Improving quantum measurements by introducing "ghost" Pauli products

TL;DR

This paper introduces a novel method to reduce measurement costs in quantum algorithms by adding 'ghost' Pauli products, which lowers variances of fragments without altering the overall expectation value, leading to fewer required measurements.

Contribution

The paper proposes a new 'ghost' Pauli product technique that minimizes individual fragment variances in quantum measurements, improving efficiency over existing methods.

Findings

Several-fold reduction in measurement numbers demonstrated

Method maintains overall observable expectation value

Numerical tests show significant variance reduction

Abstract

Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.