Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes

TL;DR

This paper introduces finite block-length quantum LDPC codes called tricycle codes, enabling efficient, fault-tolerant magic state generation with constant-depth circuits and high noise thresholds, advancing scalable quantum computing.

Contribution

The paper develops a new class of 3D homological quantum LDPC codes supporting constant-depth CCZ gates and single-shot error correction, improving magic state production efficiency.

Findings

High circuit-noise threshold of >0.5% demonstrated

Logical error rates of 6×10^{-10} achieved with small codes

Efficient syndrome extraction protocols for tricycle codes developed

Abstract

The preparation of high-fidelity non-Clifford (magic) states is an essential subroutine for universal quantum computation, but imposes substantial space-time overhead. Magic state factories based on high rate and distance quantum low-density parity check (LDPC) codes equipped with transversal non-Clifford gates can potentially reduce these overheads significantly, by circumventing the need for multiple rounds of distillation and by producing a large number of magic states in a single code-block. As a step towards realizing efficient, fault-tolerant magic state production, we introduce a class of finite block-length quantum LDPC codes which we name tricycle codes, generalizing the well-known bicycle codes to three homological dimensions. These codes can support constant-depth physical circuits that implement logical $CCZ$ gates between three code blocks. To construct these constant-depth $CCZ$ circuits, we develop new analytical and numerical techniques that apply to a broad class of three-dimensional homological and balanced product codes. We further show that tricycle codes enable single-shot state-preparation and error correction, leading to a highly efficient magic-state generation protocol. Numerical simulations of specific codes confirm robust performance under circuit-level noise, demonstrating a high circuit-noise threshold of $>0.5\%$. With modest post-selection, certain tricycle codes of block-lengths of only $50-100$ qubits are shown to achieve logical error-rates of $6\times 10^{-10}$ or lower. Finally, we construct optimal depth syndrome extraction circuits for tricycle codes and present a protocol for implementing them efficiently on a reconfigurable neutral atom platform.