readout module. Both the encoder and readout were constructed from linear layers interleaved with GELU activations, with hidden_size = 128. encode = nn . Sequential ( nn . Linear ( input_size , 1024 ) , nn ,GELU( ) , nn . Linear ( 1024 , 512 ) , nn ,GELU( ) , nn . Linear ( 512 , 256 ) , nn ,GELU( ) , nn . Linear ( 256 , hiddensize )) readout = nn . Sequential ( nn . Linear ( hidden_size * 8 , 512 ) , nn ,GELU( ) , nn . Linear ( 512 , 256 ) , nn ,GELU( ) , nn . Linear ( 256 , 128 ) , nn ,GELU( ) , nn . Linear ( 128 , out_size ))The convolutional block applied between the encoder and readout modules was defined as: conv = nn . Sequential ( nn . Conv2d ( hidden_size, hidden_size * 2 , kemel_size =3 , padding= ’ same ’ ) , nn ,GELU( ) , nn . BatchNorm2d ( hidden_size * 2 ) , nn . Conv2d ( hidden_size *2, hidden_size *4 , kemel_size =3 , padding= ’ same ’ ) , nn ,GELU( ) , nn . BatchNorm2d ( hidden_size *4 ) , nn . Conv2d ( hidden_size *4, hidden_size* 8 , kemel_size =3 , padding= ’ same ’ ) ,- 64 -#14172487v2Attorney Docket No. : H0776.70181 WO00 nn ,GELU( ) , nn . BatchNorm2d ( hidden_size *8 ) ,)

[0242] For ID cluster state decoders, the 2D convolutions were replaced with ID convolutions and the batch normalization layers were omitted.

[0243] Training was performed using circuit- level simulations. The decoder was tasked with inferring the signs of the logical cluster state stabilizers, which were of the form X on a given qubit and Z on its neighbors. By performing measurements in alternating X and Z bases, half of the stabilizers were reconstructed. The remaining stabilizers were recovered by repeating the experiment with the measurement bases swapped. The decoder predicts the stabilizer signs by inferring the initial state of the qubits measured in the X basis. All logical qubits were initialized in the |+L) state, and software logical Z flips were applied with probability 1 / 2 to those measured in the X basis, to generate a balanced training set.

[0244] The decoder input included the raw measurement outcomes (0, 1, or loss), detector values computed from the measurements, and the raw logical operator values, similar to the input format used in the surface code decoder. The four distinct decoders were trained: one for each combination of code type ([[7,1,3]], [[16,6,4]]) and cluster state geometry (ID, 2D). Each model was trained on over 100 million simulated shots.

[0245] For further details on this decoder architecture, and on ML-based decoders for general quantum algorithms.Benchmarking surface code performance

[0246] NZNZ stabilizer gate pattern. The effective distance is described herein, defined as the minimum number of physical errors required to create a logical error, of d rounds of repeated syndrome extraction using alternating “N” or “Z” movement patterns (FIG. 17D). By alternating gate orderings, the effective distance was close to the optimal value. To see this, without alternating orderings, the effective code distance in the rotated surface code was reduced by a factor of two due to hook errors [6] . A hook error is a physical error on the ancilla qubit halfway through the stabilizer measurement cycle that propagated onto two data qubits oriented parallel to the corresponding logical operator (e.g., X 'or a physical X error). One of these propagated data qubit errors was immediately detected, while the other was detected in the following round by the next-nearest stabilizer along the direction of error- 65 -#14172487v2Attorney Docket No. : H0776.70181 WO00 propagation. As a result, if the same gate ordering was used for each round of stabilizer measurements, a sequence of [~~] hook errors, one occurring in each round along the direction of error propagation, generated a logical error upon correction. This issue was circumvented by alternating gate orderings between rounds, as only every other round hadthe unfavorable propagation. In this case, physical errors on consecutive rounds wereneeded to generate a logical error.

[0247] In these experiments in particular, an ordering of N Z ZrNrwas selected, where r represented performing the reverse ordering (see FIG. 17D). In addition to such structuring helping preserve fault- tolerance against hook errors, the dominance of Z-type errors showed that most errors did not lead to propagated errors between the middle two CZ gate layers.

[0248] Simulations. Simulations using the Stim simulation package

[0088] were performed. Both Pauli errors and qubit losses were sampled. Pauli errors were generated using Stim’s sampling routines, based on circuit-level noise models. Qubit losses were sampled according to the loss probabilities associated with each instruction, and when a loss occurred, subsequent gates acting on the lost qubit were removed to reflect the absence of the qubit. The simulations detected {0, 1, loss] during qubit readout, like the present example . For each set of physical parameters, the logical error rate was estimated via Monte Carlo sampling. Logical errors were declared when the decoder’s prediction for the logical observable differed from the true value.

[0249] Analysis of below -threshold performance for deep circuits. Four rounds of repeated QEC was performed as a benchmark in FIGs. 9A-9G. However, increasing the circuit depth affected the threshold in various ways, depending on the particular circuit. FIG. 19A shows how the LEPR ratio r changed for a single logical qubit as the number of QEC rounds was increased using an error model, showing a roughly 17% decrease in r from 4 rounds to 20. Similarly, FIG. 19B plots the same quantity for a single logical qubit with an approximate experimental error model, showing an analogous 9% decrease in r from 4 rounds to 50. Further, by interspersing 1 transversal gate every 1 QEC round under an approximate experimental error model, it was found that the ratio r changes by 2% at 25 QEC rounds. Similarly, prior work with an error model has shown that the threshold can change by ~ 10% with 1 gate per QEC round

[0041] . These simulations indicated that the benchmark studied in FIGs. 9A-9G was representative, but depending on context, may be different on the scale of ~ 15% for deep circuits. However, in transversal architectures, the prevalence of logical - 66 -#14172487v2Attorney Docket No. : H0776.70181 WO00 gate teleportations (e.g., in magic state distillation and angle synthesis) was such that there were typically only several stabilizer measurement rounds before transversal measurement.

