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  • Open Access

Dense packing of the surface code: Code deformation procedures and hook-error-avoiding gate scheduling

Kohei Fujiu1, Shota Nagayama2, Shin Nishio1,3, Hideaki Kawaguchi1, and Takahiko Satoh4

  • 1Graduate School of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan
  • 2Graduate School of Media Design, Keio University, Yokohama, Kanagawa 223-0061, Japan
  • 3Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
  • 4Faculty of Science and Technology, Keio University, Yokohama, Kanagawa 223-8522, Japan

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Phys. Rev. A 113, 042412 – Published 3 April, 2026

DOI: https://doi.org/10.1103/7lm4-3bnh

Abstract

The surface code is one of the leading quantum error correction codes for realizing large-scale fault-tolerant quantum computing (FTQC). One major challenge in realizing surface-code-based FTQC is the extremely large number of qubits required. To mitigate this problem, fusing multiple code words of the surface code into a densely packed configuration has been proposed. It is known that by using dense packing, the number of physical qubits required per logical qubit can be reduced to approximately three-fourths compared to simply placing surface-code patches side by side. Despite its potential, concrete deformation procedures and quantitative error-rate analyses have remained largely unexplored. In this work, we present a detailed code-deformation procedure that transforms multiple standard surface-code patches into a densely packed, connected configuration, along with a conceptual microarchitecture to utilize this dense packing. We also propose a cnot gate scheduling for stabilizer measurement circuits that suppresses hook errors in the densely packed surface code. We performed circuit-level Monte Carlo noise simulation of densely packed surface codes using this gate scheduling. The numerical results demonstrate that as the code distance of the densely packed surface code increases and the physical error rate decreases, the logical error rate of the densely packed surface code becomes lower than that of the standard surface code. Furthermore, we find that only when employing hook-error-avoiding syndrome extraction can the densely packed surface code achieve a lower logical error rate than the standard surface code, while simultaneously reducing the space overhead.

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Phys. Rev. A 113, 042412– Published 3 April, 2026
  • Received 9 November 2025
  • Revised 29 January 2026
  • Accepted 24 February 2026
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