基于可重构原子阵列的逻辑量子处理器

　　我们的实验设备（扩展数据图1A）先前在参考文献中描述。7,8,30 。为了进行这些实验，已经进行了几个关键升级 ，可以在物理和逻辑量子位上启用可编程量子电路
。将含有数百万个冷87RB原子的云层装在玻璃真空电池内的一个磁光陷阱中
，然后将其随机加载到可编程的，可编程的 ，静态排列的852 nm陷阱中
，由SLM产生，并用850 nm的移动traps产生 ，由一对crossed aods-400-400-ddttttttttttttt aa aa ai traps重新排列
，无缺陷阵列30,31,62。将原子用0.65-NA物镜（特殊光学元件）成像，以用于快速电子读取时间的CMOS摄像头（Hamamatsu orca-Quest C15550-20UP） 。量子状态在87RB地面歧管中以MF = 0的超精细时钟状态编码 ，其T2> 1s（参考文献7）
，快速，高保真的单品控制由Twip-Photon Raman激发7,63执行（扩展数据图1B）
。整个阵列的全局拉曼路径用于整体旋转（大约1 MHz的Rabi频率 ，用复合脉冲技术7）以及整个电路的动态解耦（通常为每个运动的全局π脉冲）。完全可编程的本地单量旋转是用相同的拉曼光实现的，但通过本地路径重定向
，该路径通过额外的2D AOD集中在靶向原子上 。使用420-nm和1,013 nm的Rydberg梁在n = 53 rydberg状态下，使用420 nm和1,013 nm的rydberg梁进行纠缠的门（270 ns的持续时间） ，并使用时间优势的两Q Qubit Gate Pulse使用420-NM和1,013 nm的Rydberg梁进行纠缠
。8。在计算过程中
，原子与AOD陷阱重新排列以实现任意连接7,66,67 。中路读数是通过从侧面用当地聚焦的780 nm成像光束从侧面照亮的， 在CMOS摄像机上收集的散射光子 ，并通过FPGA（Xilinx ZCU102）实时处理
，并带有前馈控制信号输出
。 　　如在扩展数据中所示，使用五个任意波形发生器（AWGS）（AWGS）（AWGS）（AWGS）组成的对照基础架构对量子电路进行编程 ，如图1C所示
，同步为 <10 ns jitter. The two-channel ‘Rearrangement AWG’ is used for rearranging into defect-free arrangements30 before the circuit, the one channel of the ‘Rydberg AWG’ is used for entangling-gate pulses, the four channels of the ‘Raman AWG’ are used for IQ (in-phase and quadrature) control of a 6.8-GHz source7,63 (the global phase reference for all qubits) and pulse-shaping of the global and local Raman driving, the two channels of the ‘Raman AOD AWG’ are used for generating tones that create the programmable light grids for local single-qubit control and the two channels of the ‘Moving AOD AWG’ are used for controlling the positions of all atoms during the circuit. AODs are central to our methods of efficient control62, in which the two voltage waveforms (one for the x axis and one for the y axis) control many physical or logical qubits in parallel: each row and column of the grid simply corresponds to a single frequency tone, and these tones are then superimposed in the waveform delivered to the AOD (amplified by Mini Circuits ZHL-5W-1+). The phase relationship between tones is chosen to minimize interference. 　　Most of the system parameters used in our approach do not have hard limits but instead result from possible trade-offs. Next, we detail some design decisions made for the circuits used in this work. 　　For simplicity, we keep the entangling zone fixed for all experiments. This conveniently allows us to switch between, for example, surface code and [[8,3,2]] code experiments, without further calibrations. We choose our entangling-zone profile, realized by 420-nm and 1,013-nm Rydberg ‘top-hat’ beams generated by SLM phase profiles30, to be homogeneous over a 35-μm-tall region. As the Rydberg beams propagate longitudinally, the entangling zone is longer than it is tall. We optimize top hats to be homogeneous over roughly 250-μm horizontal extent. Taller regions are also achievable, with a trade-off with reduced laser intensity and greater challenge in homogenization. The 250-μm width of the zones used here is set by the bandwidth of our AOD deflection efficiency. We position the readout zone on the other side of the storage zone to further minimize decoherence on entangling-zone atoms. 　　Our two-qubit gate parameters are similar to our previous work in ref. 8. During two-qubit Rydberg (n = 53) gates, we place atoms 2 μm apart within a ‘gate site’, resulting in 450 MHz interaction strength between pairs, much larger than the Rabi frequency of 4.6 MHz. Notably, owing to the use of the Rydberg blockade32,68, the gate is largely independent of the exact distance between atoms. Hence, precise inter-atom positioning is not required. Gate sites are separated such that atoms in different gate sites are no closer than 10 μm during the gate, resulting in negligible long-range interactions. Throughout this work, we use four gate sites vertically (five for the surface-code experiment) and 20 horizontally, performing gates on as many as 160 qubits simultaneously (see Extended Data Fig. 1d). Under various conditions, with proper calibration, we measure two-qubit gate fidelities in the range F = 99.3–99.5%. We do not observe any error on storage-zone atoms when Rydberg gates are executed in the entangling zone. Even though the tail of the top-hat Rydberg excitation beams is only suppressed to about 10% intensity, the two-photon drive is far off-resonant owing to the approximately 20 MHz 1013 light shift detuning that is present for the entangling-zone atoms8. We natively realize physical CZ gates; when implementing CNOTs, we add physical H gates. We find minimal two-qubit cross-talk between gate sites, as examined with long benchmarking sequences in ref. 8. Although ref. 8 seems to find some small cross-talk seemingly originating from decay into Rydberg P states, this should be considerably suppressed in the practical operation here owing to the approximately 200 μs duration between gates, during which time Rydberg atoms should either fly away or decay back to the ground state. 　　The SLM tweezers can have arbitrary positions but are static. The AOD tweezers are mobile but have several constraints7,69. In particular, the AOD array creates rectangular grids (but not all sites need to be filled). During the atom-moving operations, they are only used for stretches, compressions and translations of the AOD trap array, that is, atoms move in rows and columns, and rows and columns never cross7,69. Arbitrary qubit motions and permutation is achieved by shuttling atoms around in AOD tweezers and then transferring atoms between AOD and SLM tweezers as appropriate. We perform gates on pairs of atoms in both AOD–AOD traps and AOD–SLM traps, with no observed difference for gate performance as measured by randomized benchmarking8. 　　We find that free-space shuttling of atoms (that is, no transfers) in AOD tweezers comes essentially with no fidelity cost (other than time overhead), consistent with our previous work7. Two further improvements here are the use of a photodiode to calibrate and homogenize the 2D deflection efficiency of our 2D AODs to percent-level homogeneity across our used region and engineering atomic trajectories and echo sequences to cancel out residual-path-dependent inhomogeneities. For example, we move an atom 100 μm away to realize a distant entangling gate and then, before returning the atom, we perform a Raman π pulse, so that differential light shifts accumulated during the return trip cancel with the first trip. Motion is realized with a cubic profile as in ref. 7, the characteristic free-space movement time between gates is roughly 200 μs and acoustic-lensing effects from the AOD are estimated to be negligible. We pulse the 1013 laser off during motion to remove loss effects from the large light shifts. Note that the 1013-induced differential light shift on the hyperfine qubit is only on the kHz scale but we still ensure its effects are properly echoed out. 　　