Designing the Quantum Channels Induced by Diagonal Gates

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.

Jingzhen Hu and Qingzhong Liang contribute equally to this work. The paper was accepted to Quantum on 31 Aug 2022

References in corpus (10)

• Restrictions on Transversal Encoded Quantum Gate Sets
• Magic state distillation with low overhead
• Magic state distillation in all prime dimensions using quantum Reed-Muller codes
• Improved magic states distillation for quantum universality
• Diagonal gates in the Clifford hierarchy
• Morphing quantum codes
• Order 3 Symmetry in the Clifford Hierarchy
• Classification of Small Triorthogonal Codes
• Optimizing Stabilizer Parities for Improved Logical Qubit Memories
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers

Cited by in corpus (5)

• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers
• Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code
• Climbing the Diagonal Clifford Hierarchy
• Geometric structure and transversal logic of quantum Reed-Muller codes
• Transversal Clifford and T-gate codes of short length and high distance