Climbing the Diagonal Clifford Hierarchy

• URL: http://arxiv.org/abs/2110.11923v2
• Date: Wed, 27 Oct 2021 16:30:04 GMT
• Title: Climbing the Diagonal Clifford Hierarchy
• Authors: Jingzhen Hu, Qingzhong Liang, Robert Calderbank
• Abstract summary: We introduce a method of synthesizing codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace.
• Score: 0.6445605125467572
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level $l$ inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level $l+1$ induces the original logical gate. The next step is judicious removal of $Z$-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level $l$ to level $l+1$, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of $Z$-stabilizers may reduce distance, and the purpose of the third basic operation, addition of $X$-stabilizers, is to compensate for such losses. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace by simply applying Pauli $X$ matrices. The approach to logical gate synthesis taken in prior work focuses on the code states, and results in sufficient conditions for a CSS code to be fixed by a transversal $Z$-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach is demonstrated by two proofs of concept: the $[[2^{l+1}-2,2,2]]$ triorthogonal code family, and the $[[2^m,\binom{m}{r},2^{\min\{r,m-r\}}]]$ quantum Reed-Muller code family.

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