[Paper Review] Practical optimization for hybrid quantum-classical algorithms

Gian Giacomo Guerreschi, Mikhail Smelyanskiy|arXiv (Cornell University)|Jan 5, 2017

Quantum Computing Algorithms and Architecture102 citations

TL;DR

This paper analyzes how finite precision and choice of optimization method affect the performance of variational hybrid quantum-classical algorithms, introduces quasi-Newton methods, and applies them to QAOA with concrete cost estimates.

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.

Motivation & Objective

• Motivate and characterize hybrid quantum-classical algorithms where a classical optimizer tunes quantum gate parameters
• Quantify how finite precision in objective function evaluations and gradients affects repetition costs and optimization performance
• Introduce quasi-Newton optimization approaches in this context and compare gradient-based vs derivative-free methods
• Provide analytical and finite-difference gradient formulations for efficient parameter updates within QAOA
• Demonstrate the framework on the Quantum Approximate Optimization Algorithm (QAOA) for MAX-CUT and discuss practical implications

Proposed method

• Describe the hybrid algorithm structure with state preparation, measurement, and classical optimization
• Express the objective as an observable C and its measurement via a linear combination of Pauli terms
• Quantify repetition cost M needed to estimate F_p(γ) to precision ε and its gradient to precision ε′
• Present finite-difference and analytical gradient evaluation for the objective, including error terms
• Derive gradient components for γ- and β-type parameters in QAOA using circuit-based estimators
• Apply quasi-Newton (BFGS) optimization and compare with derivative-free Nelder-Mead under finite precision
• Specialize formulas and costs for the QAOA algorithm applied to MAX-CUT on random 3-regular graphs

Experimental results

Research questions

• RQ1 How do finite precision and measurement repetitions influence the efficiency of gradient-based optimization in hybrid quantum-classical algorithms?
• RQ2 What are the relative costs and benefits of finite-difference vs analytical gradient evaluation in this setting?
• RQ3 Can quasi-Newton methods provide practical advantages over derivative-free methods for variational quantum circuits?
• RQ4 How does the QAOA performance (in terms of objective value) scale with p and problem instances like MAX-CUT on random graphs?
• RQ5 What are concrete repetition-cost estimates for obtaining gradients and objective evaluations in QAOA?

Key findings

• Gradient-based optimization can outperform derivative-free methods under finite-precision constraints, given accurate gradient information
• Analytical gradient formulas enable efficient estimation with circuit-based measurements and can reduce repetition costs
• Finite-difference gradient estimates introduce additional repetition costs that depend on δ and ε′, influencing overall optimization efficiency
• A lower bound on gradient estimation precision (ε′ ≥ (1/10)ε) keeps the gradient estimation cost from exploding
• QAOA applied to MAX-CUT on random 3-regular graphs demonstrates the framework's applicability and highlights initialization considerations
• Numerical experiments show how precision and initial conditions affect the quality of the found solution and the observed repetition costs