[0250] These benchmark results were comparable to those in Ref. [7]. For instance, although a one-to-one comparison was not direct due to the presence of loss information, using the supercheck metric revealed a 9.04% mean detector error, comparable to the 8.5- 8.7% mean detector error in Ref. [7].

[0251] Error budget and path to lOx below threshold. To get to algorithmically relevant error rates of ~ IO-10[97, 98], a factor of 5-10x below threshold achieved the required errors with several hundred qubits in a code block

[0099] . This performance was captured by the error budget in FIG. 9F, which is described in further detail.

[0252] Listed below are illustrative single-qubit errors:• Local single-qubit gates have approximately 99.9% fidelity, arising from a 0.05% scattering error and residual miscalibrations.•The coherence time in 852-nm traps is approximately 1-2 seconds depending on the dynamical decoupling sequence applied.• A total loss from movement of roughly 1% on the ancilla atoms was experienced, arising from transfers and moves between and within the zones. The repeated QEC sequence also experienced 0.6% background loss from vacuum.• The lattice readout operated with a loss rate of 0.3% and a 99.5% bit-flip error rate.Processor clock speed

[0253] In the present work, clock speed was not optimized, and slower speeds were often selected for components so that they could function reliably without detailed characterization on existing infrastructure. Multiple measurements of these circuit durations is provided herein.

[0254] In the repeated surface code experiments in FIGs. 9A-9G, each QEC round was 4.45 ms. This originated from 0.47 ms time between gates, and a total of 2.57 ms from moving the ancilla atoms to the storage zone and bringing in the next group to the entangling zone.In the transversal CNOT experiments in FIGs. 10A-10D, the overall circuit duration (independent of number of CNOTs) was fixed at 17.7 ms, corresponding to the time of the longest circuit of 27 total transversal CNOTs. This corresponds to 0.655 ms per transversal- 67 -#14172487v2Attorney Docket No. : H0776.70181 WO00CNOT on average. In the deep circuit experiments in FIGs. 13A-13I, the cycle rate was bottlenecked through the use of desktop computers for all data processing for the mid-circuit image analysis and rearrangement, and times were not reduced. For this reason each logical teleportation layer was 41.9 ms.

[0255] In the repeated Rabi calibration in FIG. 12B, speed was optimized and achieved a cycle time of 4 ms. Although the imaging here was global as a demonstration of fast calibration, such speeds may also be achieved in a zoned manner.

[0256] In a planar architecture, such a logical teleportation step involved multiple logical gates and required multiple hundreds of QEC rounds for large-distance codes, e.g. 200-300 stabilizer measurement rounds. As such, it was estimated that the present methods were slower by a factor of ~ 10 - 20 relative to a conventional planar architecture with I ps speed per stabilizer measurement cycle [7, 97].Physical entropy removal

[0257] Types of entropy. QEC enabled removing entropy from the physical qubits, and this entropy took on many different forms. As discussed above, error correction such as stabilizer measurement served the role of converting generic quantum errors, such as coherent ones, into incoherent bit- and phase-flip errors. Detecting and tracking these errors further removed entropy from the system. Finally, physical systems such as atoms had entropy in other degrees of freedom such as loss, leakage, or atom heating.

[0258] Overview of entropy removal methods. The ancilla-based stabilizer extraction used in FIGs. 4A-4B, 8-10, was one form of entropy removal, in which stabilizer information was mapped onto the ancilla and then the ancilla was measured. Shor- style syndrome extraction operated via entangling ancillas into a GHZ state and extracting the stabilizer in a single step [1], Steane- style syndrome extraction operated via creating an ancilla logical qubit and extracting stabilizers via a transversal CNOT

[0100] , and measurement based quantum computing (MBQC)-style syndrome extraction operated via sequential entanglement with adjacent layers [19, 20, 101]. Leveraging teleportation native to the algorithm was another related method of entropy removal without ever ‘directly’ correcting the initial logical qubit block after it was used in computation. While these methods all varied in their specific implementations, their core mechanism of entropy removal was similar. These methods may be used interchangeably depending on specific practical considerations, such as those discussed in the next section.- 68 -#14172487v2Attorney Docket No. : H0776.70181 WO00

[0259] Use of teleportation for ensuring error removal. Logical teleportations were a method to ensure an architecture natively removed physical errors such as bit- and phaseflips, but also physical errors such as loss, leakage, and heating

[0019] . In particular, by teleporting a logical qubit from one block to another, the logical information propagated but the physical errors — both Pauli-type and other complex errors — were all left behind. This method ensured errors of all types were removed. Due to transversal gates leading to only 0(1) QEC rounds per logic gate, algorithms may be composed of a high density of logical gate teleportations. FIGs. 13A-13I shows that teleportation can perform logical operations while natively removing all such errors without additional overhead. An alternative method for removing physical errors was teleportation at the physical level - specifically, by swapping quantum information between a physical data qubit and a physical ancilla. This approach underlies various implementations of leakage reduction units [9, 20, 102-104]. However, it necessitated pairing each data qubit with a dedicated ancilla, which can present challenges. In general, the number of unpaired data qubits in each round of error correction is lower bounded by k = # data qubits - # independent checks, where k is the number of encoded qubits. For example, in hypergraph product codes constructed from (w, v)-biregular expanders - bipartite graphs in which checks have degree u and bits have degree v - the compact rearrangement scheme of Ref.

[0105] implies that there will be O((y - u)d) unpaired data qubits per error-correction cycle, where d is the distance of the code. In contrast, logical-level teleportation was directly accessible in all CSS codes (as they all had a transversal CNOT), as demonstrated in the high-rate [[16,6,4]] code in FIGs. 13A-13I. Such analysis highlights that leveraging the transversal teleportations native to an algorithm lends to a robust, low-overhead procedure that ensured all physical errors were removed independent of the specific code.

