Expectation value estimation with parametrized quantum circuits

TL;DR

This paper introduces a framework for efficiently estimating expectation values of quantum observables using shallow parametrized circuits, combining decomposition algorithms and importance sampling to reduce sample complexity.

Contribution

It presents a novel approach that optimizes sample complexity for expectation estimation with shallow circuits, including new decomposition algorithms and a fundamental lower bound analysis.

Findings

Demonstrates improved expectation value estimation for specific Hamiltonians.

Shows advantages over conventional methods in numerical simulations.

Derives a lower bound on sample complexity for shallow quantum circuits.

Abstract

Estimating properties of quantum states, such as fidelities, molecular energies, and correlation functions, is a fundamental task in quantum information science. Due to the limitation of practical quantum devices, including limited circuit depth and connectivity, estimating even linear properties encounters high sample complexity. To address this inefficiency, we propose a framework that optimizes sample complexity for estimating the expectation value of any observable using a shallow parameterized quantum circuit. Within this framework, we introduce two decomposition algorithms, a tensor network approach and a greedy projection approach that decompose the target observable into a linear combination of multiple observables, each of which can be diagonalized with the shallow circuit. Using this decomposition, we then apply an importance sampling algorithm to estimate the expectation value of the target observable. We numerically demonstrate the performance of our algorithm by estimating the expectation values of some specific Hamiltonians and inner product of a Slater determinant with a pure state, highlighting advantages compared to some conventional methods. Additionally, we derive the fundamental lower bound for the sample complexity required to estimate a target observable using a given shallow quantum circuit, thereby enhancing our understanding of the capabilities of shallow circuits in quantum learning tasks.