Best-approximation error for parametric quantum circuits

TL;DR

This paper analyzes the best-approximation error in parametric quantum circuits, proposing methods to optimize circuit design and estimate errors, which are crucial for effective variational quantum simulations.

Contribution

It introduces an inductive construction for candidate circuits, estimates the best-approximation error using Voronoi diagrams, and discusses a hybrid algorithm for error estimation and its implications.

Findings

Voronoi diagrams effectively characterize approximation errors.

Underparametrized circuits face obstacles in variational quantum simulations.

Hybrid algorithms can estimate worst-case approximation errors.

Abstract

In Variational Quantum Simulations, the construction of a suitable parametric quantum circuit is subject to two counteracting effects. The number of parameters should be small for the device noise to be manageable, but also large enough for the circuit to be able to represent the solution. Dimensional expressivity analysis can optimize a candidate circuit considering both aspects. In this article, we will first discuss an inductive construction for such candidate circuits. Furthermore, it is sometimes necessary to choose a circuit with fewer parameters than necessary to represent all relevant states. To characterize such circuits, we estimate the best-approximation error using Voronoi diagrams. Moreover, we discuss a hybrid quantum-classical algorithm to estimate the worst-case best-approximation error, its complexity, and its scaling in state space dimensionality. This allows us to identify some obstacles for variational quantum simulations with local optimizers and underparametrized circuits, and we discuss possible remedies.