[0260] Feedforward in universal processing. Once bit-or phase-flip errors were detected, a natural question was if they needed to be physically corrected in-hardware in order to return back to a configuration with all stabilizers = +1. For conventional computation based on transversal (or planar) Clifford gates, stabilizer measurements, and universality achieved via teleportation of \TL) states (realized via physical Clifford gates), one does not have to apply such physical qubit corrections. This was most directly seen by the fact that universal computation on the logical-qubit level was realized via physical Clifford gates [18, 99], and- 69 -#14172487v2Attomey Docket No. : H0776.70181 WO00 so the Pauli corrections were deterministically tracked in-software as a Pauli frame update without additional overhead on the decoding.

[0261] When transversal non-Cliffords were realized, such as the transversal T gate in the [[15,1,3]] Reed-Muller code

[0106] , X Pauli corrections did not commute through, and so in such a case the stabilizers did need to be returned to a deterministic +1 eigenvalue. However, in the results here, for example, the deterministic initialization of the Reed-Muller code was realized with +1 eigenvalues as a method of ensuring constant entropy operation, and in such a case even here mid-circuit correction of individual physical qubits was not required.

[0262] In both of these settings, feedforward was indeed required, but only on the logical- qubit level (feedforward .S' for T teleportation, and feedforward X for H teleportation). This logical feedforward was implemented in FIGs. 11A-11D of Ref.

[0011] where feedforward logical .S' gates were realized to entangle two qubits that did not directly interact.

[0263] Transversal logic with 0(1) stabilizer measurements per gate. Since in the transversal setting, the role of stabilizer measurements was simply to remove entropy, the conventional d rounds of stabilizer measurement per logic gate was not required, as shown in FIGs. 10A-10D. These techniques directly applied for universal computation. Concretely, universal computation was implemented by a transversal Clifford circuit, where T gates were realized via a transversal teleportation circuit with \TL) inputs that had already been prepared fault- tolerantly. It was shown that this universal processing could proceed with 0(1) stabilizer measurements per transversal gate, and that the decoding was done efficiently with a decoding complexity that could be in fact even less than the conventional lattice surgery setting [12, 41, 44, 107-109]. As such, these experimental results directly applied to universal computation, under the assumption that the \TL) inputs were prepared to high quality.Methods of universality

[0264] Universality, transversal gates, and Eastin-Knill. Universality means that any unitary can be closely approximated by using sequences of gates from a universal gate set

[0024] . An example universal gate set is {H, T, CNOT} . 2D topological codes can have a discrete gate set of {H, S, CNOT}, but cannot transversally implement the T gate. 3D topological codes can in fact have a transversal T gate

[0110] , and the [[15,1,3]] 3D color code in particular has a transversal gate set of {CZ, CCZ, CNOT, T}. The Eastin-Knill- 70 -#14172487v2Attorney Docket No. : H0776.70181 WOOO theorem forbids having a unitary transversal gate set which is universal

[0045] . This would, for example, allow realizing a transversal logical 9 rotation by a sequence of transversal operations on the underlying physical qubits, and thereby could not be protected as it would be sensitive to small imperfections in the physical rotation.

[0265] The Eastin-Knill theorem was circumvented simply by the introduction of logical measurement, which broke unitarity and enables universality. This was directly achieved with 3D codes, as realizing a CZ gate between state |I]JL) and |+L), followed by logical measurement and feedforward, teleported a Hadamard gate directly onto |I[FL). As such, X- basis preparation and X-basis measurements (guaranteed in all CSS codes), combined with transversal CZ gates, were used to straightforwardly implement a universal gate set of {H, T, CNOT} using fully transversal operations. This was the basis behind the implementation of universality in FIGs. 11A-11D. In many protocols, universality is directly generated via the measurement of a 3D code. Code switching is one example, where one switches between codes that have T and H transversal gates

[0111] . For example, a code switching protocol was realized in FIG. 20E, where a logical T from a 3D [[15,1,3]] color code was teleported onto a 2D [[7,1,3]] color code. Such operations between codes of different dimensionality can often be realized, and here just involved entangling the 2D surface of the 3D pyramid with the 2D color code face. While teleportation onto the 2D color code admits transversal H gates, this was anyway already accomplished via the transversal measurement and feedforward from the 3D code.

[0266] Connection to magic state distillation. The protocols described in this example are similar to those underlying magic state distillation [106, 112, 113]. In the conventional 15- to-1 magic state distillation, 16 surface code logical qubits are entangled in a manner where the first surface code qubit is entangled with the logical qubit of a [[15,1,3]] code made out of surface codes. Subsequently, noisy T gates with some error p are realized on the surface codes via teleportation, which the outer [[15,1,3]] code distills into ~ p2with correction or ~ p3with postselection

[0114] . By measuring the Reed-Muller code, the resulting distilled \T) state was teleported onto the first surface code.

[0267] The protocol shown in FIGs. 11A-11D was a more compact representation of the same magic state distillation circuit, but with replacing the inner surface codes with unencoded physical qubits. While in conventional distillation the \T) is teleported onto the surface code, for example, in small-angle synthesis with sequential HT HT... gates, one can- 71 -#14172487v2Attorney Docket No. : H0776.70181 WO00 leave the \T) encoded in the Reed-Muller code and then realize a transversal CZ gate between the two concatenated Reed-Muller codes.Physical resources for QEC

[0268] This work investigated the relationships of many different physical resources, and how they were used in quantum error correction. An overview of these observations is provided herein.

