Climbing the Diagonal Clifford Hierarchy

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 to logical gate synthesis is demonstrated by two proofs of concept: the [[2 l+1 − 2, 2, 2]] triorthogonal code family, and the [[2 m , m r , 2 min{r,m−r} ]] quantum Reed-Muller code family.

Transversal Diagonal Logical Operators for Stabiliser Codes

“…A method for generating CSS codes with transversal diagonal logical operators at increasing levels of the Clifford hierarchy is presented in Ref. [19], along with a method to increase the Z-distance of such codes. In Ref.…”

Transversal diagonal logical operators for stabiliser codes

“…In [18], a method of generating circuits using multi-qubit gates which implement arbitrary logical operators at level 2 of the Clifford hierarchy is presented. A method for generating CSS codes with transversal diagonal logical operators at increasing levels of the Clifford hierarchy is presented in [19], along with a method to increase the Z-distance of such codes. In [14], we demonstrated an algorithm for finding all diagonal logical operators composed of single-qubit phase gates which, for CSS codes, involves taking the kernel modulo N of a matrix with n + 2 k columns where n and k are the number of physical and logical qubits respectively.…”