Variational Quantum Eigensolver with Constraints (VQEC): Solving Constrained Optimization Problems via VQE

TL;DR

VQEC extends the variational quantum eigensolver to constrained optimization by coupling a quantum-variate objective with a classical perturbed primal–dual update over both primal parameters $\boldsymbol{\theta}$ and dual multipliers $\boldsymbol{\lambda}$. The approach leverages the parameter-shift rule to compute gradients and, when cost and constraint observables are diagonal and simultaneously measurable, incurs only modest quantum overhead relative to VQE. The paper provides theoretical bounds on the duality gap and the impact of approximating PMFs with a VQC, and demonstrates through numerical experiments on QCBO, stochastic PMFs, chance constraints, and large LPs that VQEC can yield high-quality, feasible solutions on NISQ-like hardware. This framework broadens the applicability of near-term quantum devices to practical constrained optimization tasks in ML, communications, and resource allocation, with future work aimed at non-diagonal observables and richer problem classes.

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.

Figures (13)

• Figure 1: Coordination between a quantum and a classical computer while running the PD or PPD method to solve a constrained optimization problem through a variational quantum approach. The VQC is encoded by unitary matrix $\mathbf{S}(\boldsymbol{\theta})$. The proposed method features minimal computational overhead over the standard, unconstrained VQE. If all involved quantum observables $\{F_m(\boldsymbol{\theta})\}_{m=0}^M$ can be measured simultaneously, the overhead is insignificant and is confined only to the classical computer.
• Figure 2: The two-local VQC operating on an $n$-qubit system. Top: One layer of the two-local VQC consists of the parameterized block $\mathbf{W}(\boldsymbol{\theta}_1)$ followed by the full entanglement block $\mathbf{U}_\text{ent}$. Bottom: A $d$-layered two-local VQC.
• Figure 3: Convergence of primal/dual variables for PD and PPD methods over the number of primal/dual iterations under setup S1) using a quantum state simulator (no measurement noise). Top: Convergence of dual variable $\lambda$. Because the constraint is active, the optimal $\lambda$ is nonzero. Bottom: Convergence of entries 5, 15, 25, 35 of the primal variable $\boldsymbol{\theta}$.
• Figure 4: Convergence of constraint function value $F_1(\boldsymbol{\theta}^t)$ (top) and relative cost error $|(P_\theta^t-P^*)/P^*|$ (bottom) for the PD and PPD methods over the number of primal-dual iterations under setup S1) using a quantum state simulator (no measurement noise).
• Figure 5: Convergence of constraint function value $F_1(\boldsymbol{\theta}^t)$ (top) and relative cost error $|(P_\theta^t-P^*)/P^*|$ (bottom) for PPD over the number of primal-dual iterations under S1) using different values of measurement shots $S$. For $S\in\{1, 25, 50\}$, PPD was repeated 8 times to account for the randomness in iterations. The plots display confidence intervals within one standard deviation around the mean per iteration.

Theorems & Definitions (5)

• Remark 1
• Theorem 1
• Lemma 1
• Theorem 2
• Remark 2