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

arXiv: https://arxiv.org/abs/2311.08502 · HTML: https://mlsanity.com/paper/2311.08502/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 80 references that are in the map.
- First posted on arXiv 14 Nov 2023.

## Builds on (most cited first, 19 of 19)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Ising formulations of many NP problems](https://mlsanity.com/paper/1302.5843/) (2013)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [Learning Optimal Resource Allocations in Wireless Systems](https://mlsanity.com/paper/1807.08088/) (2018)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [An Introduction to Quantum Machine Learning for Engineers](https://mlsanity.com/paper/2205.09510/) (2022)
- [Variational Quantum Algorithms for Semidefinite Programming](https://mlsanity.com/paper/2112.08859/) (2021)
- [Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method](https://mlsanity.com/paper/1304.5590/) (2013)
- [Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications](https://mlsanity.com/paper/1411.7564/) (2014)
- [Controlling Smart Inverters using Proxies: A Chance-Constrained DNN-based Approach](https://mlsanity.com/paper/2105.00429/) (2021)
- [Multi-block ADMM Heuristics for Mixed-Binary Optimization on Classical and Quantum Computers](https://mlsanity.com/paper/2001.02069/) (2020)
- [A Quantum Approach for Stochastic Constrained Binary Optimization](https://mlsanity.com/paper/2301.01443/) (2023)
- [Solving constrained quadratic binary problems via quantum adiabatic evolution](https://mlsanity.com/paper/1509.05001/) (2015)
- [Nearly Optimal Linear Convergence of Stochastic Primal-Dual Methods for Linear Programming](https://mlsanity.com/paper/2111.05530/) (2021)

## Abstract (arXiv)

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.