Co-design of CSS Codes and Diagonal Gates
The challenge of quantum computing is to combine error resilience with universal computation. There are many finite sets of gates that are universal, and a standard choice is to augment the set of Clifford gates by a non-Clifford unitary such as the T gate. Given a CSS code, we introduce a method of synthesizing all possible diagonal physical gates that preserve the codespace and induce a target logical gate. We denote an [[n, k = k1 - k2, d]] CSS code C by CSS (X, C 2;Z, C1⊥}), where the [n, k2] binary code C 2 determines the X-stabilizers in C, and the [n, n-k1] binary code C1 ⊥ determines the Z-stabilizers in C. The diagonal entries of a diagonal physical gate are indexed by binary vectors in F 2n. We show that a diagonal physical gate preserves the CSS codespace if and only if entries from the same coset of C 2 in C1 (same X-logical) are identical. We also show that the target logical operator only specifies 2k1 out of 2n diagonal entries of the diagonal physical gate. The remaining degrees of freedom can be used to optimize implementation of the physical gate within a particular quantum computing infrastructure. This encompasses optimization with respect to locality of the physical gate, a criterion that is essential to fault tolerance. When the target logical operator is the identity, the physical gates that preserve the CSS code represent noise operators to which the codespace is oblivious. We illustrate our method by providing several examples of code-gate pairs for which the target logical gate is a non-Clifford unitary. The framework is extended to stabilizer codes in https://arxiv.org/abs/2109.13481.
