Abstract Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code’s measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.
Abstract Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code’s measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.
Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code's measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.
Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their X components and are therefore single shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of 3.08(4)% and 2.90(2)% for three-dimensional (3D) surface and toric codes, respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed family of 3D homological product codes.
We first provide a simple proof showing that any single-shot code is self-correcting. We then show that any [[n,k]] CSS quantum low-density parity-check (QLDPC) code can be transformed into an [[nm,k]] CSS QLDPC code with single-shot (SS) quantum error correction and an efficient local decoder which runs in O(logm) parallel time. Below a constant threshold pRG>0 for joint local stochastic physical and measurement noise, the logical failure probability over T correction rounds is bounded by \begin{equation*}
O(T n)\exp[-Ω(m^α)] \end{equation*} for a constant α>0; we say that the \textit{threshold} is pRG and \textit{error suppression} is Ω(mα). Moreover, if the input code is equipped with a single-shot decoder with threshold pc>0 and error suppression Ω(nβ), then the output code \textit{enhances} the error suppression to Ω(mαnβ), while maintaining the decoder-independent threshold pRG, and thus \textit{enhanced} if pc<pRG.
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requiresΩ(rd2)parity check measurements for a distance-dcode defined byrparity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using onlyrmeasurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error ratep. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved usingO(dlog(d))measurements for any code with distanced≥Ω(nα)for some constantα>0. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer thanrmeasurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.
Abstract Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.
Quantum processors often exhibit biased noise (dominantZerrors) and noisy readout, both of which degrade reliability and reproducibility. This work unifies two complementary strategies: bias tailoring, which realigns stabilizers to the dominant error basis, and single-shot (SS) decoding, which leverages metachecks to identify measurement faults from a single noisy round. We instantiate this in a four-dimensional lifted hypergraph-product (4D-LHP) LDPC code built from quasi-cyclic protograph seeds. Simulations show that tailoring reduces word-error rate (WER) by 20-60% across realisticZ:Xbias ratios (1:1 to 103:1), with the largest gains at moderate bias. Under realisticq=4%measurement noise, a single SS round recovers∼33%of the performance lost to readout faults; metachecks detect > 99.8% of faulty syndromes, ffering near-complete fault visibility despite modest correction capability. These results indicate that 4D-LHP codes offer practical resilience under realistic noise, making them well-suited for integration into orchestrated QPU-CPU workflows.