Transferring atoms between tweezers9 presents further challenges. We measure the infidelity of each transfer, encompassing both dephasing and loss, to be 0.1%. To achieve this performance, in our transfer from SLM to AOD, we ramp up the intensity of the AOD tones (with quadratic intensity profile when possible) corresponding to the appropriate sites over a time of 100–200 μs to a trap depth about two times larger than the SLM trap depth, and then move the AOD trap 1–2 μm away over a time of 50–100 μs. These timescales can probably be shortened considerably while suppressing errors using optimal control techniques. During subsequent motion, we leave the AOD trap depth at this 2× value. To transfer an atom AOD to SLM, we perform the reversed process. During these transfer processes, the differential light shifts on the transferred atoms are dynamically changing and can result in large unechoed phase shifts. As such, whenever possible, we engineer circuits such that pairs of transfers will echo with appropriately chosen π pulses. When echoing pairs of transfers is not possible, we perform one cycle of XY4 or XY8 dynamical decoupling during the transfer. Finally, we note that low-loss transfer is highly sensitive to alignment of the AOD and SLM grid. We fix small optical distortions between the AOD and SLM tweezer grids by fine adjustment of individual SLM grid tweezers, which can be arbitrarily positioned, to overlap with the AOD traps as seen on an image plane reference camera. It is important to adjust the SLM and not the AOD, as small adjustments of individual AOD tones deviating from a frequency comb causes beating and atom loss. 　　In our circuit design, we engineer our echo sequences to cancel out as many deleterious aspects as possible. We ensure that, in our dynamical decoupling, we have an odd number of π pulses between CZ gates (whenever possible), as this echoes out both systematic and spurious contributions to the single-qubit phase7,8. We apply appropriate X(π) and Z(π) rotations between local addressing with the local Raman to cancel out errors induced by the global π/2 pulses, as well as between pulses of the 420-nm laser (when used for entangling-zone single-qubit rotations7) to echo out small cross-talk experienced in the storage zone by the tail of the 420-nm beam. For our global decoupling pulses, we use both BB1 pulses70 and SCROFULOUS pulses71. To benchmark and optimize coherence during our complex circuits, we perform a Ramsey fringe measurement encompassing the entire movement and single-qubit gate sequence and optimize the observed contrast7. When performing properly, our total single-qubit error is consistent with state preparation and measurement (SPAM)8, an effective coherence time of 1–2 s and the Raman scattering error of all the Raman pulses7,63. We note that these measured coherence times include the movement within and between zones; although we use fewer pulses (typically one per movement) than the XY16-128 sequence used to benchmark 1.5 s coherence in ref. 7, the coherence times here are naturally longer because of further-detuned tweezers used (852 nm rather than 830 nm). 　　Local single-qubit gates34,72 with the Raman AOD are realized in arbitrary positions in space on both AOD and SLM atoms. Targeted logical qubit blocks are addressed by a grid illumination of the logical block. Arbitrary patterns of rotations on the qubit grid (for example, during colour-code preparation) are realized with row-by-row serializing, with the targeted x coordinates in each row simultaneously illuminated. The duration of each row is 5–8 μs (corresponding to several tens of μs for an arbitrary pattern of rotations), which can be sped up considerably, as discussed in the next section. For simplicity, we carefully calibrate rotations on 80–160 specific sites across the array, but also perform rotations in arbitrary spots using the nearest calibrated values. 　　With the local single-qubit gates and entangling-zone two-qubit gates calibrated, the entire circuit is simply defined by the appropriate trapping SLM phase profile and waveforms for our several AWG channels and TTL pulse generator. These several channels then program complex, varied circuits on hundreds of physical qubits. Animations of all of our programmed circuits are attached as Supplementary Videos. 　　To enable individual single-qubit gates, we use the same Raman laser system as our global rotation scheme and illuminate only chosen atoms using a pair of crossed AODs. The focused beam waist in the plane of the atoms is 1.9 μm, which is large enough to be robust to fluctuations in atomic positions and small enough to prevent cross-talk to neighbouring atoms separated by 6 μm. For Raman excitation, polarization needs to be carefully considered. Unlike the global path, the local beam-propagation direction is perpendicular to the atom-quantization axis (set by the external magnetic field). Therefore, the fictitious magnetic field responsible for driving the transitions, as described in ref. 63, preferentially drives σ± hyperfine transitions rather than the desired π clock transition73. There exist two possible approaches to single-qubit gates, as illustrated in Extended Data Fig. 2a. First, off-resonant σ± dressing generates differential light shifts between qubit states, enabling fast local Z(θ) gates. Global π/2 rotations convert these to local X(θ) gates. Second, we can directly apply local X(θ) gates with direct π transitions by slightly rotating the quantization axis towards the local beam direction; this could be achieved with an external field but, conveniently, has a DC component that naturally rotates the axis. Note that, if the local beam is quickly turned on, this same fictitious DC field causes leakage out of the mF = 0 subspace, therefore Gaussian-smoothed pulses are used throughout this work. 　　