[0269] Logical entanglement and. physical entanglement. In a transversal gate setting, logical entanglement was generated using only physical entanglement between the code blocks. This was in contrast to lattice surgery, where entanglement within the blocks was necessary to mediate interaction between non-overlapping logical operators, and a robust entanglement was required. This was the origin of the sensitivity to measurement errors in the lattice surgery context, and the insensitivity to measurement errors in the transversal gate context. Logical entanglement within the code block also played arole. The [[16,6,4]] codes, for example, contained many logical qubits within the block, which were entangled, but only with a sufficient degree of physical entanglement present (discussion below).

[0270] Motivated by these observations, one way to re-frame efficient encodings was to find methods that generated the target logical entanglement with the minimum amount of physical entanglement. To this end, it was proved that the amount of logical entanglement - even generated with techniques such as permutation gates - could not exceed the amount of physical entanglement.

[0271] Operator entanglement quantified the maximum entanglement a gate could produce on separable inputs. For any gate acting on k qubits, the operator entanglement was bounded above by [k / 2] . Thus, for a quantum code with parameters [[n, k, d]], the logical operator entanglement satisfies SL0< [k / 2] . To lower bound the physical entanglement entropy, the logical operators could not be supported on any set of d - 1 or fewer physical qubits. Therefore, for any region A with |A| < d — 1, all logical codewords yielded identical reduced density matrices on A (not necessarily maximally mixed). For the maximally mixed logical state pL, the reduced density matrix on A has rank 2min^k'd -1implying Sps> min(L d -1). If k < d - 1, then Sps> k, so Sps> SL0always holds. If k > d - 1, then Sps> d -1. Here, SL0< Spsas long as d - 1 > [k / 2], i.e., k < 2d.- 72 -#14172487v2Attorney Docket No. : H0776.70181 WOOO

[0272] Thus, for any [[n, k, d\ \ code with k < Id, the logical operator entanglement could not exceed the physical entanglement available in any region of size d - 1.

[0273] These observations had applicability to finding efficient algorithm compilations with high-rate codes and transversal operations, both of which were observed here could reduce the amount of physical entanglement to realize the target logical entanglement. For instance, each transversal CNOT in the [[16,6,4]] code generated 16 physical CNOTs worth of entanglement and 6 logical CNOTs worth of entanglement, but realizing in-block permutation CNOTs could not generate an additional 4 x 2 logical CNOTs, totaling 14 logical CNOTs worth of entanglement, close to the bound of 16 physical CNOTs.

[0274] Physical entanglement and logical magic. While physical entanglement was the underpinning of logical entanglement, it was also the underpinning of logical magic. In particular, it was found that states with logical magic required more in-block entanglement than states without any logical magic. This could be understood by the fact that, while logical Pauli states such as | +L) were represented by operators XL = X1X2X3... (in CSS codes), which is atensorproduct of physical operators, states such as \TL) were represented by (X 1X2X3... + Y1Y2Y3...), involving a macroscopic superposition of operators spanning the code that was necessarily entangled [115, 116]. Analogously, any physical product state that had deterministic X-type stabilizers had zero expectation value for YL. The need for well-defined stabilizers in both bases is thereby another way to see that the code must be entangled. Similarly, CSS codes were constructed from two classical codes [1, 2, 23, 117], and Pauli states were ‘classical’ in that they store 1 bit of information in one of the two classical codes (and 0 bits in the other), whereas \TL) states truly required both codes.

[0275] These observations suggested a potentially more fundamental mechanism of what algorithmic outputs do and don’t need full protection. For example, consider making a remote entangled Bell pair. To probe its fidelity with XLXL and ZLZL entanglement witnesses, then with correlated decoding methods one does not need a high degree of entanglement within the individual code blocks - just between the blocks. However, if one would instead like to perform an error-corrected Bell inequality test

[0047] , to provide evidence that quantum mechanics is real, then now one has to measure in the \T) basis and requires the full entanglement within the block. It has been argued that so-called quantum contextuality, which arises from measurements in non-Pauli bases, is the core aspect of quantum mechanics that cannot be described by classical theories

[0118] . Relatedly, theoretical work- 73 -#14172487v2Attorney Docket No. : H0776.70181 WO00 has shown a connection between contextuality and computational hardness

[0052] , and in this work, it was found that both of these were also linked to the minimum amount of entanglement required to perform the requisite error correction. An experimental error- corrected Bell inequality test is shown in FIG. 20F.

[0276] Logical gate fidelity and physical entropy. With physical qubits, which are two-level systems, fidelity is a descriptive and accurate concept as noise can often be decomposed into either realizing the correct operation or the exact opposite (e.g., a bit-flip error). Conversely, logical qubits are many-level systems, and so this property does not hold. This fact is related to the observation in FIG. 10D that the error per logical operation was not constant as a function of the number of applied logical gates. Instead, there was a logical fidelity associated with the probability of decoding correctly

[0099] , which depended on the internal density of errors p.

[0277] The per-step logical error from decoding may scale approximately as PLoc (.P / Pth)d + 1^2

[0099] . The results shown in FIG. 10D clearly indicated that quantifying logical gate performance should encapsulate {FLpdet')lLpdet], which captured how the logical fidelity FLdepended on the internal density of errors p, as well as the gate’s increase to local error density Ap. This was investigated quantitatively in FIGs. 18A-18E.Additional experiment and data analysis details

[0278] Figures 4A-4B and8A-8D. In FIGs. 8C-8D, eitherwere prepared and up to five rounds of stabilizer measurement were applied followed by measurement in the X or Z basis, respectively. A global Z(0) rotation was applied to the data qubits at every gate layer (20 time-steps in total). For fewer than five stabilizer measurement rounds, the relevant CZ gates were removed but single qubit rotations still applied. 1 QEC round had gates in the first round only, 2 QEC rounds had gates in the first and fourth rounds, and 5 QEC rounds had gates in all five rounds. The data was averaged over both initial states and used MLE decoding with a 50% acceptance fraction for visual clarity, as well as preselection on perfect initial qubit filling. The right plot shows an injected error of 0 / 2n = 0.016 and additional error rates are shown in FIG. 18B.

