We discuss single-shot decoding of quantum Calderbank-Shor-Steane codes with faulty syndrome measurements. We state the problem as a joint source-channel coding problem. By adding redundant rows to the code’s parity-check matrix, we obtain an additional syndrome error correcting code which addresses faulty syndrome measurements. Thereby, the redundant rows are chosen to obtain good syndrome error correction capabilities while keeping the stabilizer weights low. Optimal joint decoding rules are derived which, though too complex for general codes, can be evaluated for short quantum codes.
We discuss single-shot decoding of quantum Calderbank-Shor-Steane codes with faulty syndrome measurements. We state the problem as a joint source-channel coding problem. By adding redundant rows to the code’s parity-check matrix, we obtain an additional syndrome error correcting code which addresses faulty syndrome measurements. Thereby, the redundant rows are chosen to obtain good syndrome error correction capabilities while keeping the stabilizer weights low. Optimal joint decoding rules are derived which, though too complex for general codes, can be evaluated for short quantum codes.
Abstract The syndrome decoding problem is known to be NP-complete. The goal of the decoder is to find an error of low weight that corresponds to a given syndrome obtained from a parity-check matrix. We use the quantum approximate optimization algorithm (QAOA) to address the syndrome decoding problem with elegantly-designed reward Hamiltonians based on both generator and check matrices for classical and quantum codes.We evaluate the level-4 check-based QAOA decoding of the [7,4,3] Hamming code, as well as the level-4 generator-based QAOA decoding of the [[5,1,3]] quantum code.Remarkably, the simulation results demonstrate that the decoding performances match those of the maximum likelihood decoding. Moreover, we explore the possibility of enhancing QAOA by introducing additional redundant clauses to a combinatorial optimization problem while keeping the number of qubits unchanged. Finally, we study QAOA decoding of degenerate quantum codes.Typically, conventional decoders aim to find a unique error of minimum weight that matches a given syndrome. However, our observations reveal that QAOA has the intriguing ability to identify degenerate errors of comparable weight, providing multiple potential solutions that match the given syndrome with comparable probabilities.This is illustrated through simulations of the generator-based QAOA decoding of the [[9,1,3]] Shor code on specific error syndromes.
We optimize fault-tolerant quantum error correction to reduce the number of syndrome bit measurements. Speeding up error correction will also speed up an encoded quantum computation, and should reduce its effective error rate. We give both code-specific and general methods, using a variety of techniques and in a variety of settings. We design new quantum error-correcting codes specifically for efficient error correction, e.g., allowing single-shot error correction. For codes with multiple logical qubits, we give methods for combining error correction with partial logical measurements. There are tradeoffs in choosing a code and error-correction technique. While to date most work has concentrated on optimizing the syndrome-extraction procedure, we show that there are also substantial benefits to optimizing how the measured syndromes are chosen and used. As an example, we design single-shot measurement sequences for fault-tolerant quantum error correction with the 16-qubit extended Hamming code. Our scheme uses 10 syndrome bit measurements, compared to 40 measurements with the Shor scheme. We design single-shot logical measurements as well: any logical Z measurement can be made together with fault-tolerant error correction using only 11 measurements. For comparison, using the Shor scheme a basic implementation of such a non-destructive logical measurement uses 63 measurements. We also offer ten open problems, the solutions of which could lead to substantial improvements of fault-tolerant error correction.
Quantum stabilizer codes often struggle with syndrome errors due to measurement imperfections. Typically, multiple rounds of syndrome extraction are employed to ensure reliable error information. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between extractions, and each measurement can be faulty. We introduce generalized quantum data-syndrome codes along with a generalized check matrix that integrates both quaternary and binary alphabets to represent diverse error sources. This results in a Tanner graph with mixed variable nodes, enabling the design of belief propagation (BP) decoding algorithms that effectively handle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations, we achieve an error threshold of more than 3% for quantum memory protected by rotated toric codes, using solely BP without post-processing. Our results indicate that d rounds of syndrome extraction are sufficient for a toric code of distance d. We observe that at high error rates, fewer rounds of syndrome extraction tend to perform better, while more rounds improve performance at lower error rates. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Our simulations show that BP decoding remains highly effective even with a high syndrome error rate.
We explore the effect of Shor-state construction methods on logical state encoding and quantum error-correction for the [7,1,3] Calderbank-Shor-Steane quantum error-correction code in a nonequiprobable-error environment. We determine the optimum number of verification steps to be used in Shor-state construction and whether Shor states without verification are usable for practical quantum computation. These results are compared to the same processes of encoding and error correction where Shor states are not used. We demonstrate that the construction of logical zero states with no first-order error terms may not require the complete edifice of quantum fault tolerance. With respect to error correction, we show for a particular initial state that error correction using a single qubit for the syndrome measurement yields a similar output-state accuracy to error correction using Shor states as syndrome qubits. In addition, we demonstrate that error correction with Shor states has an inherent sensitivity to bit-flip errors. Finally, we suggest that in this type of error-correction scenario one should always repeat a syndrome measurement until attaining an all-zero readout (twice in row).
Quantum stabilizer codes often face the challenge of syndrome errors due to error-prone measurements and multiple rounds of syndrome extraction are typically employed. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between two syndrome extractions, and each syndrome measurement can be faulty. To handle these diverse error sources, we define a generalized check matrix over mixed quaternary and binary alphabets to characterize their error syndromes. This generalized check matrix leads to the creation of a Tanner graph comprising quaternary and binary variable nodes, which facilitates the development of belief propagation (BP) decoding algorithms to tackle phenomenological errors. Additionally, our BP decoders are applicable to general sparse quantum codes.
Quantum errors are primarily detected and corrected using the measurement of syndrome information which itself is an unreliable step in practical error correction implementations. Typically, such faulty or noisy syndrome measurements are modeled as a binary measurement outcome flipped with some probability. However, the measured syndrome is in fact a discretized value of the continuous voltage or current values obtained in the physical implementation of the syndrome extraction. In this paper, we use this "soft" or analog information without the conventional discretization step to benefit the iterative decoders for decoding quantum low-density parity-check (QLDPC) codes. Syndrome-based iterative belief propagation decoders are modified to utilize the syndrome-soft information to successfully correct both data and syndrome errors simultaneously, without repeated measurements. We demonstrate the advantages of extracting the soft information from the syndrome in our improved decoders, not only in terms of comparison of thresholds and logical error rates for quasi-cyclic lifted-product QLDPC code families, but also for faster convergence of iterative decoders. In particular, the new BP decoder with noisy syndrome performs as good as the standard BP decoder under ideal syndrome.