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 thatmore »
Classical Coding Problem from Transversal T Gates
Universal quantum computation requires the implementation of a logical nonClifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T † gates. For example, this could enable magic state distillation with nonCSS codes and, thus, provide better parameters than CSSbased protocols. However, among nondegenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purelyclassical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum ReedMuller codes. This result is generalized to finer angle Zrotations in https://arxiv.org/abs/1910.09333.
 Award ID(s):
 1908730
 Publication Date:
 NSFPAR ID:
 10283649
 Journal Name:
 2020 IEEE International Symposium on Information Theory (ISIT)
 Page Range or eLocationID:
 1891 to 1896
 Sponsoring Org:
 National Science Foundation
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