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IEEE International Conference on Quantum Computing and Engineering (QCE)

Optimal Decoding of 2D Compass Codes Under Coherent Noise

Bálint Pató, Qiang Miao, Kenneth R. Brown

Published Sep 15, 2024• 0 citations
Publisher

Abstract

Coherent noise on the physical qubits can lead to coherent error on the logical qubits in a quantum error-corrected system, even if the best recovery operator is found. An optimal decoder under coherent noise finds the minimal angle of the logical, coherent rotation for a given physical rotation angle under the uniform, coherent noise model. We show that one can optimally decode$2\mathrm{D}$compass codes without an explicit optimal decoder in polynomial complexity by simply flipping the sampled logical angle using a Majorana simulator and a sub-optimal decoder. Estimating the threshold requires tools different from MWPM decoding. As the infidelity curves don't intersect, we use finite-size scaling to evaluate optimal decoding performance by estimating the logical infidelity at the infinite distance limit. We motivate our finite size scaling ansatz based on repetition code-like compass codes for which we can exactly calculate the logical channel up to a distance of up to 1021. Under optimal decoding, our finite-size scaling methods predict a threshold of$(0.274\pm 0.010)\pi$for the surface code and a threshold of$(0.252\pm 0.001)\pi$for the m=2 Z-stacked Shor code. We compare our results to previous predictions and find a slight deviation in the case of the Z-stacked Shor code and a slight decrease in the case of the surface code.

DOI
10.1109/qce60285.2024.10349

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Logical coherence in two-dimensional compass codes

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What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent $X$ rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching $1/2$ relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below $1/2$, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's $X$ distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.

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Coherent noise on the physical qubits can lead to coherent error on the logical qubits in a quantum error-corrected system, even if the best recovery operator is found. An optimal decoder under coherent noise finds the minimal angle of the logical, coherent rotation for a given physical rotation angle under the uniform, coherent noise model. We show that one can optimally decode2D2\mathrm{D}2Dcompass codes without an explicit optimal decoder in polynomial complexity by simply flipping the sampled logical angle using a Majorana simulator and a sub-optimal decoder. Estimating the threshold requires tools different from MWPM decoding. As the infidelity curves don't intersect, we use finite-size scaling to evaluate optimal decoding performance by estimating the logical infidelity at the infinite distance limit. We motivate our finite size scaling ansatz based on repetition code-like compass codes for which we can exactly calculate the logical channel up to a distance of up to 1021. Under optimal decoding, our finite-size scaling methods predict a threshold of(0.274±0.010)π(0.274\pm 0.010)\pi(0.274±0.010)πfor the surface code and a threshold of(0.252±0.001)π(0.252\pm 0.001)\pi(0.252±0.001)πfor the m=2 Z-stacked Shor code. We compare our results to previous predictions and find a slight deviation in the case of the Z-stacked Shor code and a slight decrease in the case of the surface code.
2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the XXX-Shor and ZZZ-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance LLL. However, the current analytical lower bound on the threshold for the rotation angle θθθ is ∣sin⁡(θ)∣<1/L|\sin(θ)| < 1/L∣sin(θ)∣<1/L, which linearly vanishes in LLL instead of being constant. We show that this lower bound is achievable by the ZZZ-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the ZZZ-Shor codes and the XXX-Shor codes.
Two-dimensional compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the XXX-Shor and ZZZ-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance LLL. However, the current analytical lower bound on the threshold for the rotation angle $\ensuremath{\theta}$ is $|sin(\ensuremath{\theta})|<1/L$, which linearly vanishes in LLL instead of being constant. We show that this lower bound is achievable by the ZZZ-Shor code, which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the ZZZ-Shor codes and the XXX-Shor codes.
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and ddd times that infidelity at leading order, for logical dimension ddd. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is βr=E[sin⁡2r(θ/2)]β_r=\mathbb E[\sin^{2r}(θ/2)]βr​=E[sin2r(θ/2)], with θθθ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(βr)O(β_r)O(βr​) exactly when generator powers through degree r−1r-1r−1 are correctable. When the first uncorrectable power has degree m<rm<rm<r, the optimal loss scales as βrm/rβ_r^{m/r}βrm/r​, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
Abstract We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted ZZZ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent ZZZ-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent XXX rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching 1/21/21/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below 1/21/21/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's XXX distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.