Although we realize both the π and σ± versions above, in these experiments, we use the off-resonant σ± dressing procedure because of reduced polarization sensitivity, as our polarization homogeneity was affected by the sharp wavelength edge of a dichroic after the AOD. Furthermore, as for most circuits, we perform local rotations row by row (only one Y tone at a time); this enables arbitrary fine-tuning of X coordinates and powers at each site for homogenizing and calibrating rotations (Extended Data Fig. 2b). We calibrate using the procedure in Extended Data Fig. 2c and find that these calibrations are stable on month timescales. 　　To quantify the fidelity, we perform randomized benchmarking using 0, 10, 20, 30, 40 and 50 local Z(π/2) rotations (per site) on 16 sites, obtaining , as shown in Extended Data Fig. 2d (note that the single-qubit gates we execute globally have fidelity closer to 99.99% (refs. 7,8)). This approaches the Raman scattering limit for our σ± scheme (error of about 7 × 10−4 per π/2 pulse), but when not well calibrated is limited by inhomogeneity, in particular, associated with distortions of the y position of the rows. In the future, the performance can be further improved by using X(θ) gates, which enables robust composite sequences such as BB1 (ref. 70), has an improved Raman scattering contribution and is faster (roughly 1 μs duration). 　　To perform mid-circuit readout10,11,12,13,14,15 of selected qubits without affecting the others, we use a local imaging beam focused on the readout zone that is roughly 100 μm spatially separated from the entangling zone7,35. The local imaging beam consists of 780-nm circularly polarized light, with a near-resonant component from F = 2 to F′ = 3 and a small repump component. This beam is sent through the side of our glass vacuum cell, co-propagating with the global Raman and 1,013-nm Rydberg beams (Extended Data Fig. 1a). We use cylindrical lenses to shape the beam, with focused beam waists of 30 μm in the plane of the atom array and 80 μm out of the plane. After moving some of the atoms to this readout zone, we first perform local pushout of population in the F = 2 ground-state manifold (by turning off the repump laser frequency), followed by local imaging of the remaining F = 1 population. 　　As depicted in Extended Data Fig. 3a, we collect an average of about 50 photons per imaged atom. To avoid losing the atoms too quickly during mid-circuit imaging (which, unlike our global imaging scheme, does not have multi-axis cooling), we use deep (roughly 5-mK) traps (helping retain the atoms) and stroboscopically pulse them on and off out of phase of the local imaging light to avoid deleterious effects of the deep traps, such as inhomogeneous light shifts and fluctuating dipole force heating74 (Extended Data Fig. 3b). From a double-Gaussian fit to the two distributions in Fig. 3a, we extract an imaging fidelity of more than 99.9%. Because this fit can lead to an overestimate of the imaging fidelity (for example, owing to atom loss during imaging), we compare the total SPAM error (measured by the amplitude of the Ramsey fringe) with local imaging versus with global imaging for the same state-preparation sequence, extracting 0.14(5)% higher error with local imaging; with these considerations, we conservatively estimate a local imaging fidelity of around 99.8%. 　　Various design considerations facilitate local imaging in the readout zone while preserving coherence of the data qubits in the entangling zone35 (Extended Data Fig. 3e–g). The main sources of decoherence are rescattering of photons from the locally imaged atoms, as well as beam reflections and tails of the local imaging beam hitting the data qubits. As shown in Fig. 1c, for the 500-μs mid-circuit imaging used in this work, we are able to achieve unchanged coherence (identical within the error bars) of the data qubits with the local imaging light on as without it. To understand these effects more quantitatively, we measure the error probability of the data qubits in the entangling zone while the local imaging beam is on in the readout zone for up to 20 ms and with higher intensities than used for local imaging in this work. We suppress decoherence by light shifting the 780-nm transition of the data qubits to be different from that of the locally imaged qubits by several tens of MHz, as studied in Extended Data Fig. 3f–g. Data qubit decoherence is further suppressed by the large spatial separation between the readout zone and the entangling zone, in which intensity from the Gaussian tail of the local imaging beam should theoretically fall off rapidly. Even at large separations, we find that stray beam reflections (for example, from the glass cell window and other optical elements) can hit the data qubit region. To mitigate this effect, we displace reflections away from the atom array by angling the local imaging beam as it hits the glass cell window. The estimated effects of rescattered photons from the imaged atoms, especially with the added relative detuning, is negligible. With all these considerations, we find that we are able to suppress data qubit decoherence rates to 0.1% per 500 μs of local imaging exposure, as illustrated in Extended Data Fig. 3h. 　　The full mid-circuit readout and feedforward cycle occurs in slightly less than 1 ms, including local pushout, local imaging, readout of the camera pixels, decoding of the logical qubit state on the FPGA and a local Raman pulse, which is gated on or off by a conditional trigger (Extended Data Fig. 3d). In future work, this approach to mid-circuit readout and feedforward can be considerably improved to enable mid-circuit readout close to the 100-μs scale75. This method can be directly extended to perform many rounds of measurement and feedforward, in which groups of ancilla atoms are consecutively brought to the readout zone throughout a deep quantum circuit. 　　During transversal CNOT operations, physical CNOT gates are applied between the corresponding data qubits of two logical qubits. These physical CNOT gates propagate errors between the data qubits in a deterministic way: X errors on the control qubit are copied to the target qubit and Z errors on the target qubit are copied to the control qubit (see Extended Data Fig. 4b). As a result, the syndrome of a particular logical qubit can contain information about the errors that have occurred on another logical qubit, at the point in time in which the pair underwent a transversal CNOT operation. We can use the information about these correlations and improve the circuit fidelity by jointly decoding the logical qubits involved in the algorithm. We note that this is closely related to other recent developments in decoding entire circuits, or so-called space-time decoding76,77,78,79. It is also related to Steane error correction80, for which errors are intentionally propagated from a data logical qubit onto an ancilla logical qubit, which is then projectively measured to extract the syndrome of the data logical qubit. 　　