[Paper Review] Variational Quantum Eigensolver with Constraints (VQEC): Solving Constrained Optimization Problems via VQE

Thinh Viet Le, Vassilis Kekatos|arXiv (Cornell University)|Nov 14, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes VQEC, a hybrid quantum-classical algorithm that extends the Variational Quantum Eigensolver (VQE) to solve constrained optimization problems by jointly optimizing primal quantum circuit parameters and dual Lagrange multipliers via a perturbed primal-dual method. It achieves high-quality solutions for QCBO, stochastic binary policies, and large-scale linear programs on the probability simplex, with numerical results showing improved convergence and solution quality using repeated parameter circuits.

ABSTRACT

Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks. Nonetheless, enforcing constraints in a disciplined fashion has been largely unexplored. To address this gap, this work proposes a hybrid quantum-classical algorithmic paradigm termed VQEC that extends the celebrated VQE to handle optimization with constraints. As with the standard VQE, the vector of optimization variables is captured by the state of a variational quantum circuit (VQC). To deal with constraints, VQEC optimizes a Lagrangian function classically over both the VQC parameters as well as the dual variables associated with constraints. To comply with the quantum setup, variables are updated via a perturbed primal-dual method leveraging the parameter shift rule. Among a wide gamut of potential applications, we showcase how VQEC can approximately solve quadratically-constrained binary optimization (QCBO) problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs (LP) over the probability simplex. Under an assumption on the error for the VQC to approximate an arbitrary probability mass function (PMF), we provide bounds on the optimality gap attained by a VQC. Numerical tests on a quantum simulator investigate the effect of various parameters and corroborate that VQEC can generate high-quality solutions.

Motivation & Objective

• To address the lack of systematic constraint enforcement in variational quantum algorithms, especially for NISQ-era quantum hardware.
• To develop a principled method for incorporating constraints into VQE by leveraging Lagrangian duality and primal-dual optimization.
• To enable the solution of constrained problems such as quadratically-constrained binary optimization (QCBO), stochastic policy design, and large-scale linear programs over the probability simplex.
• To analyze the performance gap between the quantum variational solution and the true optimal solution under VQC approximation error assumptions.
• To evaluate the algorithm's robustness and convergence using quantum simulation on IBM's simulator across diverse problem classes.

Proposed method

• VQEC formulates the constrained optimization problem using a Lagrangian function that combines the objective and constraints via dual variables.
• It performs classical optimization over both the variational quantum circuit (VQC) parameters and the dual Lagrange multipliers using a perturbed primal-dual (PPD) method.
• The PPD method updates primal and dual variables at perturbed points to ensure convergence without requiring strict convexity/concavity of the Lagrangian.
• Quantum measurements of cost and constraint observables are performed simultaneously to minimize overhead, leveraging the parameter shift rule for gradient estimation.
• The method supports repeated parameterization (L > 1) of the VQC to enhance expressivity and solution quality.
• The algorithm is applied to diagonal observable problems, including QCBO, chance-constrained binary policies, and large-scale linear programs over the probability simplex.

Experimental results

Research questions

• RQ1 Can a hybrid quantum-classical algorithm effectively enforce constraints in variational quantum optimization without relying on penalty methods?
• RQ2 How does the perturbed primal-dual method improve convergence and solution quality compared to standard primal-dual or penalty-based approaches in constrained VQE?
• RQ3 To what extent does repeated parameterization of the VQC enhance the solution quality of VQEC for constrained problems?
• RQ4 What is the optimality gap of VQEC solutions under the assumption that the VQC can approximate the target probability mass function within a given error bound?
• RQ5 How does VQEC perform on large-scale linear programs over the probability simplex using a quantum simulator?

Key findings

• VQEC with L=3 repeated VQC parameters achieved relative errors below 20% on 8 out of 9 tested instances of large-scale LPs, significantly outperforming L=1.
• The perturbed primal-dual (PPD) method reduced relative error by up to 50% compared to standard PD methods with only one additional measurement shot per iteration.
• VQEC successfully generated high-quality solutions for QCBO and stochastic QCBO problems, including satisfying constraints in expectation and in probability.
• Solution quality and convergence rate were sensitive to VQC parameterization: under- and over-parameterized circuits degraded performance.
• The method demonstrated robustness across 10 different large-scale LP instances, with fixed step sizes and simulator seed, showing consistent convergence after 500 iterations.
• Theoretical analysis shows that the optimality gap of VQEC scales with the approximation error of the VQC in representing the target probability mass function, under Assumption 3.