A framework to evaluate the performance of Variational Quantum Algorithms

TL;DR

The paper addresses the challenge of benchmarking Variational Quantum Algorithms (VQAs) for Quadratic Unconstrained Binary Optimization (QUBO) on NISQ devices by introducing a framework that combines feasibility, quality, and reproducibility into a single, decision-guiding toolkit. It formalizes VQAs as stochastic maps yielding distributions on a quality diagram and defines quantitative metrics: feasibility $\mathcal{F}[\rho]$, reproducibility $\mathcal{R}[\rho]$ via Shannon entropy $S[\rho]$, and a quality measure $\mathcal{Q}[\rho]$ based on a Heaviside-weighted distance to an ideal point, with a CVaR-based cost function used in the demonstration. The methodology is validated through a proof-of-concept study on a fixed 16-qubit QUBO using RealAmplitudes, CVaR costs with multiple $\alpha$ values, and varying shot counts, showing that higher $\alpha$ and larger $s$ typically improve feasibility and quality, while reproducibility remains relatively stable; the final selection highlights eight high-performing $(\alpha,s)$ configurations. Overall, the framework enables systematic, budget-aware benchmarking and supports adaptive, hybrid quantum-classical workflows for selecting VQAs under resource constraints. The work lays groundwork for broader testing across additional QUBO instances, ansätze, optimizers, and hardware to realize practical, scalable VQA benchmarking in the NISQ era.

Abstract

Variational Quantum Algorithms (VQAs) are promising methods for solving combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices. However, benchmarking VQAs is difficult due to their stochastic behavior and the lack of standardized performance criteria. This work introduces a general framework for evaluating VQAs applied to Quadratic Unconstrained Binary Optimization (QUBO) problems. The framework uses three complementary metrics: feasibility, quality, and reproducibility. It also introduces a quality diagram that visualizes trade-offs between success probability and computational resources. Reproducibility is formalized using Shannon entropy, and a decision rule is defined for selecting algorithms under resource constraints. As a demonstration, the framework is applied to several VQAs using Conditional Value at Risk (CVaR) cost functions and different shot counts on a 16-qubit QUBO instance. The results show how the framework supports systematic benchmarking and provides a foundation for adaptive algorithm selection in hybrid quantum-classical workflows.

Figures (4)

• Figure 1: On the left, the distribution of results obtained by running 400 VQAs to solve a QUBO problem as specified in Section 3. The orange line represents the probability threshold, while the red lines correspond to the level curves of the quality function for $q = 1$, $q = 2$, and $q = 4$ (see Subsection 2.3). On the right, a contour plot is shown for illustrative purposes, representing the underlying distribution of VQA outcomes.
• Figure 2: Scaling of feasibility with $s$ for every cost function CVaR$_\alpha$ considered.
• Figure 3: Scaling of quality with $s$ for every cost function CVaR$_\alpha$ considered.
• Figure 4: Scaling of reproducibility with $s$ for every cost function CVaR$_\alpha$ considered.