Climbing the diagonal Clifford hierarchy

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. The approach to logical gate synthesis taken in prior work focuses on the code states, resulting 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.