Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of classical linear codes can be applied to quantum stabilizer codes after some modifications. Some necessary and/or sufficient conditions for the stabilizer code over GF(2) is degenerate or nondegenerate for Pauli channel based on the relationship between the error syndrome matrices and the check matrix was presented. A new way to find the minimum distance of the quantum stabilizer codes based on their check matrices was presented, and followed from which we proved that the performance of degenerate quantum code outperform (at least have the same performance) nondegenerate quantum code for Pauli channel.
Stabilizer codes are a fundamental class of error-correcting codes for quantum information that allow for syndrome decoding in the quantum domain. One of the substantial challenges in quantum error correction is that the quantum gates that perform error correction themselves are faulty in practice, which makes fault-tolerant implementation a vital component. Recently, a coding theoretic technique is proposed that makes it possible for syndrome decoding to help correct imperfectly extracted syndromes instead of fully relying on an external fault-tolerant mechanism. In particular, it was proved that any single-error-correcting stabilizer code can be made robust against single errors on either a data qubit or a syndrome bit by using at most one more stabilizer operator than necessary for standard syndrome decoding, while analogous overhead for making double-error-correcting stabilizer codes robust against double errors involving data qubits and/or syndrome bits was shown to be at most logarithmic. We generalize this result to t-error-correcting stabilizer codes and show that the overhead for achieving analogously defined global t-error correction for data qubits and/or syndrome bits is also at most logarithmic. The proof exploits combinatorial arrays that may be seen as parity-check matrices that detect errors of even weight but may overlook errors of odd weight.
In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not take into account error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes is computationally much harder than optimal decoding of classical linear codes, it is #P.
In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t*-m, where the parameterm, 0 ≤m≤ 2t*, depends on a property of the parity-check matrix. Asmdecreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed-Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters.
High-fidelity decoding of quantum error correction codes relies on an accurate experimental model of the physical errors occurring in the device. Because error probabilities can depend on the context of the applied operations, the error model is ideally calibrated using the same circuit as is used for the error correction experiment. Here, we present an experimental approach guided by an analytical formula to characterize the probability of independent errors using correlations in the syndrome data generated by executing the error correction circuit. Using the method on a distance-three surface code, we analyze error channels that flip an arbitrary number of syndrome elements, including Pauli Y ̂ errors, hook errors, multiqubit errors, and leakage, in addition to standard Pauli X ̂ and Z ̂ errors. We use the method to find the optimal weights for a minimum-weight perfect matching decoder without relying on a theoretical error model. Additionally, we investigate whether improved knowledge of the Pauli Y ̂ error channel, based on correlating the X- and Z-type error syndromes, can be exploited to enhance matching decoding. Furthermore, we find correlated errors that flip many syndrome elements over up to eight cycles, potentially caused by leakage of the data qubits out of the computational subspace. The presented method provides the tools for accurately calibrating a broad family of decoders, beyond the minimum-weight perfect matching decoder, without relying on prior knowledge of the error model.
Dissipative cat qubits exponentially suppress one Pauli error channel with the mean photon number, leaving the conjugate bit-flip error as the dominant failure mode. This strong noise bias makes the full machinery of general quantum error correction unnecessary: a code need only protect against a single error type, and any classical linear code can be promoted to a Clifford stabilizer code that does exactly this. We use this observation to build a bit-flip-only quantum Reed-Solomon (RS) code. Starting from the maximum-distance-separable RS code [7, 3, 5] over GF(
23), we binary-expand it to the linear code [21, 9, 6] over GF(2) and realize it as a [[21, 9,
dX=6,dZ=1]] bit-flip code whose stabilizers are products of
Z operators. Because no phase-flip correction is attempted, the construction discards the redundancy that standard quantum RS codes spend on correcting
Z errors -- which a strongly biased cat qubit renders unnecessary -- and yields a shallow Clifford circuit that samples directly in Stim. Errors are decoded by an optimal bounded-distance syndrome-lookup table. We then introduce a Tornado architecture: a two-layer concatenation that wraps every position of the outer RS code in an inner distance-three repetition code, yielding a [[63, 9, 18]] code decoded by a two-stage inner majority vote and outer lookup decoder. Monte-Carlo simulation shows that at a physical bit-flip rate
p=0.1 the Tornado code reaches a logical error rate
pL≈5.3×10−3, below both parent codes, and that its logical error rate scales as
pL∝p6 at low
p, in contrast to
p2 for the repetition code and
p3 for the standalone RS code. We give the exact construction, the error and circuit model, an asymptotic scaling analysis, and an honest account of the overhead cost and single-shot assumptions.