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  • Open Access

Low-Overhead Magic State Distillation with Color Codes

Seok-Hyung Lee1,*, Felix Thomsen1, Nicholas Fazio1, Benjamin J. Brown2,3, and Stephen D. Bartlett1,†

  • 1Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
  • 2T. J. Watson Research Center, IBM Quantum, Yorktown Heights, New York 10598, USA
  • 3IBM Denmark, Sundkrogsgade 11, Copenhagen 2100, Denmark

  • *Contact author: seokhyung.lee@sydney.edu.au
  • †Contact author: stephen.bartlett@sydney.edu.au
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PRX Quantum 6, 030317 – Published 30 July, 2025

DOI: https://doi.org/10.1103/ch5r-cnfq

Abstract

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O(p3) for physical noise strength p. To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2×10−16 at p=10−3), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.

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Popular Summary

Quantum computers promise to solve certain problems exponentially faster than classical computers, but they face a major challenge: quantum information is extremely fragile and prone to errors. To protect against these errors, we use quantum error-correcting codes, which encode logical quantum bits into many physical ones. However, implementing certain crucial operations (called non-Clifford gates) while maintaining error protection has been notoriously resource intensive. Our work presents new methods that dramatically reduce these resource requirements for a promising type of error correction called color codes, making fault-tolerant quantum computing significantly more practical.

We developed two schemes for preparing “magic states” (special quantum states that enable these non-Clifford gates) through distillation. Think of magic states as prepackaged quantum resources that can be consumed to perform complex gates reliably. Our first scheme uses a technique called faulty T measurement, which is simple but has limitations on how pure the magic states can be. Our second scheme combines this with “cultivation” (a method for creating high-quality magic states, which is highly resource efficient but not scalable) achieving extremely low error rates. Remarkably, both schemes require about 100 times fewer resources than previous approaches for color codes.

While our schemes make color codes more competitive, they still require more resources than surface codes, the current leading approach. A key challenge lies in the decoding complexity of color codes arising from their complicated structure, which needs further investigation in future work.

Article Text

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Outline

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PRX Quantum 6, 030317– Published 30 July, 2025
  • Received 30 September 2024
  • Revised 21 March 2025
  • Accepted 30 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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