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

• URL: http://arxiv.org/abs/2411.03683v1
• Date: Wed, 06 Nov 2024 06:06:36 GMT
• Title: Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes
• Authors: Shiro Tamiya, Masato Koashi, Hayata Yamasaki,
• Abstract summary: A major challenge in fault-tolerant quantum computation is to reduce both space overhead and time overhead. We show that a protocol using non-vanishing-rate quantum low-density parity-check codes achieves constant space overhead and polylogarithmic time overhead.
• Score: 2.048226951354646
• License:
• 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.

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