# Designing the Quantum Channels Induced by Diagonal Gates

Jingzhen Hu1, Qingzhong Liang1, and Robert Calderbank1,2

1Department of Mathematics, Duke University, Durham, NC 27708, USA
2Department of Electrical and Computer Engineering, Department of Computer Science, Duke University, NC 27708, USA

### Abstract

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.

We have introduced 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. The generator coefficient mathematical framework describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. This interaction depends strongly on the signs of $Z$-stabilizers in a CSS code.

We have derived necessary and sufficient conditions for a diagonal gate to preserve the code space of a CSS code and have provided an explicit expression of its induced logical operator. When the diagonal gate is a transversal $Z$-rotation through an angle $\theta$, we derived a simple global condition that can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. When all signs in the CSS code are positive, we have proved the necessary and sufficient conditions for Reed-Muller component codes to construct families of CSS codes invariant under transversal $Z$-rotation through $\pi/2^l$ for some integer $l$.

The generator coefficient framework provides a tool to analyze the evolution under any given diagonal gate of stabilizer codes with arbitrary signs, and helps to characterize more possible CSS codes can be used in magic state distillation.

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### Cited by

[1] Jingzhen Hu, Qingzhong Liang, Narayanan Rengaswamy, and Robert Calderbank, "Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers", arXiv:2011.00197, (2020).

[2] Jingzhen Hu, Qingzhong Liang, and Robert Calderbank, "Climbing the Diagonal Clifford Hierarchy", arXiv:2110.11923, (2021).

[3] Jingzhen Hu, Qingzhong Liang, and Robert Calderbank, "Divisible Codes for Quantum Computation", arXiv:2204.13176, (2022).

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