Quantum Error Correction Using Low-Density Parity-Check Codes and Erasure Qubits

Citation

Gu, Shouzhen (2024) Quantum Error Correction Using Low-Density Parity-Check Codes and Erasure Qubits. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5pj8-wv34. https://resolver.caltech.edu/CaltechTHESIS:05172024-212658426

Abstract

Quantum error correction is a method to reduce the effective error rate on quantum computers so that they can be used to carry out useful computation. In this thesis, we study two main problems: decoding quantum low-density parity-check codes and using erasure qubits to implement error correction protocols.

In the first part of this thesis, we focus on quantum low-density parity-check codes, which are a promising approach to reducing the spacetime overhead associated with error correction. We show that certain families of codes with constant rate and linear distance can be decoded efficiently. In particular, we propose a linear-time algorithm that will correct any error affecting at most a constant fraction of the qubits.

We also analyze the setting where the measurement outcomes given to the decoder can be corrupted. In this more realistic scenario, the decoder is shown to have the single-shot property. Using one round of noisy syndrome data, it can output a correction that is close to the data error as long as at most a constant fraction of the data qubits and syndrome bits are flipped. As a consequence, the decoder can operate under a stochastic noise model where errors occur with sufficiently small but constant probability.

In the second part of the thesis, we analyze quantum error-correcting codes implemented using erasure qubits. The idea behind erasure qubits is to bias the noise into a form where likely locations of errors are known, for example, by converting the dominant noise source into detectable leakage from the computational subspace. We provide a formalism for simulating and decoding stabilizer circuits with erasures, erasure checks, and resets. Using this formalism, we study the performance of Floquet codes and show that the benefits of knowing error locations outweigh the cost of extra noise due to erasure checks.

Lastly, we optimize erasure check schedules in the context of the surface code. By performing simulations with one, two, or four erasure checks per syndrome extraction round, we find different error parameter regimes where it is optimal to use each schedule. Additionally, we provide a simplified way of decoding erasure circuits suitable for circuits with infrequent erasure checks.

Thesis Files