To perform correlated decoding, we solve the problem of finding the most likely error given the measured syndrome. We start by constructing a decoding hypergraph based on a description of the logical algorithm, which describes how each physical error mechanism (for example, a Pauli-error channel after a two-qubit gate) propagates onto the measured stabilizers76,81. The hypergraph vertices correspond to the stabilizer measurement results. Each edge or hyperedge corresponds to a physical error mechanism that affects the stabilizers it connects, with an edge weight related to the probability of that error. Each hyperedge can connect stabilizers both within and between logical qubit blocks (see Fig. 2b). We then run a decoding algorithm that uses this hypergraph, along with each experimental snapshot, to find the most likely physical error consistent with the measurements. This correction is then applied in software (with the exception of Fig. 4e, which is decoded in real time). 　　Concretely, to construct the hypergraph for a given logical circuit, we perform the following procedure. For each logical algorithm (in this section, considering only Clifford gates), we identify a set of N detectors (vertices of the hypergraph) Di  {0, 1} for i = 1,…, N, which are sensitive to physical errors occurring during the logical circuit. A detector is either on (1) or off (0) to indicate the presence of an error. For the general case, we let Di = 0 if the ith stabilizer measurement matches the measurement of its backwards-propagated Pauli operator at a previous time and 1 otherwise (the latter indicates that an error has occurred). In particular, for our surface-code experiments, detectors in the final projective measurement are computed by comparing the final projective measurement of the stabilizers with the value of the ancilla-based stabilizer measurement that occurred before the CNOT (note that, owing to our state-preparation procedure, the initial stabilizer measurement is randomly ±1, but the detector is deterministically zero in the absence of noise). For our 2D colour-code experiments, the initial stabilizers are deterministically +1, so each detector is equal to zero if the corresponding stabilizer in the final projective measurement is +1. To construct the concrete hypergraph and hyperedge weights, we then use Stim76 to identify the probability pj (j = 1,…, M) of each error mechanism Ej in the circuit using a Pauli-channel noise model with approximate experimental error rates, along with the detectors that are affected by Ej. 　　To find the most likely physical error, we encode it as the optimal solution of a mixed-integer program, a canonical problem in optimization with commercial solvers readily available82, similar to previous work in ref. 83. We associate each error mechanism Ej with a binary variable that is equal to 1 if that error occurred and 0 otherwise. Our goal is then to find the error assignment {0, 1}M with maximum total error probability (alternatively, the error with the minimum total weight, in which the weight of error i is wi = log[(1 − pi)/pi]), subject to the constraint that the error is consistent with the measured detectors. To be consistent with the measured detectors, the parity of the error variables for all the hyperedges connected to a given detector should match the parity of that detector. Concretely, let f be a map from each detector Di to the subset of error mechanisms that flip its parity. The most likely error is then the optimal solution to the following mixed-integer program: 　　The objective function evaluates to the logarithm of the probability of the assigned error configuration, and each variable Ki ensures that the sum of the error variables in f(Di) matches Di, modulo 2. Finally, we solve the mixed-integer program to optimality using Gurobi, a state-of-the-art solver82, and apply the correction string associated with the error indices j for which Ej = 1 in the optimal assignment. We explore this correlated decoding in more detail, including its consequences on error-corrected circuits and the asymptotic runtimes of different decoders (M.C. et al., manuscript in preparation). See sections ‘Surface code and its implementation’ and ‘Correlated decoding in the surface code’ for further discussion on the surface code in particular. 　　One challenge with logical qubit circuits is that convenient probes that are accessible with physical qubits may no longer be accessible. The GHZ state studied here provides such an example, as conventional parity-oscillation measurements cannot be performed84. Instead, we use a technique known as direct fidelity estimation39, which can be understood as follows. The target state ψ is the simultaneous eigenstate of the N stabilizer generators {Si} and, so, the projector onto the target state is (which is 1 if Si = 1  i and 0 otherwise). Thereby, we can directly measure fidelity by measuring the expectation values of all terms in this product, which—in other words—refers to measuring the expectation values of all elements of the stabilizer group given by the exponentially many products of all the Si. The logical GHZ fidelity is defined as the average expectation value of all measured elements of the stabilizer group. With our four-qubit GHZ state, with four stabilizer generators {XXXX, ZZII, IZZI, IIZZ}, the 16-element stabilizer group is given by all possible products: {IIII, ZZII, IZZI, IIZZ, ZIIZ, IZIZ, ZIZI, ZZZZ, XXXX, XYYX, YXXY, XXYY, YYXX, YXYX, XYXY, YYYY}. We measure the expectation values of all 16 of these operators; for each element, we simply rotate each logical qubit into the appropriate logical basis and then calculate the average parity of the four logical qubits in this measurement configuration. We then directly average all 16 elements equally (with appropriate signs, as some of the stabilizer products should have −1 values) and, in this way, compute the logical GHZ state fidelity. This is an exact measurement of the logical state fidelity39. Scaling to larger states can be achieved by measuring elements of the stabilizer group at random39. To perform full tomography in Fig. 3e, we measure in all 34 = 81 bases, thereby measuring the expectation values of all 256 logical Pauli strings, and reconstruct the density matrix by solving the system of equations with optimization methods. 　　