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<p>In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates is usually more challenging and resource-intensive than Clifford gates. Methods such as distilling a high-fidelity magic state from noisy copies or transforming to higher-dimensional codes require a significant qubit count overhead. In this work, we propose a new protocol that combines magic state preparation and code transformation to realize logical non-Clifford operations, which has the potential for reducing the qubit count overhead, provided an efficient decoding algorithm exists for the intermediate non-Abelian code involved. Our approach begins with a special logical state in the <InlineEquation ID="IEq1"><EquationSource Format="TEX">\({{\mathbb{Z}}}_{4}\)</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></EquationSource></InlineEquation> surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian <i>D</i><sub>4</sub> quantum double model. This process ultimately produces a magic state encoded in the <InlineEquation ID="IEq2"><EquationSource Format="TEX">\({{\mathbb{Z}}}_{2}\)</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></EquationSource></InlineEquation> surface code. A logical <i>T</i> gate can be implemented in the standard <InlineEquation ID="IEq3"><EquationSource Format="TEX">\({{\mathbb{Z}}}_{2}\)</EquationSource><EquationSource Format="MATHML"><math><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></EquationSource></InlineEquation> surface code by gate teleportation. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding transformations between topological codes.</p>

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Generating logical magic states with the aid of non-Abelian topological order

  • 期刊论文

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  • 发表日期：

    2026年7月16日

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  • Sheng-Jie Huang,
  • Yanzhu Chen

摘要

In fault-tolerant quantum computing with the surface code, non-Clifford gates are crucial for universal computation. However, implementing these gates is usually more challenging and resource-intensive than Clifford gates. Methods such as distilling a high-fidelity magic state from noisy copies or transforming to higher-dimensional codes require a significant qubit count overhead. In this work, we propose a new protocol that combines magic state preparation and code transformation to realize logical non-Clifford operations, which has the potential for reducing the qubit count overhead, provided an efficient decoding algorithm exists for the intermediate non-Abelian code involved. Our approach begins with a special logical state in the \({{\mathbb{Z}}}_{4}\)Z4 surface code. By applying a sequence of transformations, the system goes through different topological codes, including the non-Abelian D4 quantum double model. This process ultimately produces a magic state encoded in the \({{\mathbb{Z}}}_{2}\)Z2 surface code. A logical T gate can be implemented in the standard \({{\mathbb{Z}}}_{2}\)Z2 surface code by gate teleportation. In our analysis, we employ a framework where the topological codes are represented by their topological orders and all the transformations are considered as topological manipulations such as gauging symmetries and condensing anyons. This perspective is particularly useful for understanding transformations between topological codes.