Glassy Noise in Quantum Phase Estimation Algorithm

Abstract

Quantum phase estimation is a subroutine that allows us to determine the eigenvalue - a phase - corresponding to an eigenvector of a unitary operator, with applications in several essential quantum algorithms. We investigate the response to noise, in the form of glassy disorder, present in the circuit elements, in the success probability of this algorithm. We examine the behavior of the probability of estimating the correct phase in response to the inflicted disorder. We prove that when a large number of auxiliary qubits are involved, the probability does not depend on the actual form of disorder distribution but only on the strength of disorder. As examples, we analyze our model for three disorder distributions: Haar-uniform with a finite circular cut-off, Haar-uniform with an elliptical (squeezed) cutoff, and spherical normal. There is generally a depreciation of the quenched-average success probability in response to the disorder incorporation. In presence of disorder, using a higher number of qubits may not help in getting a better precision, unlike in the clean case. However, the effect of noise is more prominent for a larger number of auxiliary qubits. We find a concave to convex transition in the dependence of probability on the degree of disorder. A log-log dependence is witnessed between the strength of disorder at the point of inflection and the number of auxiliary qubits used. Furthermore, we find that the phase estimation process does not depend on the direction of squeezing of the region of glassy disorder on the Bloch sphere.