Here we provide more information about the sliding-scale error-detection protocol applied for Figs. 3, 5 and 6. Typically, error detection refers to discarding (or postselecting) measurements in which any stabilizer errors occurred. In the context of an algorithm, however, discarding the result of an entire algorithm if just one physical qubit error occurred may be too wasteful and we may want to only discard measurements in which many physical qubits fail and the probability of algorithm success is greatly reduced. For this reason, for the algorithms here, we explore error detection on a sliding scale, for which we can set a desired ‘confidence threshold’ such that, on the basis of the syndrome outcomes, we determine whether to accept a given measurement. Sliding this confidence threshold enables a continuous trade-off (in data analysis) between the fidelity of the algorithm and the acceptance probability. When sliding-scale error detection is applied, in all applicable cases, we also apply error correction to return to the codespace. 　　We apply such a sliding-scale error detection for the colour-code logical GHZ fidelity measurements in Fig. 3d. One possible method would be to discard measurements based on the number of detected stabilizer errors. However, this is suboptimal, both because on the colour code a single physical qubit error can result from anywhere between 1 and 3 stabilizer errors and also because errors deterministically propagate between codes during the transversal CNOT gates, such that a single physical error on one code can lead to detected errors on all codes, but which are still all correctable errors. As such, we perform the sliding-scale error detection using the correlated decoding technique and set the confidence threshold as a threshold weight of the overall correction weight on the decoding hypergraph. For example, in the colour code GHZ experiment, a stabilizer error on all four logical qubits that is just consistent with a single physical qubit error that propagated to all four logical qubits is in fact a low-weight (or high-probability) error, as it corresponds to just a single physical qubit error. If the weight of hypergraph correction (inversely related to the log of the probability that a given error mechanism would have occurred leading to the observed syndrome outcome) is below the cut-off threshold weight, then the measurement is accepted; otherwise, it is rejected. For each threshold, we then calculate the average algorithm result (y axis of Fig. 3d), as well as the fraction of accepted data (x axis of Fig. 3d). 　　In Fig. 5 with [[8,3,2]] codes, for 3, 6, 24 and 48 logical qubits, we apply our sliding-scale detection simply as given by the total number of stabilizer errors detected, although—as illustrated above—this can probably be improved by considering which stabilizer error patterns are more likely to cause an algorithmic failure. For the 12 logical qubits, to have a more fine-grained sliding scale, for each of the 24 = 16 possible stabilizer outcomes, we calculate the XEB to rank the likelihood that each of the observed stabilizer outcomes leads to an algorithmic failure and then use this ranking when deciding whether a given measurement is above/below the cut-off threshold. In Fig. 6b, we set the threshold by the number of stabilizer errors and in Fig. 6d, to have more fine-grained sliding-scale information, we take different subsets of stabilizer outcome events that are all below the threshold of the allowed number of stabilizer errors and calculate the y axis (Pauli expectation value) and x axis (purity) for all of them. Broadly, there are many ways to perform this sliding-scale error detection, and this can be useful both as continuous trade-offs between fidelity and acceptance probability, as well as for use in techniques such as zero-noise extrapolation in data analysis (Fig. 6d). 　　Here we provide a brief overview of key QEC methods used in our work. 　　[[n,k,d]] notation describes a code with several physical qubits n, several logical qubits k and a code distance d. The code distance d sets how many errors a code can detect or correct. The code distance is the minimum Hamming distance between valid codewords (logical states), that is, the weight of the smallest logical operator85. In the case of the 2D surface and colour codes studied here, d is equivalent to the linear dimension of the system24. 　　Following this definition, quantum codes of distance d can detect any arbitrary error of weight up to d − 1. Such errors cause stabilizer violations, indicating that errors occurred. Postselecting on the results with no such stabilizer violations corresponds to performing error detection, which protects the quantum information up to d − 1 errors at the cost of postselection overhead. Conversely, codes can correct fewer errors than they detect (but without any postselection overhead). The correction procedure brings the system back to the closest logical state (codeword); thus, if more than d/2 errors occur, the resulting state may be closer to a codeword different from the initial one, resulting in a logical error85. For this reason, codes of distance d can correct any arbitrary error of weight up to (d − 1)/2 (rounded down if d is even24). The process of decoding refers to analysing the observed pattern of errors and determining what correction to apply to return to the original code state and undo the physical errors created. In many cases, such as with the 2D surface and colour codes, one does not need to apply the correction in hardware (physically flipping the qubits); instead, it is sufficient to undo an unintended XL/ZL operator that was applied by hardware errors by simply applying a ‘software’ XL/ZL operator24, also described as Pauli frame tracking86. 　　