Universal Fault Tolerance with Non-Transversal Clifford Gates

We propose a construction 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^2log(nd^2logn)) ancilla qubits and O(nd^2log(nd^2logn)) extra CNOTs. Beyond asymptotic analysis, we demonstrate our construction by implementing the non-transversal logical Hadamard gate for the [[15,1,3]] 3D color code, which admits a transversal implementation of logical 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.