Milad Marvian

It has previously been shown that fault-tolerant syndrome can be enabled with flag gadgets. In essence, this relies upon the fact that syndrome extraction circuits are Clifford circuits. In this work we extend our previous framework to produce flag gadgets for syndrome extraction to a framework to flag any Clifford circuit. The construction we present allows a Clifford circuit including n two-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. This framework opens new pathways to universal fault-tolerance, for instance by allowing T gates to be implemented transversally while (a subset of) Clifford gates are fault-tolerantly implemented using flags, or by creating higher-reliability magic states using a flagged preparation circuit.

Abstract We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.