Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes

TL;DR

This paper demonstrates a fault-tolerant quantum computation protocol that uses quantum LDPC codes and concatenated Steane codes to achieve constant space overhead and polylogarithmic time overhead, improving efficiency.

Contribution

It introduces a novel fault-tolerance protocol combining quantum LDPC and Steane codes with a new partial circuit reduction technique, completing the proof of the threshold theorem.

Findings

Achieves constant space overhead in FTQC.

Attains polylogarithmic time overhead even with classical computation.

Provides a comprehensive comparison of quantum LDPC and code-concatenation approaches.

Abstract

A major challenge in fault-tolerant quantum computation (FTQC) is to reduce both space overhead -- the large number of physical qubits per logical qubit -- and time overhead -- the long physical gate sequences per logical gate. We prove that a protocol using non-vanishing-rate quantum low-density parity-check (LDPC) codes, combined with concatenated Steane codes, achieves constant space overhead and polylogarithmic time overhead, even when accounting for non-zero classical computation time. This protocol offers an improvement over existing constant-space-overhead protocols, which have polynomial time overhead using quantum LDPC codes and quasi-polylogarithmic time overhead using concatenated quantum Hamming codes. To ensure the completeness of this proof, we develop a technique called partial circuit reduction, which enables error analysis for the entire fault-tolerant circuit by examining smaller parts composed of a few gadgets. With this technique, we resolve a previously unaddressed logical gap in the existing arguments and complete the proof of the threshold theorem for the constant-space-overhead protocol with quantum LDPC codes. Our work highlights that the quantum-LDPC-code approach can realize FTQC with a negligibly small slowdown and a bounded overhead of physical qubits, similar to the code-concatenation approach, underscoring the importance of a comprehensive comparison of the future realizability of these two approaches.