具有横向T门的改进量子码
Improved Quantum Codes with Transversal T Gates
AI总结：
本工作开发可除递减单项式码框架，构造出具有横向$T$门、恒定码率与增长码距且魔法态蒸馏开销指数趋近于0的改进量子码，拓宽了相关码族的参数范围。
AI中文摘要：
本工作研究具有横向$T$门的量子CSS码，此处的$T$门横向性取最强定义：对每个物理量子比特施加物理$T$门，无需Clifford修正即可得到每个逻辑量子比特上的逻辑$T$门。尽管$T$门在容错量子计算中至关重要，但自Hastings与Haah 2017年的工作及Haah 2018年的工作以来，这类码的渐近族参数未得到改进。本工作显著拓宽了具有横向$T$门的量子码族的可实现参数，既扩展了可实现多项式码率与码距的范围，又构造了具有恒定码率和增长码距的此类码；即使允许在横向$T$门后进行Clifford修正，这也是首次实现后一目标。这些码也是首次实现具有横向$T$门的码的$\boldsymbol{\to}0$（其中$\boldsymbol{\to}$为魔法态蒸馏的开销指数）。为实现这一点，我们开发了可除递减单项式码框架，在布尔超立方体上的向下闭集处打孔以创建逻辑量子比特。我们证明了此类在该集合处打孔的码的码距的闭式表达式，这可能具有独立意义。我们首先基于加权Reed-Muller码进行实例化，在低汉明重量点处打孔，然后进行随机构造，其中一小部分随机点受打孔保护以节省量子码距，从而实现改进的参数。
英文摘要：
In this work, we study quantum CSS codes with transversal $T$ gates. Here, $T$ gate transversality is meant in the strongest sense; the application of physical $T$ to every physical qubit yields logical $T$ on every logical qubit, without Clifford corrections. Despite the importance of the $T$ gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal $T$ gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal $T$ gate. These are also the first codes achieving $γ\to 0$ for a code with a transversal $T$ gate, where $γ$ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.
