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- W3209600147 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." @default.
- W3209600147 created "2021-11-08" @default.
- W3209600147 creator A5009473808 @default.
- W3209600147 creator A5016897646 @default.
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- W3209600147 date "2021-10-22" @default.
- W3209600147 modified "2023-10-16" @default.
- W3209600147 title "Climbing the Diagonal Clifford Hierarchy" @default.
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- W3209600147 doi "https://doi.org/10.48550/arxiv.2110.11923" @default.
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