[0279] Figures 9A-9G. No postselection was used in the analysis of the surface code. Preselection of initial qubit rearrangement (standard in the literature) was used. Data in FIGs. 9B-9D was averaged over |+L) and |0L) , and the distributions in FIGs. 9E and 9G aggregated the two bases. FIGs. 9B-9C plot the detector error probability averaged over all - 74 -#14172487v2Attorney Docket No. : H0776.70181 WO00 rounds. FIG. 9B uses the same data set with shots binned according to data qubit loss. The four metric were (i) ‘bare’, where loss was converted to qubit state 0 (ii) ‘detect loss’, where projective measurements whose value was ‘loss’ were not counted erroneously (iii) ‘supercheck’, where stabilizers with a lost data qubit were formed into superchecks for all prior rounds, and (iv) ‘postselected’, where detectors involving any lost atoms were ignored. The plots show the mean error of all deterministic detectors (96 per basis). The supercheck error was calculated over all samples per round per basis, and the mean of these 8 values plotted. Superchecks paired to the boundary were removed from the averaging as these returned no error by construction; if included, the mean error decreased from 9.0% to 8.8%. The contribution of each supercheck was normalised by the supercheck weight, e.g., a weight-6 supercheck contributed an error of 4 / 6 (to account for the greater amount of information in the check - e.g., multiplying checks even in absence of loss raised the detector error without reducing the amount of information). Without reweighting, the error probability increased to 9.6%.

[0280] In FIGs. 9D-9E, the logical error per round was calculated as LEPR where pLwas the final logical error after r = 4 rounds, same asthe definition in Ref. [7].

[0281] The d = 5 dataset contained 9021 shots in the X basis and 5834 in the Z basis. The d = 3 dataset contained 2523 shots for X and 2534 for Z (on average per quadrant). To make a d = 3 surface code in each of four possible quadrants, atoms were removed and the circuit was not modified. The specific circuit for the repeated stabilizer measurement is shown in FIG. 17C. Not shown are local Y(K) and Y(K / 2) gates on the boundary ancillas. Additionally, local detunings were applied to the lowest row of gate sites (where there were only isolated ancilla qubits) to mitigate inhomogeneity in the 1013-nm lightshift during entangling gates.

[0282] In FIG. 9F, the error budget shows the contributions to the detector error (with loss detection) and was obtained by removing error sources individually from the simulation error model. A similar error model breakdown was obtained by simulating the relative contribution to the logical error. FIG. 9G plots the detector distribution with loss detection. FIG. 17F compares the bare detector error to simulation.- 75 -#14172487v2Attorney Docket No. : H0776.70181 WOOOSee Ref.

[0042] for all raw experimental shots, the analysis notebook, and trained machine learning decoders.

[0283] Figures 10A-10D. All data used MLE decoding and was pre-selected on perfect initial qubit filling. In FIGs. 10C-10D, logical Bell states were prepared using either transversal gates or lattice surgery, and the mean error was measured in the resulting XX and ZZ parities. The error per logical operation was defined as £ = 0.5 1 — (1 —for N transversal gates per round and Bell state infidelity pL, and a = pL, for the lattice surgery logical product measurement. In FIG. 10C, the transversal CNOT is shown for 3 CNOTs per QEC round, and the injected measurement error was applied in-software with probability p independently to every ancilla qubit. An acceptance fraction of 1 was used for the transversal CNOT plots unless otherwise stated. In FIG. 10D, the lattice surgery point used error detection on the middle three ancillas each having the same value in both rounds of stabilizer measurement, to compensate for having fewer than d rounds of repeated syndrome measurement for this result. It was found in numerical simulations using the experimental error model that the optimal number of QEC rounds for this circuit was approximately 3 (as opposed to 5), and that by using error detection with 2 rounds a similar performance was recovered to this optimal value found in numerics.

[0284] Figures 11A-11D. To modify the stabilizer signs in FIG. 11A, local 7t pulses were applied at the end of the encoding circuit. “Negative” stabilizers corresponds to flipped qubits on the four comers of the Reed-Muller tetrahedron. A lookup table for decoding was used and all three 3D color code curves were plotted with an acceptance fraction of 46% and the 2D color code with 74%, corresponding to a rescaling by the number of physical qubits in the code. For postselection, the shots were ordered by the weight of the detected error. The curves were further normalized to highlight key features, with unnormalized data shown in FIG. 20A. FIG. 11C shows error detection and plots the angles for < N T gates. All plots were postselected on no loss and perfect initial qubit filling.

[0285] Figures 12A-12D. In FIG. 12C, the atom temperature and loss as a function of cycle was investigated using the circuit for state preparation of Steane codes (FIG. 13B) with only entangling gates removed. In the fifth cycle, all imaging and cooling light was turned off. For comparison, the same measurements were repeated with conventional 3D PGC imaging and cooling in place of the local techniques. To extract the atom temperature shown in the- 76 -#14172487v2Attorney Docket No. : H0776.70181 WOOO upper panel, a drop-recapture measurement was used after N cycles and fit the resulting loss to a Monte-Carlo simulation

[0076] . Shaded regions indicate the range of fitted temperatures due to uncertainty in trap parameters.

[0286] Figures 13A-13I. The [[7,1,3]] and [[16,6,4]] codes in FIGs. 13A-13I, as well as the [[15,1,3]] in FIGs. 11A-11D, were members of the family of quantum Reed-Muller codes based on the hypercube encoding circuit illustrated in FIG. 21 A

[0120] . For each code, a different pattern of local Y(7t / 2) pulses was applied while the entangling gate structure was the same; for the 2D [[7,1,3]] code, the fourth layer of gates was turned off. In FIG. 13B, groups of sixteen independent [[7,1,3]] codes were prepared in parallel in each time layer, repeated for 27 layers. The stabilizer error probability as a function of layer was plotted for no loss in the code block.