As the size of an error correcting code and the corresponding code distance is increased, so are the opportunities for errors to occur as the number of physical qubits increases. This leads to a threshold behaviour in QEC: if the density of errors p is above a (possibly circuit-dependent) characteristic error rate pth, then increasing code distance will worsen performance. However, if p < pth, then increasing code distance will improve performance24. Theoretically, because we require (d + 1)/2 errors to create a logical error, the logical error rate will be exponentially suppressed as (p/pth)(d+1)/2 at sufficiently low error rates24. The performance improvement with increasing code distance, observed for the preparation and entangling operation in Fig. 2, implies that we surpass the threshold of this circuit. We note that, in this regime, improving fidelities by, for example, a factor of 2× can then lead to an error reduction of 24 = 16× for the distance-7 code studied and further exponential suppression with increasing code distance. This rapid suppression of errors with reduced error rate and increased code distance is the theoretical basis for realizing large-scale computation. We emphasize that thresholds can be circuit-dependent, as discussed in detail in the surface-code section below. 　　A common definition of fault tolerance in quantum circuits85 (which we use in this work) is that a weight-1 error (that is, an error affecting one physical qubit) cannot propagate into a weight-2 error (now affecting two physical qubits) within a logical block. This property implies that errors cannot spread within a logical block and thereby prevents a single error from growing uncontrollably and causing a logical error. 　　Distance-3 codes, which are of notable historical importance3,87, can correct any weight-1 error. Fault tolerance is particularly important for these codes because otherwise a weight-1 error can lead to a weight-2 error and thereby cause a logical fault. An important characteristic of a fault-tolerant circuit that uses distance-3 codes85 is that (in the low-error-rate regime) physical errors of probability p lead to logical errors with probability p2. We emphasize that the notion of fault tolerance refers to circuit structuring to control propagation of errors, but a circuit can be fault-tolerant with low fidelity or non-fault-tolerant with high fidelity. For example, even if a weight-1 error can lead to a weight-2 error but the code has high distance, or if this error-propagation sequence is possible but highly unlikely, then this property may not be of practical importance (for this reason, definitions of fault tolerance may vary). In practice, the goal of QEC is to execute specific algorithms with high fidelity, and fault-tolerant structuring of a circuit is one of many tools in the design and execution of high-fidelity logical algorithms. 　　Transversal gates, defined here as being composed of independent gates on the qubits within the code block (that is, entangling gates are not performed between qubits within the same code block)42, constitute a direct approach to ensure fault-tolerant structuring of a logical algorithm. Because transversal gates imply performing independent operations on the physical constituents of a code block, errors cannot spread within the block and fault tolerance is guaranteed. In this work, all logical circuits we realize (following the logical state preparation) are fault-tolerant, as all logical operations we perform are transversal. Note, in particular, that even though the transversal CNOT allows errors to propagate between code blocks, this is still fault-tolerant, as it does not lead to a higher-weight error within the block and, thereby, a single physical error can neither lead to a logical failure nor an algorithmic failure. Notably, the large family of codes referred to as Calderbank–Shor–Steane (CSS) codes all have a transversal CNOT (ref. 2), all of which can be implemented with the single-step, parallel-transport approach here. 　　Although all the logical circuits we implement are fault-tolerant, the logical qubit state preparation is fault-tolerant for our d = 3 colour code (Figs. 3 and 4) and d = 3 surface code (part of Fig. 2), but is non-fault-tolerant for the state preparation of our d = 5, 7 surface codes and [[8,3,2]] codes. Thus, all of our experiments with the d = 3 colour codes are fault-tolerant from beginning to end, and so the entire algorithm is fault-tolerant and theoretically has a failure probability that scales as p2. However, we note that having a fault-tolerant algorithm also does not imply that errors do not build up during execution of the circuit. For this reason, deep circuits require repetitive error correction6,88 to constantly remove errors and continuously benefit from, for example, the p2 suppression. 　　Our logical GHZ state theoretically has a failure probability scaling as p2. Nevertheless, the error build-up (increasing p) during the operations of the circuit and the spreading of errors through transversal gates limits our logical GHZ fidelity to 72%. This is consistent with numerical modelling. Similar to the surface-code modelling (Extended Data Fig. 4), we use empirical error rates consistent with 99.4% two-qubit gate fidelity, as well as roughly 4% data qubit decoherence error (including SPAM) over the entire circuit. We simulate the experimental circuit (including the fault-tolerant state preparation with the ancilla logical flag) and measurements of all 16 elements of the stabilizer group (see the ‘Direct fidelity estimation and tomography’ section), and extract a simulated logical GHZ fidelity of 79%. This is slightly higher than our measured 72% logical GHZ fidelity, possibly originating from imperfect experimental calibration. This modelling indicates that our logical GHZ fidelity is limited by residual physical errors, which will be reduced quadratically as p2 with reduction in physical error rate p, in particular by reducing residual single-qubit errors, which were larger during this measurement and are dominating the error budget here. 　　In 2D planar architectures, such as those associated with superconducting qubits6,88, stabilizer measurement is the most important building block of error-corrected circuits24. In such systems, stabilizers need to be constantly measured to correct qubit dephasing and increase coherence time, as demonstrated recently6. Logic operations are implemented by changing stabilizer measurement patterns, enabling realization of techniques such as braiding24 and lattice surgery89. Similar techniques can be used to move logical degrees of freedom to implement nonlocal logical gates23. Owing to this gate-execution strategy, d rounds of stabilizer measurement are required for each entangling gate for ensuring fault tolerance24. 　　Neutral-atom quantum computers feature different challenges and opportunities. Specifically, they feature long qubit coherence times (T2 >1s），可以通过建立良好的技术将其进一步提高到数十秒至数百秒的比例72。通过使用存储区 ，可以长时间无需重复稳定器测量
