Upper bounds on the noise threshold for fault-tolerant quantum computing

Abstract

We prove new upper bounds on the tolerable level of noise in a quantum circuit. Weconsider circuits consisting of unitary k-qubit gates each of whose input wires is subject todepolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentiallynoise-free. We assume that the output of the circuit is the result of measuring somedesignated qubit in the final state. Our main result is that for p > 1 - Θ(1/√k), theoutput of any such circuit of large enough depth is essentially independent of its input,thereby making the circuit useless. For the important special case of k = 2, our bound isp > 35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubitCNOT gate, then our bound becomes 29.3%. These bounds on p are numerically betterthan previous bounds, yet are incomparable because of the somewhat different circuitmodel that we are using. Our main technique is the use of a Pauli basis decomposition,in which the effects of depolarizing noise are very easy to describe.