[0287] To characterize the propagation of physical and logical information in deep circuits, the codes into ID and 2D cluster states were further entangled. Starting with two groups of logical qubits, group A and group B in FIG. 13A, these were entangled to form the first two time layers of the cluster state. Due to the local entanglement structure of a cluster state, group A underwent no further entangling gates and was idle up until its measurement (in the appropriate basis), and thereby the measurement was performed and the same physical qubits were re-used to form the third layer of the cluster state (in typical MBQC fashion). Group B was then measured and re-used to form the fourth layer of the cluster state, and so on. This alternating structure is typical in MBQC using cluster states

[0019] .

[0288] The physical correlations in FIGs. 13C, 13D, and 13G were calculated as the covariance between errors (stabilizer = -1) on the same stabilizer between codes at different coordinates in the cluster state. The covariance was then averaged across all co-propagating cluster states and the different stabilizers (three for [[7,1,3]] and five for [[16,6,4]]). The logical correlations in FIGs. 13C and 13G were calculated as the appropriate product of cluster state stabilizers between the two target coordinates (cluster states had stabilizers corresponding to XilTj Zj where z was a specific site and j was its neighbors). For example, this was defined as, ZQZ4) = (ZQ^I^) ’ (Z2X3Z4)). The single-qubit expectation values (Zi) were calculated using a lookup table decoder for [[7,1,3]] and raw values for [[16,6,4]]. Time layers were truncated where the reservoir began to be depleted. This corresponds to 16 layers for FIG. 13C, 13 layers for FIG. 13D (correlations plot only), and 12 layers for FIG. 13G. This had only a small effect on the measured logical correlations but otherwise- 77 -#14172487v2Attorney Docket No. : H0776.70181 WO00 led to a longer-tail of physical correlations because atoms were not properly refilled once the reservoir began depleting.

[0289] All logical operators in FIGs. 13A-13I were decoded with machine learning which directly predicted the cluster state stabilizers. The 2D cluster state stabilizers in FIG. 13D used a global acceptance fraction of 0.24%. In FIGs. 13C and 13G a global confidence threshold for each curve was used instead, which was then converted to a mean acceptance fraction. In this way, each curve corresponds to a constant effective error rate for the logical operator independent of its weight, resulting in a reduced acceptance fraction for higher weight operators. The confidence for products of the weight-3 logical stabilizers was given as the geometric mean of the constituent confidences. FIG. 13G used a mean acceptance fraction of 3.4% (same data for both curves). The 2D [[16,6,4]] cluster state in FIG. 131 also used the confidence-based postselection, where the confidence per cluster state stabilizer was the geometric mean of the six decoded co-propagating 2D cluster states. On top of this decoding postselection, the logical stabilizer expectation value was shown as a function of the minimum number of co-propagating operators, N, with the same measurement outcome. The mean of all combinations of choosing N out of 6 such operators was used.

[0290] The permutation CNOT in FIG. 13G was applied in software and its effect here was to increase the weight of the operator connecting coordinates ti and tj , labeled as an effective separation i -j . Following the definitions in Ref.

[0025] , two permutation CNOTs (swapping a pair of rows and a pair of columns) converted the cluster state stabilizers supported on logical qubits 3 to 6 from four weight-3 to one weight-3, two weight-6 and one weight- 12 operator.REFERENCES[1] Shor, P. W. Fault- tolerant quantum computation. In Annual Symposium on Foundations of Computer Science - Proceedings, 56-65 (IEEE, 1996).[2] Steane, A. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551-2577 (1996).[5] Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate (1999). arXiv: 9906129.[6] Dennis, E., Kitaev, A., Landahi, A. & Preskill, J. Topological quantum memory. Journal of Mathematical Physics 43, 4452-4505 (2002).- 78 -#14172487v2Attorney Docket No. : H0776.70181 WO00[7] Acharya, R. et al. Quantum error correction below the surface code threshold. Nature 638, 920-926 (2025).[8] Wu, Y., Kolkowitz, S., Puri, S. & Thompson, J. D. Era- sure conversion for fault- tolerant quantum computing in alkaline earth Rydberg atom arrays. Nature Communications 13, 1-7 (2022).[9] Baranes, G. et al. Leveraging Atom Loss Errors in Eault Tolerant Quantum Algorithms (2025). arXiv:2502.20558.

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Claims

Attorney Docket No. : H0776.70181 WOOOCLAIMSWhat is claimed is:

1. A quantum computer, comprising: a computation chamber, comprising a storage zone, an entangling zone, a readout zone, and a reservoir zone; and at least one controller configured to, during qubit preparation and manipulation operations of a quantum information processing device, cause: transfer, from the readout zone of a computation chamber to the entangling zone of the computation chamber, a first plurality of qubits, wherein qubits of the first plurality om prise neutral atom qubits; entangle the first plurality of qubits in a spatial dire Qi on; transfer, from the storage zone of the computation chamber to the entangling zone, a second plurality of qubits, wherein qubits of the second plurality comprise neutral atom qubits that have previously been entangled with the first plurality of qubits; entangle the first and se ond plurality of qubits in a time dire Qi on; transfer the first plurality of qubits into the storage zone and transfer the se ond plurality of qubits into the readout zone; measure qubits of the se ond plurality; and reinitialize states of the qubits of the second plurality.

2. The quantum computer of claim 1, wherein during operation of the quantum information processing device: the storage zone is configured to hold qubits for performance of quantum operations between qubits held in the storage zone, the entangling zone is configured to hold qubits for performance of entangling operations between qubits held in the entangling zone, the readout zone is configured to hold qubits to be measured and / or reinitialized, and- 86 -#14172487v2Attorney Docket No. : H0776.70181 WOOO the reservoir zone is configured to provide a supply of qubits to one or more of the storage zone, the entangling zone, and / or the readout zone.

