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.

翻译：变分量子方法在寻找计算复杂问题的近优解方面展现出巨大潜力，但如何系统性地施加约束仍鲜有探索。为填补这一空白，本文提出一种名为VQEC的混合量子-经典算法范式，它将著名的VQE扩展到处理带约束的优化问题。与标准VQE类似，优化变量向量由变分量子电路（VQC）的状态表征。为处理约束，VQEC在经典层面同时对VQC参数和约束关联的对偶变量进行拉格朗日函数优化。为适应量子计算框架，变量通过基于参数平移规则的扰动原始-对偶方法更新。在广泛的应用场景中，我们展示了VQEC如何近似求解二次约束二元优化（QCBO）问题、寻找满足平均约束和概率约束的随机二元策略，以及求解概率单纯形上的大规模线性规划（LP）。基于VQC近似任意概率质量函数（PMF）误差的假设，我们给出了VQC所能达到的最优性间隙上界。在量子模拟器上的数值实验探究了多种参数的影响，并证实VQEC能够生成高质量解。