Practical optimization for hybrid quantum-classical algorithms

TL;DR

This paper analyzes the performance trade-offs in hybrid quantum-classical variational algorithms, focusing on the impact of measurement precision and optimization methods, and introduces quasi-Newton techniques for improved optimization.

Contribution

It introduces quasi-Newton optimization methods into hybrid variational algorithms and provides a quantitative analysis of precision and optimization choices on algorithm performance.

Findings

Higher measurement precision improves solution quality but requires more repetitions.

Quasi-Newton methods enhance convergence speed in variational optimization.

Performance depends critically on the balance between measurement precision and number of repetitions.

Abstract

A novel class of hybrid quantum-classical algorithms based on the variational approach have recently emerged from separate proposals addressing, for example, quantum chemistry and combinatorial problems. These algorithms provide an approximate solution to the problem at hand by encoding it in the state of a quantum computer. The operations used to prepare the state are not a priori fixed but, quite the opposite, are subjected to a classical optimization procedure that modifies the quantum gates and improves the quality of the approximate solution. While the quantum hardware determines the size of the problem and what states are achievable (limited, respectively, by the number of qubits and by the kind and number of possible quantum gates), it is the classical optimization procedure that determines the way in which the quantum states are explored and whether the best available solution is actually reached. In addition, the quantities required in the optimization, for example the objective function itself, have to be estimated with finite precision in any experimental implementation. While it is desirable to have very precise estimates, this comes at the cost of repeating the state preparation and measurement multiple times. Here we analyze the competing requirements of high precision and low number of repetitions and study how the overall performance of the variational algorithm is affected by the precision level and the choice of the optimization method. Finally, this study introduces quasi-Newton optimization methods in the general context of hybrid variational algorithms and presents quantitative results for the Quantum Approximate Optimization Algorithm.