3. The quantum computer of claim 1, wherein the quantum computer further comprises a spatial light modulator configured to load qubits into arrangements of qubit traps in at least one of the storage zone, the entangling zone, the readout zone, and / or the reservoir zone.

4. The quantum computer of claim 3, wherein the spatial light modulator is configured to load the qubits into arrangements of qubit traps equal to approximately 852-nm.

5. The quantum computer of claim 3, wherein the spatial light modulator is configured to maintain a position of the entangling zone such that the entangling zone is separated from the storage zone by approximately 40-pm and from the readout zone by approximately 40- pm.

6. The quantum computer of claim 3, wherein the quantum computer further comprises a first pair of crossed acousto-optical deflectors configured to move qubits between the arrangements of qubit traps.

7. The quantum computer of claim 6, wherein the first pair of crossed acousto-optical deflectors are configured to arrange qubits of the first plurality and / or the second plurality in blocks of [[7,1,3]] or [[16,6,4]] codes.

8. The quantum computer of claim 7, wherein the quantum computer further comprises a first laser configured to illuminate arrays of qubits in at least one of the storage zone, the entangling zone, the readout zone, and the reservoir zone with a global Raman beam.

9. The quantum computer of claim 8, further comprising a second pair of cross acousto- optical deflectors configured to form a local Raman beam to perform single-qubit rotations using the global Raman beam.- 87 -#14172487v2Attorney Docket No. : H0776.70181 WOOO10. The quantum computer of claim 8, wherein the global Raman beam, while illuminating qubit arrays during operation of the quantum computer, is further configured to cool qubits in the arrangements of qubit traps using Al lambda-enhanced gray molasses cooling.

11. The quantum computer of claim 8, wherein the quantum computer further comprises a second laser configured to generate a laser beam to illuminate qubits in the storage zone.

12. The quantum computer of claim 11, wherein the second laser is further configured to focus the laser beam to an elliptical waist, wherein a semi-minor axis of the elliptical waist is aligned in a vertical direction relative to a center position of the storage zone.

13. The quantum computer of claim 11, wherein the second laser is configured to generate the laser beam having a wavelength of approximately 1529-nm.

14. The quantum computer of claim 2, wherein the quantum computer further comprises a third and fourth laser configured to perform entangling gates between qubits disposed in the entangling zone using two-photon excitation.

15. The quantum computer of claim 14, wherein the third and fourth lasers are configured to emit 420-nm and 1013-nm Rydberg beams, respectively, configured to excite the qubits disposed in the entangling zone to n = 53 Rydberg states.

16. The quantum computer of claim 3, wherein the quantum computer further comprises a plurality of arbitrary waveform generators configured to control one or more lasers of the quantum computer.

17. The quantum computer of claim 16, wherein a first arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to generate entangling gate pulses to entangle qubits disposed in the entangling zone and / or to perform local detunings of the spatial light modulator.- 88 -#14172487v2Attorney Docket No. : H0776.70181 WOOO18. The quantum computer of claim 17, wherein the first arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform through-the-lens (TTL) metering.

19. The quantum computer of claim 18, wherein a second arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform in-phase and quadrature control of the qubits and / or pulse-shaping of a global and a local Raman driving.

20. The quantum computer of claim 19, wherein a third arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to perform real-time rearrangement of qubits.

21. The quantum computer of claim 20, wherein a fourth arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to control positions of the qubits during operation of the quantum computer.

22. The quantum computer of claim 21, wherein a fifth arbitrary waveform generator of the plurality of arbitrary waveform generators is configured to create light grids for local single-qubit control.

23. A method for performing a quantum information processing operation using a quantum computer, the method comprising: transferring, from a readout zone of a computation chamber of the quantum computer to an entangling zone of the computation chamber, a first plurality of qubits, wherein qubits of the first plurality comprise neutral atom qubits; entangling the first plurality of qubits in a spatial direction; transferring, from a storage zone of the computation chamber to the entangling zone, a second plurality of qubits, wherein qubits of the second plurality comprise neutral atom qubits that have previously been entangled with the first plurality of qubits; entangling the first and second plurality of qubits in a time direction; transferring the first plurality of qubits into the storage zone and transferring the second plurality of qubits into the readout zone;- 89 -#14172487v2Attorney Docket No. : H0776.70181 WOOO measuring the second plurality of qubits; and reinitializing states of the second plurality of qubits.

24. The method of claim 23, wherein measuring qubits of the second plurality comprises: illuminating the qubits with counterpropagating laser beams to form a onedimensional qubit lattice; converting qubits having state |1) to stretched dark states; converting qubits having state |0) to stretched bright states; moving the stretched dark states using acousto-optical deflector (AOD) tweezers while maintaining positions of the stretched bright states; and imaging the qubits.

25. The method of claim 24, wherein imaging the qubits of the second plurality comprises: performing a first cooling step by detuning the two counterpropagating beams emitted by a first laser; performing a second cooling step by blue-detuning the two counterpropagating beams emitted by the first laser; reducing a power of one of the two counterpropagating beams emitted by the first laser; and obtaining optical images of the qubits.

26. The method of claim 24, further comprising identifying a spin state and / or a position of the second plurality of qubits using at least one image obtained from imaging the qubits.

27. The method of claim 23, further comprising performing repetitive stabilizer measurements to blocks of qubits of the first plurality by: physically transporting blocks of the first plurality of qubits to interlace the blocks; and applying interleaved CNOT gates and stabilizer measurements to the interlaced blocks.- 90 -#14172487v2Attorney Docket No. : H0776.70181 WOOO28. The method of claim 27, further comprising arranging the blocks of the qubits of the first plurality in blocks of [[7,1,3]] or [[16,6,4]] codes.