，可以闲置且安全地存储 。因此，从实际的角度来看 ，通过使用逻辑编码来提高量子的连贯性并不能立即获得改善量子算法的收益，而提高纠缠操作的忠诚度将取得收益。此外
，逻辑门和Qubit运动不必通过稳定器测量进行
。取而代之的是，它们可以使用非本地原子传输和横向门执行。由于这种横向门具有本质上的耐断层 ，因此它们不一定需要每次操作后进行D轮校正 。在某些情况下
，即使是通过诸如Steane误差校正80的技术（类似于我们的Ancilla逻辑标志，具有图3中使用的颜色代码） ，而不是重复的稳定器测量
，甚至可以通过诸如Steane误差校正80（类似于我们的Ancilla逻辑标志）进行更好地执行综合征测量
。由于这些原因，横向CNOT是错误校正电路中最重要的构件之一。因此 ，我们在这里着重于通过缩放代码距离来改善横向CNOT 。 　　具体而言
，我们使用所谓的旋转表面代码6，该代码6具有代码参数[[D2,1 ，d]]
。我们的距离7表面代码（如图2D所示）由49个物理数据量表组成
，具有24 x稳定器（浅蓝色正方形）和24 Z稳定器（深蓝色的正方形），以及一个由抗重量-7操作员描述的编码逻辑Qubit ，是重量-7操作员，固定的XL XL
，XL方向方向上方向上方向上的ZL。X和Z稳定器与XL和ZL逻辑操作员通勤，允许测量稳定器 ，而不会扰乱潜在的逻辑自由度 。在我们的实验中
，我们在|+L中准备一个表面代码，并在| 0L中准备一个表面代码
。在第一个代码中 ，这是通过在|+中准备所有物理数据量子的实现
，从而准备XL和24 X稳定器的本征态，然后使用四个Ancilla Qubits（图2D红点）进行预测测量24 Z稳定剂（图2D红点）。第二个代码的制备方式类似 ，但所有物理量子位均以| 0初始化
，从而准备了Zl和24 Z稳定器的本征态，然后用24个Ancillas测量了24 x稳定器 。CNOT是直接横向的 ，因为这两个表面代码块具有相同的方向
，并且不需要晶格旋转以实现H
。Ancillas的射击测量值定义了稳定剂的值。在横向CNOT期间，稳定器的值也被复制到其他代码中 ，并在软件中进行跟踪 。 　　因为我们只执行一轮稳定器测量，所以我们的状态准备方案是d = 5
，7代码的NFT。例如，考虑到所有稳定器被定义为+1并且系统中没有误差的情况 ，但是表面代码晶格中间的Ancilla测量误差可产生-1的稳定器测量
。然后
，校正会导致这种明显的稳定剂违反边界的重量配对。因此，这个单一的Ancilla测量误差可能导致多个数据量误差 ，从而导致NFT操作 。d = 3代码初始化是一种特殊情况
，不会遭受本期38的困扰
。还可以考虑对稳定器测量过程中门排序给出的容错的高阶注意事项也可以考虑6。 　　嘈杂综合征提取的这些NFT错误的影响是在| 0L状态上对|+L状态和z物理错误引起X物理错误 。因此，在仅执行垃圾邮件测量值时 ，这些误差的存在将无法直接明显
，因为这些误差通过x基础中的x基础测量|+l的通勤和| 0l在z基础上
。因此，该电路将不是表面代码态制备的良好基准。相反 ，横向CNOT实验对电路的各个方面敏感
，并且表现出良好的表现 。因为我们在底部和基础上都测量了钟状状态，所以两个基部的NFT误差都会通过逻辑cnot传播 ，并在x和z底座中的两个逻辑Qubit上造成误差
。由于这些原因，与表面代码垃圾邮件测量不同
，该实验是逻辑性能的良好指示。实际上，这些NFT误差的效果是 ，如果我们只是在每个逻辑块中应用常规解码
，那么钟状状态会随着代码距离的增加而大大降级（图2D） 。 　　通过使用相关的解码技术，这种NFT制剂的效果被抑制（但并非完全消除）
。例如 ，考虑在d = 7 |+l状态的中间线的左侧的NFT诱导的明显稳定器违反
，对应于边界的三个物理X误差。这些误差将通过逻辑cnot传播到第二个逻辑量子位，并在研究稳定器时在z基础上的两个逻辑量子台的独立测量 。当独立解码时 ，如果在CNOT将稳定器违反中间线的右侧移动稳定器后的第一个块上发生了另一个X错误
，成为四个X物理错误的链条，这将导致不正确的配对并导致该代码上的独立错误 ，从而仅损坏稳定器
，并将其与两个编码之间的总权重校正相对应。但是，当与解码相关的共同解码时 ，这些误差可以有效地解码，因为它们将出现在两个逻辑量子的稳定器上
。在此示例中
，最低的配对将从两个代码中删除这三个X错误的链，并且仅在第一个块上留下剩下的X错误 ，也可以成功解码（这里的总配对权重仅为2）。因此
，我们相关的解码技术对于我们对使用代码距离提高的铃铛性能的观察至关重要 。 　　最后，我们详细介绍了对钟形错误的评估
。钟形的忠诚度是由种群和连贯的平均值给出的 ，对于物理量子
，可以将其作为ZZ种群和奇偶校验振荡的幅度来衡量。用稳定器的语言，均衡振荡幅度由xx和-yy的平均值（参考文献90）给出 。借助表面代码 ，我们无法方便地测量YL操作员容量的故障（这就是为什么我们将颜色代码用于可编程的Clifford算法和完整层析成像；请参见下一节）
。因此
，我们将逻辑连贯性估计为XLXL，然后我们平均使用用于计算钟形误差的人群。为了支持该分析的有效性 ，我们可以计算出钟形的Fidelity90上的下限
，这也显示出与增加代码距离的性能相同的改善（扩展数据图4D） 。 　　在上述讨论之后，我们提供了与表面代码横向CNOT的相关解码相关的更多见解
。考虑一个初始化完美（无噪声）表面代码的电路 ，执行横向CNOT，然后执行投影读数。如果错误在横向cnot之前发生
，则这些错误可能会传播 。例如，控件逻辑量子器上的X物理错误将传播到目标逻辑量子位 ，从而将目标逻辑量子误误的密度增加一倍
。通过将目标逻辑值的项目测量的Z稳定器乘以控制逻辑量子量值的Z稳定剂
，该传播被撤消。现在，目标逻辑量子位只需要解码其X错误的原始密度 。对于源自目标逻辑值的Z错误 ，可以考虑相同的考虑
，从而传播到控制逻辑量子量乘以。但是，如果横向CNOT之后存在误差 ，则将稳定器倍增会使此类误差的密度增加一倍
。因此
，如果仅在横向CNOT之后发生错误，则最佳解码策略是在两个代码中执行独立匹配。横向CNOT之前和之后存在误差的一般情况都不对应于情况 ，并且由我们的解码超图建模
，我们的解码超图具有连接两个逻辑Qubit的边缘和超中心，并由我们的实验误差模型告知边缘权重 。扩展数据图5探索了用不同的边缘和超根的缩放权重值连接两个逻辑量子尺的稳定器的解码性能
。这些结果表明 ，相关的解码与与Ancilla测量误差相关的NFT误差是可靠的（但并非完全不敏感）。更简单的乘法解码器也将恢复此功能，这对Ancilla测量的错误完全不敏感
， 但这是 - 否则 - 对CNOT之后的错误更加敏感 。具体而言，扩展数据图5C表明 ，我们优化的解码器不仅仅是“乘法解码器
”
，因为Ancilla测量值确实有助于校正过程，并使相关解码对解码器参数更强大
。对于给定的逻辑电路 ，我们的相关解码过程会生成解码超图
，然后我们使用最可能的误差方法求解，该方法在此处用于表面代码和颜色代码实验 ，并且可以通常应用于任何稳定器代码和Clifford CircitiTs79。M.C.将提供更多理论细节和相关解码的讨论 。等人
，准备手稿
。 　　2D颜色代码是与表面代码相似的拓扑代码91。此处使用的颜色代码通常在三角形几何形状中描绘，其中包括三种颜色的重量-4和重量-6稳定器 ，XL和ZL运算符沿代码91的边界延伸 。在这项工作中
，我们研究了2D d = 3颜色代码，如图3A所示 ，仅包含由X和Z产物给出的重量-4稳定剂，该稳定器在每个彩色plaquette的量子台上
。此d = 3颜色代码与七分位steane代码相同。但是
，我们强调，此处使用的技术直接适用于较大的颜色代码92 。 　　尽管颜色代码类似于表面代码 ，但重要的区别是
，在颜色代码中，X和Z稳定器直接位于相同量子位的顶部（而不是相互相对于双重晶格） ，并且类似地
，XL和ZL操作员彼此之间（相反，在表面代码上都在Orthogogonal code上）。换句话说 ，这里的操作员是对称的
，并且通过全球基础转换相关
。这对允许的横向门SET41,93具有重要的后果。特别是，尽管表面代码在技术上具有转换XL ZL的横向h ，但它需要代码块的物理90°旋转 。尽管使用原子运动技术可以使用这种晶格旋转
，但对于许多电路，这是不便的
。相反 ，在颜色代码中，H是横向的：它直接交换XL ZL以及X和Z稳定器。这种差异对于横向门更重要
，这对于颜色代码可能是可能的 。在这里，横向交换XL YL（YL由XL和Zl的乘积彼此置于XL和Zl的产品中给出） ，并且给定Plaquette的X稳定剂通过将同一plaquette的Z稳定器倍增
。（这与没有横向s的表面代码相反
，yl操作员是水平传播XL并垂直传播Zl（参考文献24）的产物。因为颜色代码具有{H，S ，CNOT}的整个横向门集
，并且也不需要跟踪任何晶格旋转，因此非常适合探索可编程的逻辑Clifford算法 。 　　对于D = 3颜色代码的易耐故障准备 ，我们使用参考文献中总结的计划的修改版本
。38
，在其中，我们使用的是九个栅极编码电路 ，而不是更方便地映射到系统中特定的原子运动（对应于与参考文献7相似的图形制剂）
，然后是带有Ancilla逻辑标志的横向cnot。然后将逻辑垃圾邮件保真度计算为解码后观察| 0L的概率 。我们注意到，在图3中 ，我们也可以做出五倍的GHz状态，但为了简单地执行完整的断层扫描而建立了四倍的GHz状态
。在图4中
，当报道了带有前馈的钟形忠诚度时，我们将逻辑连贯性估计为XLXL和-Ylyl的平​​均值 ，然后我们将其平均与ZLZL种群（未绘制）计算钟形忠诚度。最后
，我们注意到，图4E中的前馈状态也可以通过两个量子位中的任何一个软件zl旋转进行 ，从而可以对适当的钟形状态进行校正
，但是在这里我们在两个量子位上执行馈电S来测试我们的进料功能 。该技术与执行魔术状态传送24直接兼容。 　　2D拓扑代码（例如表面和颜色代码）具有Clifford门的横向实现（例如，{H ，S
，CNOT}）
。此门集不是通用的，也就是说 ，它不能单独用于实现任意量子计算
，并且需要一个非克利福德门，例如{t ，ccz}来实现通用计算。此外，由于Gottesman – Knill theorem44
，可以在多项式时间内模拟仅由稳定态和克利福德门组成的电路 。这可以理解为稳定器跟踪；例如，请考虑一个三量系统的系统 ，在该系统中
，状态的稳定器为x i i，使得x稳定|+状态 ，而我是身份
。应用两个CZ纠缠的门CZ1,2 CZ1,3将此稳定剂转换为X Z Z
，因为X在CZ之前翻转CZ只是更改是否将Z Flip应用于其他量子位。即使Clifford电路在Qubits之间产生了叠加和纠缠，但可以简单地跟踪状态的N初始稳定器 ，因为它们通过电路传播（所谓的操作员扩展94）
，从而可以轻松完成电路的模拟 。 　　但是，非克利福德门的影响要复杂得多
。例如 ，通过CCZ地图将稳定器X I I传递到Pauli Strings的叠加中
，也就是说，X I I→1/2（X I I+X Z I+X Z I+X I Z -X -X Z Z Z）现在更改 ，是否会更改是否将CZ操作员应用于其他QUBIT，从而在四次中应用于单个操作员
，从而追踪单个CC cc。（CZ运算符矩阵仅等于1/2 [i i+z i+i z -z z z]） 。这不仅会导致运营商传播，还会导致所谓的操作员纠缠94
。当我们应用进一步的非克利福德门时 ，要跟踪的运算符数量将成倍增长
，并最终将在计算上变得棘手。例如，最先进的Clifford+T模拟器可以处理大约16 ccz Gates50 。这是我们复杂采样电路的基础 ，其中48个逻辑量子台上的48 ccz会产生高度的乱拼图和魔术（定义下面）
，使Clifford+T模拟不切实际
。 　　在这里，我们提供了有关[[8,3,2]]电路实现的更多详细信息。[[8,3,2]]代码块在| -L ，+L
，-L状态中初始化为扩展数据中的电路，可以理解为准备两个四个Qubit GHz状态（[[[4,2,2]代码[[4,2,2] codes95） ，这些是
，以及随后在分布中添加图6a（以及图6a） 。在我们的电路实现中，对于用于采样和两拷贝测量的3-24个逻辑Qubits的系统尺寸 ，我们准备了八个块，该块编码在64个物理Qubit上。对于48个逻辑的电路（总共128个物理量子）
，我们编码8个块并纠缠它们，然后将其放入存储中；然后 ，我们从存储中拾取64个新的物理量子位
，将它们编码成纠缠区的八个块并纠缠它们
。最后，我们将原始的八个块从存储中带来 ，并将它们与纠缠区中的八个街区的第二组纠缠（扩展数据图6）（请参阅补充视频）。 　　[[8,3,2]]代码的横向门集如下所示（另请参见参考文献16,17,26,27） 。从[[8,3,2]]代码是CSS代码的事实
，块之间的横向CNOT紧随其后
。在两个逻辑Qubits Li和LJ（CZLI，LJ）之间的块内CZ门可以通过与逻辑量子lk相对应的脸部 ，S†门来实现。例如
，考虑将s，s†门的模式应用于图5中的顶部 ，也就是说
，将xl1 = x1x2x3x4转换为等于，也适用于给出 ，即，cz可以在逻辑Qubits 1和2之间实现 。该过程还可以理解为什么要了解t的模式
，t | ccc ccc ccc ccc ccc ccc ccc ccc ccc cccc cc ccc at cod a cod a cod之间的ccccc a conevery的模式很高
。CCZ门应将XL3映射到XL3 CZL1，L2。通过在图5a中应用t的模式 ，每个x面映射到本身乘以s
，s†的模式，例如 ，xl3 = x1x3x5x7映射到或以后 。这发生在所有三个XL面时
，从而意识到CCZ门
。最后，我们详细介绍了置换率 ，该列表也在参考文献中开发。27.物理上置换原子以交换4↔8和3↔7将XL1 = XL1 = x1x2x3x4带到或取而代之（也是通过乘以全局X稳定器）
