Constant overhead quantum fault-tolerance with quantum expander codes

TL;DR

This paper demonstrates that quantum expander codes can be integrated with fault-tolerance techniques to achieve constant overhead in quantum computing, significantly improving efficiency over previous polylogarithmic overhead methods.

Contribution

It introduces a fault-tolerance scheme using quantum expander codes with a robust, parallelizable, single-shot decoding algorithm, reducing overhead to a constant factor.

Findings

Achieves asymptotically constant overhead for fault-tolerant quantum computation.

Develops a decoding algorithm robust to noisy syndrome measurements.

Decoding algorithm can be parallelized to logarithmic depth.

Abstract

We prove that quantum expander codes can be combined with quantum fault-tolerance techniques to achieve constant overhead: the ratio between the total number of physical qubits required for a quantum computation with faulty hardware and the number of logical qubits involved in the ideal computation is asymptotically constant, and can even be taken arbitrarily close to 1 in the limit of small physical error rate. This improves on the polylogarithmic overhead promised by the standard threshold theorem. To achieve this, we exploit a framework introduced by Gottesman together with a family of constant rate quantum codes, quantum expander codes. Our main technical contribution is to analyze an efficient decoding algorithm for these codes and prove that it remains robust in the presence of noisy syndrome measurements, a property which is crucial for fault-tolerant circuits. We also establish two additional features of the decoding algorithm that make it attractive for quantum computation: it can be parallelized to run in logarithmic depth, and is single-shot, meaning that it only requires a single round of noisy syndrome measurement.