The Dominant Eigenvector of a Noisy Quantum State

TL;DR

This paper investigates the difference between the dominant eigenvector of a noisy quantum state and the ideal state, demonstrating that error suppression techniques can exponentially reduce the impact of this mismatch, with implications for quantum error mitigation.

Contribution

It provides a rigorous analysis of the coherent mismatch in noisy quantum states and extends mathematical bounds to eigenvectors of matrix sums in this context.

Findings

Coherent mismatch is exponentially less severe than fidelity decay.

Error suppression techniques exponentially reduce the impact of noise.

The work extends Weyl inequalities to eigenvectors of matrix sums in quantum states.

Abstract

Although near-term quantum devices have no comprehensive solution for correcting errors, numerous techniques have been proposed for achieving practical value. Two works have recently introduced the very promising Error Suppression by Derangements (ESD) and Virtual Distillation (VD) techniques. The approach exponentially suppresses errors and ultimately allows one to measure expectation values in the pure state as the dominant eigenvector of the noisy quantum state. Interestingly this dominant eigenvector is, however, different than the ideal computational state and it is the aim of the present work to comprehensively explore the following fundamental question: how significantly different are these two pure states? The motivation for this work is two-fold. First, comprehensively understanding the effect of this coherent mismatch is of fundamental importance for the successful exploitation of noisy quantum devices. As such, the present work rigorously establishes that in practically relevant scenarios the coherent mismatch is exponentially less severe than the incoherent decay of the fidelity -- where the latter can be suppressed exponentially via the ESD/VD technique. Second, the above question is closely related to central problems in mathematics, such as bounding eigenvalues of a sum of two matrices (Weyl inequalities) -- solving of which was a major breakthrough. The present work can be viewed as a first step towards extending the Weyl inequalities to eigenvectors of a sum of two matrices -- and completely resolves this problem for the special case of the considered density matrices.