29. The method of claim 23, wherein entangling the first plurality of qubits in the spatial direction comprises performing single-qubit operations between qubits of the first plurality of qubits.

30. The method of claim 23, wherein entangling the first and second plurality of qubits in the time direction comprises performing a transversal entangling gate between the first and second plurality of qubits.

31. The method of claim 23, wherein entangling the first and second plurality of qubits in the time direction comprises performing lattice surgery on both or either of the first and second plurality of qubits.

32. The method of claim 23, wherein reinitializing the states of the second plurality of qubits comprises optically pumping the second plurality of qubits.

33. The method of claim 32, wherein optically pumping coherent pulses comprises using a Raman-assisted optical pumping scheme to apply coherent 7t-pulses to the second plurality of qubits.

34. An apparatus for reading out quantum states of neutral atom qubits, comprising: a first and second laser configured to generate counter-propagating laser beams configured to form a one dimensional optical lattice of the neutral atom qubits by illuminating the neutral atom qubits; a third laser configured to convert first qubits of the neutral atom qubits having a quantum state |1) to a stretched dark state by optically pumping the first qubits; a fourth laser configured to convert second qubits of the neutral atom qubits having a quantum state |0) to a stretched bright state by causing coherent Raman transfer of the second qubits or by optically pumping the second qubits;- 91 -#14172487v2Attomey Docket No. : H0776.70181 WO00 acousto-optical deflector (AOD) tweezers configured to move the first qubits to spatially separate the first qubits from the second qubits; and a camera configured to image the first and second qubits.

35. The apparatus of claim 34, wherein the first and second laser comprises titanium: sapphire lasers configured to generate the counter-propagating laser beams with wavelength of 795-nm and a blue detuning between 50 GHz to 200 GHz from a DI line of the neutral atom qubits.

36. The apparatus of claim 35, wherein the counter-propagating laser beams are circularly polarized.

37. The apparatus of claim 34, wherein the third laser is configured to generate an optical pump resonant to an F = 2 to F’ = 3 transition of the neutral atom qubits.

38. The apparatus of claim 37, wherein the third laser is configured to generate the optical pump comprising light that is circularly polarized.

39. The apparatus of claim 34, wherein the fourth laser is configured to cause coherent Raman transfer to F = 2, mF = +2.

40. The apparatus of claim 34, wherein the fourth laser is configured to optically pump the second qubits to cause an F = 1 to F' = 2 transition of the neutral atom qubits.

41. The apparatus of claim 34, wherein the fourth laser is configured to optically pump the second qubits by generating circularly polarized light.

42. The apparatus of claim 34, wherein the AOD tweezers are configured to move the first qubits approximately 2 pm over a time period of approximately 500 ps.- 92 -#14172487v2Attorney Docket No. : H0776.70181 WOOO43. The apparatus of claim 34, wherein the third laser is configured to produce two counterpropagating beams that propagate in directions that are parallel to an applied external magnetic field.

44. The apparatus of claim 43, wherein the two counterpropagating beams are configured to be detuned by two times a Zeeman splitting of adjacent mF levels of the neutral atom qubits.

45. A method of reading out quantum states of neutral atom qubits, the method comprising: illuminating qubits, using a first and second laser, with counterpropagating laser beams to form a one-dimensional qubit lattice; converting qubits having state |1) to stretched dark states; converting qubits having state |0) to stretched bright states; moving the stretched dark states using AOD optical tweezers while maintaining positions of the stretched bright states; and imaging the qubits.

46. The method of claim 45, wherein maintaining positions of the stretched bright states comprises: ramping up the one-dimensional qubit lattice adiabatically, and pinning the stretched bright states in a lattice potential of the one-dimensional qubit lattice.

47. The method of claim 45, wherein converting the qubits having state 11 ) to stretched dark states comprises using incoherent pumping generated by a third laser and resonant to an F = 2 to F’ = 3 transition of the neutral atom qubits.

48. The method of claim 47, wherein the incoherent pumping is performed with a o'polarized 780 nm repumper.- 93 -#14172487v2Attorney Docket No. : H0776.70181 WOOO49. The method of claim 45, wherein converting the qubits having state |0) to stretched bright states comprises using incoherent pumping generated by a fourth laser resonant to an F = 1 to F’ = 2 transition of the neutral atom qubits.

50. The method of claim 49, wherein the incoherent pumping is performed with a G+- polarized 780 nm repumper.

51. The method of claim 45, wherein converting the qubits having state \F = 1; mp= 0) to stretched bright states comprises using a coherent Raman transfer.

52. The method of claim 45, wherein imaging the qubits comprises obtaining optical images of the qubits using at least one optical camera.

53. The method of claim 52, wherein obtaining optical images of the qubits comprises: performing a first cooling step by red-detuning two counterpropagating beams emitted by the first laser; performing a second cooling step by blue-detuning the two counterpropagating beams emitted by the first laser; reducing a power of one of the two counterpropagating beams emitted by the first laser; and imaging the qubits.

54. The method of claim 53, wherein, during the first cooling step, the two counterpropagating beams are detuned by two times a Zeeman splitting of adjacent mF levels of the neutral atom qubits.

55. The method of claim 53, wherein, during the first cooling step, the two counterpropagating beams are red-detuned to perform a F = 2 to F' = 3 transition of the neutral atom qubits.

56. The method of claim 53, wherein the first cooling step comprises one dimensional polarization gradient cooling (PGC).- 94 -#14172487v2Attorney Docket No. : H0776.70181 WOOO57. The method of claim 53, wherein, during the second cooling step, the two counterpropagating beams are blue-detuned to perform a F = 2 to F’ = 2 transition of the neutral atom qubits.

58. The method of claim 53, wherein the second cooling step comprises electromagnetically induced transparency (EIT).

59. The method of claim 52, further comprising identifying, using the optical images of the qubits, a spin state and / or a qubit loss of the qubits.- 95 -#14172487v2

Images