An approximate description of quantum states

TL;DR

This paper presents an efficient approximate method to describe N-qubit quantum states, enabling precise expectation value estimation of observables with minimal resources, suitable for NISQ devices, and useful for variational eigenvalue problems.

Contribution

It introduces a new approximate state description that allows expectation value estimation with error bounds independent of N, using only single-qubit operations, and demonstrates its application in variational methods.

Findings

Error in expectation value estimation decreases as 1/sqrt(number of preparations)

Method requires only single-qubit rotations and measurements

Applicable to variational determination of minimum eigenvalues

Abstract

We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of $N$. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, $N$-independent, seminorm of the observables. Building the approximate description of the $N$-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements and can be considered for implementation on today's Noisy Intermediate-Scale Quantum (NISQ) computers. The access to the expectation values of all observables for a given state leads to an efficient variational method for the determination of the minimum eigenvalue of an observable. The method represents one example of the practical significance of the approximate description of the $N$-qubit state. We conclude by briefly discussing extensions to generative modelling and with fermionic operators.