The application of weight parity error correction in quantum codes

Masahito Hayashi

Theerapat Tansuwannont, Debbie Leung

Current fault-tolerant quantum computation protocols require the preparation of a large number of auxiliary qubits. In contrast, the ``weight parity error correction'' introduced here is a remarkably qubit-efficient technique for correcting even high-weight errors in some stabilizer codes, provided that the weight parity is known.

황규호

Nicolas Delfosse, Ben W. Reichardt, Krysta M. Svore

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 requires$\Omega (r d^{2})$parity check measurements for a distance-$d$code defined by$r$parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only$r$measurements. 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 rate$p$. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using$O(d \log (d))$measurements for any code with distance$d \geq \Omega (n^\alpha)$for some constant$\alpha > 0$. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than$r$measurements. 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.

Pavel Bažant

Shi Jie Samuel Tan, Lev Stambler

Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. \cite{evra2022decodable} preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.