Beyond transversality: structure of Clifford circuits for CSS codes

Abstract

We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 136 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We find 78 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, and 12, with respective rates $2/5$, $3/4$, $1/9$, $1/5$, $2/5$, and $3/56$. We construct three families of CSS codes from bipartite grids, cut-complements, and quadrics, many of which realize the full logical Clifford group in this way. More generally, the induced logical group can be large even when it is not full logical Clifford group: it has order at least $460\,800$ for the gross code and roughly $10^{26}$ for a clustered-cyclic code.