，并且可以通过跟踪量子排列（即意识到CNOT）来查看 。最后，尽管这些3D代码没有横向H ，因为它们是CSS代码
，但可以以X或Z基础进行初始化和测量，从而有效地允许在电路的开始或结束时H门
。 　　构建逻辑上的纠缠门是按块施加的 ，并且可以实现任何块内门组合。为了概念上的简单性，我们仅在层中应用两种特定的本地拉曼模式 。第一个是栅极组合CCZL1
，L2，L3 CZL1 ，L2 CZL1
，L3 CZL2，L3 ZL1 ZL2 ZL2 ZL3 ，通过在整个物理量子块上应用T†给出
，我们应用的第二个门组合是CCZL1，L2 ，L3 CZL2
，L3 czl2，l3 czl2 ，l3 czl3 zl3排。在我们的电路中
，我们交替交替通过横向横向纠缠的大门和外部横向CNOT的层，最多4D Hyperipubes19,96,97纠缠了逻辑块（参见扩展数据图6）
。我们在整个电路中保持控件和目标量子位相同 ，以简单地进行概念上的简单性，从而可以使用块内栅极层来编译目标Quitt的局部物理H门
，但是也可以任意选择控制目标方向。我们确保应用逻辑上的纠缠大门，以免它们通过早期的纠缠栅极应用程序毫无动力地通勤并取消 。作为实验性的指出 ，我们指出的是
，对于在本工作的其他部分中实现的克利福德州，稳定器的值为+1或-1的值（例如 ，使用物理π/2旋转而不是h）
，然后将其简单地在软件中重新定义
。因为，对于我们的[[8,3,2]]电路 ，我们在物理层面上实施了非cliffords
，因此必须确保所有稳定器的初始化并保持为+1很重要；例如，如果Z稳定器为-1 ，则逻辑CCZ实现将X稳定器的期望值发送到0。这可以理解为单个位点上的物理X
，通过物理t转化为叠加，进入X稳定器的均等叠加为+1和-1 。 　　我们实施的电路等同于IQP电路98 ，这为理解为什么在经典上很难模拟我们的电路提供了一个理论基础，为此
，我们还提供了所谓的抗浓度的数值证据99,100
。IQP电路定义为在n个量子位上初始化|+N，应用对角线纠缠的统一 ，例如由{ccz
，cz，z}组成的对角线 ，然后在x Basite20,98中进行测量。创建了2N Bitsring的均匀叠加
，对角门以复杂的方式应用-1个迹象，以指数级的许多斑点 ，然后在测量之前与最终H中“撤消叠加”
，现在导致复杂的“ Speckle ”干扰模式 。可以在实现电路的量子设备上有效地进行此斑点模式的输出分布采样，但在经典设备上以成本成本 ，以实现IQP Circuits20的某些选择
。如上所述
，[[8,3,2]]代码的横向门集包含对角门门{CCZ，CZ ，Z}，这些对角线将-1个符号应用于BitStrings
，但通过施加CNOTS使得非二进制，而CNOTS则将bittrings缩小。因为此斑点排列不会打破IQP框架 ，所以这些电路等同于有效的IQP电路
，但它更为复杂：例如，具有48 cczs和96个CNOTS映射的电路将大约1,000 ccz gates的有效IQP电路映射到有效的IQP电路 。然而 ，由于IQP电路是一个很好理解的框架
，因此我们可以与此工具集讨论我们的电路属性
。 　　我们通过XEB18的实验探索这些电路，该电路定义为NL是逻辑量子数的数量 ，是我们逻辑Qubits的测量概率分布
，是计算出的概率分布。在这里，我们通过其理想值将XEB归一化 ，使得无噪声电路的XEB为1 。在典型的情况下
，如果噪声压倒了电路，则测得的分布将为均匀18 ，而测得的XEB为0。 　　IQP电路是量子优势型实验的好设置，因为已知很难在经典上很难模拟具有随机应用{CCZ
，CZ，CZ ，Z}门的IQP电路（随机应用{CCZ
，CZ，Z} Gates（随机Leger-3多项式）的斑点分布
。在M.K.等等 ，在制备中
，我们表明，随机在此处探索的随机超立方体IQP电路的合奏会收敛于统一的IQP集合 ，并且随着HyperCube的深度和大小的增加
，它会收敛于IQP集合。在扩展数据图8a中，我们表明 ，超立方体IQP电路具有随机的块内操作
，并且在外部封锁层上的随机对照目标（实现了超单行）抗凝聚力迅速随着超立方体的尺寸增加，随着XEB最终达到2.均匀的concc glifford concc glifford ccc g glifford cc cc g liff tosecc cc g liff tosecc cc g liff to cck gliff cck gliff con 。在这里 ，进一步提高了抗解特性，因为我们观察到理想的实验电路XEB也接近2
，即使没有太多随机化也是如此
。 　　此外，与通用随机电路采样设置（例如HAAR-MANDOM电路）相比 ，XEB是IQP电路的更好基准。102,103,104 。对于IQP
，XEB接近多体忠诚度，理论上可以在合理的噪声假设下进行差异（M.K.等人 ，在准备中手稿）
。直觉上
，这一事实与IQP电路的对角线结构有关，这使XEB能够以更接近Fidelity的方式捕获错误 ，尽管仅在计算基础上定义。换句话说
，Z错误总是会损坏X-BASIS测量值，而X错误（在测量之前立即发生）将创建新的Z错误 ，从而损坏X-BASIS测量 。因此
，在完全发出的定制制度中，在电路末端的误差在逻辑错误中很好地描述了 ，我们希望XEB是一个很好的保真度
。我们进一步指出，在这里介绍的[[8,3,2]]的有效生成[[8,3,2]]的GATE设置可以实现由{CCZ
，CZ，Z} Gates105组成的任意IQP电路。内置{ccz ，cz
，z}操作可以通过指出组合块内和外块CNOT的组合使我们能够组成针对性单个逻辑矩之间的任意横向交换操作 。 　　为了计算评估XEB并基准我们电路所需的逻辑斑点概率，我们使用混合模拟方法结合了波函数和Tensor-Network106方法
。它只有在一次执行HyperCube的所有纠缠门时才效果最好 ，并依赖于最终的CNOTS紧随其后进行测量
，简化网络收缩的事实。具体而言，对于D维逻辑超立方体 ，由2D-1块组成的两个子系统独立模拟
，然后将CNOTS和块内操作的最终层与测量结果（兴趣的点）结合在一起，从而在两个较大的数据中（参见扩展数据）（参见扩展数据） 。与使用空间的全波函数模拟相比 ，这是内存需求的平方根减少。理想的XEB值是通过从理想的输出分布中抽样的斑点来计算的
，然后平均相应的概率
。使用边缘采样算法对Bitsring进行采样，该算法使用上述相同的收缩方案。 　　接下来 ，我们考虑是否可以通过预言实施电路的精确模拟并使用具有更少资源的经典算法来轻松地“欺骗
”此问题中的有限XEB分数，精神与参考文献中引入的算法相似 。102
。对于图5中研究的电路
，仅包含HyperCube上一层大门，只有一轮连接两个2d-1-block分区的CNOT。因此 ，将它们从电路中删除并从两个独立的一半采样可能不会大大减少XEB
，同时减少记忆要求 。在扩展数据图8C中，我们研究了这种欺骗攻击的性能 ，并发现所获得的XEB一旦引入了进一步的栅极层
，即可迅速减少我们电路的特定扩展
。 　　上面的收缩方案，用于理想的模拟和XEB欺骗 ，与量子数的数量成倍缩放。但是
，通过使用以下事实，高立方体电路可以自然划分为较小的块 ，而在电路末端只有一个单独的cnots层
，可以大大减少指数 。因此，如图5D所述 ，如果我们在分区间之后引入额外的CNOT层（在单个分区中），则这种仿真方法的效率降低了
。应用L = {0
，…，d-1}进一步的内部内部cnot层迫使扩展数据中的cnot张量图。图8b被阻止成2升组 ，这导致执行时间大致扩展
，因为该分子来自张张量的复杂性，而张紧器的复杂性和分解器占合同数量的构成数量 ，从而构成了缩小数量的成分 。图5D中引用的显式时间是额外的cnot层的函数
，基于上述矩阵 - 刺激估计值并拟合，使得深-1超立方体时间匹配1.44 s ，与我们的实现相对应
。实际上
，由于硬件和软件优化以及其他因素（例如张量排列的成本），实际运行时间可能会有所不同。但是 ，我们预计总体趋势将会产生 。最后
，如果要直接存储2L挡块的张量，则该方法的内存需求将随着恢复L = D-1的全部记忆复杂性而增长。 　　在这项工作中 ，我们使用这些电路和XEB结果来对我们的逻辑编码进行基准测试，这需要模拟这些电路的能力
。未来的逻辑算法实验可以探索量子 - 优势18,48,107,108,109测试
，并用编码量子尺寸进行测试，这将在M.K中详细介绍。等 ，准备手稿 。 　　为了将我们的逻辑量算法与物理Qubt上的类似电路进行比较
，我们使用相同的物理门集Clifford+T（在逻辑电路中使用）在物理Qubt上进行了采样/扰动电路的具体实现，然后我们也尝试实现实验
。我们用三个物理Qubit块替换每个[[8,3,2]]块 ，将“块” CCZ门分解为六个CNOTS和七个{T
，T，T†}门 ，并直接在三个Qubit块之间实现“横向 ” CNOTS。我们注意到
，CZ可以编译到CCZ实施中，但这对我们的分析和估计有很小的影响 。这些物理电路很复杂：48个QCZ和228个两倍的门（正如我们的逻辑Qubits所实现的）分解为有效的516个两倍的门（如果CZ Gates汇编成CCZS）
。在试图在实践中实施这些电路时 ，相干错误的积累导致了我们的物理电路的消失。这些实验清楚地表明
，逻辑电路等效物极大地超过了物理电路，因此提供了直接的证据 ，表明我们的逻辑算法在该特定采样电路上优于我们的物理算法 。 　　更定量的是，通过具体的物理实现
，我们通过假设乐观的性能来计算上限
。我们假设我们的最佳保真度：垃圾邮件为99.4％（参考文献8），局部单量门储蓄率为99.91％（扩展数据图2） ，两分Quibent Gate Fidelity为99.55％（参考文献8）和T2 = 2s。然后
，我们计算CZ门的纠缠栅极脉冲的总数，编译的局部单量门门的总数和估计的电路持续时间 ，并使用它们来计算图5F中显示的估计值 。我们进一步确认了小规模电路实施的分析
。对于短的三度电路
，我们将物理电路的XEB基准为大约0.87，低于估计的三分之一上限约0.92。我们注意到 ，在图5F中
，我们绘制了物理问题保真度的估计值，而不是XEB ，但我们期望XEB和Fidelity与之前讨论的相关 。 　　我们注意到在比较这些复杂电路的物理和逻辑实现时做出了一些观察。首先
，从经验上讲，逻辑电路似乎对相干错误更容易宽容49,110,111 ，并且理解这是继续研究的主题
。具体而言，逻辑电路似乎意识到固有的数字操作
，为此，小相干误差并不能实质上移动/扭曲斑点分布 ，而只是减少了整体富达49,110（例如
，参见在扩展数据中的一致图7a）。这与物理实现相反，物理实现形成了相干误差 ，从而实质性地改变了布特特林分布的形状
，例如改变相对振幅 。其次，我们注意到 ，我们仅通过优化稳定器期望值而不是直接优化XEB或两次拷贝结果来优化[[8,3,2]]电路
。在运行复杂电路时
，稳定器可以用作有用的中间保真度基准，以优化电路设计和确保正确执行 ，尤其是在无法计算出输出分布或其他可观察到的制度中。总体而言
，我们发现这些复杂的电路似乎与物理Qub相比，使用逻辑Qubit的表现要好得多 。 　　提取各种兴趣的一种有力的方法是同一状态的两个副本之间的铃铛测量结果为21,22,52
。首先 ，我们使用这些测量值来计算所得状态的纯度或纠缠熵7,21,52,112,113。测量单线状态的发生（应用最终成对纠缠操作后我们的测量结果| 11结果）探测了给定位点i的交换操作员的特征值 。这反过来又通过观察到任何子系统A的纯度
。因此，平均纯度可以通过a内的平均均衡性估算
，因此也可以通过二阶Rényi纠缠熵来估算。 　　纠缠熵计算仅涉及单线结果 。通过使用完整的结果分布，我们还可以从一个数据集中评估所有4N Pauli字符串的绝对值 ，其中n是nate22副本中涉及的量子数的数量
。更具体地说
，考虑一个给定的Pauli字符串O = ∏IPI，其中Pi {Xi ，Yi
，Zi，ii}是站点I（以及身份）上的单个Pauli操作员 ，以及给定的观察到的Bittstring
，在其中标记了控件和目标副本中的结果。可以通过考虑铃铛状态所映射的计算状态，并考虑哪些XX ，YY和ZZ的操作员在各种铃铛状态下具有+1或-1特征值的计算状态来制定通过这些铃铛的botring弦的规则 。我们明确列出了分析过程：对于Pauli Term XI
，如果Ai = 0和-1否则，我们分配了Parity +1；对于Pauli术语 ，如果Ai≠Bi和-1，我们将平等分配+1。对于Pauli术语Zi
，如果BI = 0和-1，我们为奇偶校验+1分配；对于II ，我们总是分配奇偶校验+1
。然后
，分布的贡献由各个平等的产物给出。 　　我们可以执行相同的分析作为所应用的误差检测量的函数 。如扩展数据图9a所示，随着更多的误差检测 ，Pauli期望值的分布预期为零
，而非零的期望值进一步分开
。这也提供了一种自然方法，可以通过零噪声推断执行误差缓解：通过执行滑动尺度误差检测 ，我们可以为具有相同期望值的Pauli Strings组提取Pauli期望值平方
，这是逻辑纯度的函数。我们执行Pauli期望值平方与逻辑纯度的线性拟合，并推断出与零噪声相对应的纯度 ，以估计误差减少值 。线性拟合的选择是由于密度矩阵的功率2的比例2
。我们希望
，使用有关每个操作员的重量的知识，以及每个镜头中检测到的错误与给定的Pauli操作员的重叠是否可以进一步改善减轻错误结果。 　　我们还可以使用参考文献中的Addive Bell Magic Measure来计算与稳定器状态（也称为“魔术
”）的距离的度量 。53 ，仅需要O（1）样品数量和O（n）经典后处理时间
。为此，我们随机对四个测量的贝尔 - 巴西斯斑点r
，r'，q ，q'进行采样
，并使用参考的核对方法来计算其对贝尔魔术的贡献。53：，当两个保利弦通勤时为0 ，否则为2 。表示两个斑点之间的位xor
。p（r）是观察bittring r的概率。Pauli字符串σr为长度为n
，当目标和控制量子位分别读取00、01 、10或11时，ITH元素是i ，x
，z或y 。我们通过公式将此结果转换为添加铃魔法。我们使用大约107个示例来估算每个数据集的添加铃魔法
。估计的添加钟魔法的结果与应用的非克利福德门数的函数（图9F中显示的电路）如图6C所示。这些结果还使用了同一数据集中的纯度估计值，该数据集用于缓解误差 ，如Ref的等式（13） - （15）中所述 。53
。显示的所有添加钟魔术数据均在应用完整的错误检测中。 　　我们在这里执行的相同实验也可以解释为物理铃铛测量 。使用此见解
，在扩展数据图9C，D中 ，我们在分析数据作为物理钟形测量结果并应用不同级别的基于稳定器的后选择后，显示了不同子系统大小的纠缠熵
。值得注意的是
，当对所有稳定器进行后选择时，全系统的奇偶校验是正确的 ，将结果分析为物理电路或逻辑电路时相同。这是因为
，在此限制下，可以将物理电路分析的结果视为采用（不完美的）逻辑状态并运行完美的编码电路 ，因此给出了相同的结果 。