Designing Quantum Channels Induced by Diagonal Gates

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. We introduce 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). The framework 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 on the signs of $Z$-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. We derive necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provide an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate, the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find applications in magic state distillation and elsewhere. When all the signs are positive, we characterize all possible CSS codes, invariant under transversal $Z$-rotation through $\pi/2^l$, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on $l$. According to the divisibility conditions, we construct new families of CSS codes using cosets of the first order Reed-Muller code defined by quadratic forms. The generator coefficient framework extends to arbitrary stabilizer codes but the more general class of non-degenerate stabilizer codes does not bring advantages when designing the code parameters.

Relying on the generator coefficient framework, we introduce a method of synthesizing CSS codes that realizes 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. 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 $[[2^m, {m \choose r} ,2^{\min\{r,m-r\}}]]$ quantum Reed-Muller code family.