Universal Fault Tolerance with Non-Transversal Clifford Gates

We propose a scheme for the fault-tolerant implementation of arbitrary Clifford circuits. To achieve this, we extend previous work on flag gadgets for syndrome extraction to a general framework that flags any Clifford circuit. This framework opens new pathways toward universal fault tolerance by allowing transversal implementation of $T$ gates alongside fault-tolerant realization of selected non-transversal Clifford gates using flags. The construction we present allows a Clifford circuit consisting of $n$ two-qubit gates and $O(n)$ single-qubit gates acting upon physical qubits in a code of distance $d$ to be made fault tolerant to distance $d$ using $O(d^2 \log(nd^2\log n))$ ancilla qubits and $O(nd^2 \log(nd^2 \log n))$ extra CNOTs. Beyond asymptotic analysis, we demonstrate our construction by implementing the non-transversal logical Hadamard gate for the [[15,1,3]] code, which has transversal T, and compare to alternative approaches for universality using this code. We also apply our construction to magic-state preparation, general state preparation using Clifford circuits, and data-syndrome codes.