AI 中文总结
本研究采用代数几何技术构造二元扩展域上的码，提出多量子比特门封装方案与蒸馏协议，其在多数场景下的性能优于现有技术。
AI 中文摘要
容错量子计算架构常因制备高保真魔法态的开销而面临瓶颈。本研究采用代数几何技术构造二元扩展域$\boldsymbol{\text{F}}_{2^s}$上的码，从而发现量子比特魔法态的蒸馏新协议，研究重点聚焦于实用化的基于量子比特的量子计算架构。为此，我们表明$\text{CS}$、$\text{CCZ}$和$\text{TOF}\boldsymbol{\text{\textrm{\num}}}=\text{CCZ}_{123}\text{CCZ}_{345}$等受关注的多量子比特门可被封装为更大域上的简单门，并推导了扩展域中可实现这些门蒸馏的简单代数条件。由于这些门源自伽罗瓦量子位，对应的量子比特码可自然处理这类多量子比特态上存在的相关误差。此外，我们发现的协议极为紧凑：例如，仅用4个逻辑量子比特即可将4个$\text{CS}$态蒸馏为1个距离为2的$\text{CS}$态。作为案例研究，我们考虑从注入的$\text{T}$态和$\text{CS}$态蒸馏$\text{CS}$态与$\text{CCZ}$态。当针对单位时间魔法态产量或逻辑时空体积优化时，我们发现无论是在输入误差率$10^{-3}$（直接注入）还是$10^{-6}$（允许一定预注入培育）的情况下，所提协议在几乎所有场景下均优于现有技术。
英文摘要
Fault-tolerant quantum computation architectures are frequently bottlenecked by the overhead of producing high-fidelity magic states. In this work, we use algebraic geometric techniques to construct codes over binary extension fields $\mathbb{F}_{2^s}$, thus discovering new protocols for the distillation of qubit magic states, where our focus is on the regime of practical qubit-based quantum computing architectures. To do this, we show that multi-qubit gates of interest such as $\text{CS}$, $\text{CCZ}$, and $\text{TOF}\# = \text{CCZ}_{123}\text{CCZ}_{345}$, can be packaged into simple gates over the larger fields, and we derive simple algebraic conditions in the extension fields allowing the distillation of these gates. Because they are derived from Galois qudits, the corresponding qubits codes naturally handle the correlated errors present on such multi-qubit states. Moreover, the protocols we discover are extremely compact; for example, we show that 4 $\text{CS}$ states can be distilled to 1 $\text{CS}$ state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of $\text{CS}$ and $\text{CCZ}$ states from injected $\text{T}$ and $\text{CS}$ states. When optimized for magic state production per unit time, or logical spacetime volume, we find that our protocols outperform the state-of-the-art in almost every situation, both at input error rates $10^{-3}$ (direct injection), and $10^{-6}$ (allowing some cultivation pre-injection).
