Upper Bounds on the Noise Threshold for Fault-tolerant Quantum Computing

TL;DR

This paper establishes new upper bounds on the maximum noise level tolerable in fault-tolerant quantum circuits, showing that beyond certain noise thresholds, quantum computations become effectively useless.

Contribution

It introduces improved upper bounds on noise thresholds for quantum circuits with depolarizing noise, using a Pauli basis decomposition technique.

Findings

For k=2, noise threshold p > 35.7%.

For circuits with only CNOT gates, threshold p > 29.3%.

Bounds are tighter than previous results within the considered model.

Abstract

We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p>1-\Theta(1/\sqrt{k}), the output 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 is p>35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds.