The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal $T$ gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of $Z$-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal $Z$-rotation through $\pi/2^l$, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on $l$. The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes.

中文翻译：

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设计由对角门诱导的量子通道

量子计算的挑战是将错误弹性与通用计算相结合。诸如横向$T$门之类的对角门在实现一组通用的量子操作中起着重要作用。本文介绍了一个框架，该框架描述了准备代码状态、应用对角物理门、测量代码校正子以及应用可能取决于测量校正子（由任意对角门引起的平均逻辑信道）的泡利校正的过程. 它侧重于 CSS 代码，并根据由诱导逻辑运算符确定的生成器系数来描述代码状态和​​物理门的交互。代码状态和​​对角门的交互在很大程度上取决于 CSS 代码中 $Z$-stabilizers 的符号，并且提出的生成器系数框架明确地包括了这种自由度。推导了任意对角门保持稳定器码的码空间的充要条件，并给出了诱导逻辑算子的显式表达。当对角门是二次形式的对角门（由 Rengaswamy 等人引入）时，条件可以用确定 CSS 代码的两个经典代码中的权重可分性来表示。这些代码在魔法状态蒸馏和其他地方都有应用。当所有符号为正时，本文描述了所有可能的 CSS 代码，在通过 $\pi/2^l$ 的横向 $Z$-rotation 下不变，这些 CSS 代码由经典 Reed-Muller 代码通过推导必要和充分约束来构造$l$。