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Magic Tricycles: Efficient Magic-State Generation with Finite Block-Length Quantum LDPC Codes

Varun Menon*,†, J. Pablo Bonilla Ataides*, Rohan Mehta, Andi Gu, Daniel Bochen Tan, and Mikhail D. Lukin

  • *These authors contributed equally to this work.
  • †Contact author: varunmenon@g.harvard.edu
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Phys. Rev. X 16, 021014 – Published 15 April, 2026

DOI: https://doi.org/10.1103/ghhp-cytl

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 toward 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 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 postselection, certain tricycle codes of block lengths of only approximately 50–100 qubits are shown to produce |CCZ¯⟩ magic states with logical error rates ranging from 2×10−8 to below 3×10−11. 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.

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Generating high-fidelity non-Clifford magic states remains a primary bottleneck for universal fault-tolerant quantum computation due to the substantial space-time overhead required by traditional distillation protocols. We introduce tricycle codes, a class of quantum low-density parity-check codes generated from a three-dimensional homological product with high finite block-length performance. By leveraging mathematical properties inherent to the structure of these codes, we show that tricycle codes support constant-depth physical circuits that implement logical CCZ gates between code blocks. Our simulations demonstrate a high circuit-noise threshold exceeding 0.5% and the ability to produce magic states with logical error rates as low as 3×10−11 using blocks of only 108 qubits. We also provide a hardware-efficient protocol for single-shot state preparation on reconfigurable neutral-atom platforms. These high-rate factories offer a scalable pathway to reducing the resource costs of universal quantum algorithms.

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Phys. Rev. X 16, 021014– Published 15 April, 2026
  • Received 13 November 2025
  • Revised 23 February 2026
  • Accepted 3 March 2026
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