This paper introduces the use of tailored variational forms for variational quantum eigensolver that have properties of representing certain constraints on the search domain of a linear constrained quadratic binary optimization problem solution. Four constraints that usually appear in several optimization problems are modeled. The main advantage of the proposed methodology is that the number of parameters on the variational form remain constant and depend on the number of variables that appear on the constraints. Moreover, this variational form always produces feasible solutions for the represented constraints differing from penalization techniques commonly used to translate constrained problems into unconstrained one. The methodology is implemented in a real quantum computer for two known optimization problems: the Facility Location Problem and the Set Packing Problem. The results obtained for this two problems with VQE using 2-Local variational form and a general QAOA implementation are compared, and indicate that less quantum gates and parameters were used, leading to a faster convergence.
This paper introduces the use of tailored variational forms for variational quantum eigensolver that have properties of representing certain constraints on the search domain of a linear constrained quadratic binary optimization problem solution. Four constraints that usually appear in several optimization problems are modeled. The main advantage of the proposed methodology is that the number of parameters on the variational form remain constant and depend on the number of variables that appear on the constraints. Moreover, this variational form always produces feasible solutions for the represented constraints differing from penalization techniques commonly used to translate constrained problems into unconstrained one. The methodology is implemented in a real quantum computer for two known optimization problems: the Facility Location Problem and the Set Packing Problem. The results obtained for this two problems with VQE using 2-Local variational form and a general QAOA implementation are compared, and indicate that less quantum gates and parameters were used, leading to a faster convergence.
Variational quantum approaches have shown great promise in finding near-optimal solutions to computationally challenging tasks, including solving optimization problems. Nonetheless, optimization problems with constraints may not have been handled in a disciplined fashion thus far. To address this gap, this work proposes a hybrid quantum-classical algorithmic paradigm termed the variational quantum eigensolver with constraints (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 problems, find stochastic binary policies satisfying quadratic constraints on the average and in probability, and solve large-scale linear programs over the probability simplex. Under an assumption on the error for the VQC to approximate an arbitrary probability mass function, 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.
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.
In finance, portfolio optimization aims at finding optimal investments maximizing a trade-off between return and risks, given some constraints. Classical formulations of this quadratic optimization problem have exact or heuristic solutions, but the complexity scales up as the market dimension increases. Recently, researchers are evaluating the possibility of facing the complexity scaling issue by employing quantum computing. In this paper, the problem is solved using the Variational Quantum Eigensolver (VQE), which in principle is very efficient. The main outcome of this work consists of the definition of the best hyperparameters to set, in order to perform Portfolio Optimization by VQE on real quantum computers. In particular, a quite general formulation of the constrained quadratic problem is considered, which is translated into Quadratic Unconstrained Binary Optimization by the binary encoding of variables and by including constraints in the objective function. This is converted into a set of quantum operators (Ising Hamiltonian), whose minimum eigenvalue is found by VQE and corresponds to the optimal solution. In this work, different hyperparameters of the procedure are analyzed, including different ansatzes and optimization methods by means of experiments on both simulators and real quantum computers. Experiments show that there is a strong dependence of solutions quality on the sufficiently sized quantum computer and correct hyperparameters, and with the best choices, the quantum algorithm run on real quantum devices reaches solutions very close to the exact one, with a strong convergence rate towards the classical solution, even without error-mitigation techniques. Moreover, results obtained on different real quantum devices, for a small-sized example, show the relation between the quality of the solution and the dimension of the quantum processor. Evidences allow concluding which are the best ways to solve real Portfolio Optimization problems by VQE on quantum devices, and confirm the possibility to solve them with higher efficiency, with respect to existing methods, as soon as the size of quantum hardware will be sufficiently high.
Analytical and practical evidence indicates the advantage of quantum computing solutions over classical alternatives. Quantum-based heuristics relying on the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) have been shown numerically to generate high-quality solutions to hard combinatorial problems, yet incorporating constraints to such problems has been elusive. To this end, this work puts forth a quantum heuristic to cope with stochastic binary quadratically constrained quadratic programs (QCQP). Identifying the strength of quantum circuits to efficiently generate samples from probability distributions that are otherwise hard to sample from, the variational quantum circuit is trained to generate binaryvalued vectors to approximately solve the aforesaid stochastic program. The method builds upon dual decomposition and entails solving a sequence of judiciously modified standard VQE tasks. Tests on several synthetic problem instances using a quantum simulator corroborate the near-optimality and feasibility of the method, and its potential to generate feasible solutions for the deterministic QCQP too.
Variational Quantum Algorithms (VQAs) provide a promising framework for tackling complex optimization problems on near-term quantum hardware. Here, we demonstrate that hybrid qubit-qumode quantum devices offer an efficient route to solving Quadratic Unconstrained Binary Optimization (QUBO) problems using the Echoed Conditional Displacement Variational Quantum Eigensolver (ECD-VQE). Leveraging circuit quantum electrodynamics (cQED) architectures, we encode QUBO instances across multiple qumodes weakly coupled to a single qubit and extract binary solutions directly from photon-number measurements. We apply ECD-VQE to the Binary Knapsack Problem and show that it outperforms the Quantum Approximate Optimization Algorithm (QAOA) implemented on conventional qubit circuits, achieving higher-quality solutions with dramatically fewer resources. We also demonstrate that ECD-VQE can be extended to chemically motivated tasks such as active-space selection for multireference electronic structure methods. These results highlight the utility of hybrid qubit-qumode platforms for a broad class of NP-hard and chemistry-related optimization problems and demonstrate that variational ECD ansatz can realize expressive state preparation with significantly shallower circuits than qubit-only architectures, positioning qubit-qumode gates as compelling candidates for constrained optimization in early fault-tolerant quantum computing.
Variational quantum optimization algorithms, such as the Variational Quantum Eigensolver (VQE) or the Quantum Approximate Optimization Algorithm (QAOA), are among the most studied quantum algorithms. In our work, we evaluate and improve an algorithm based on VQE, which uses exponentially fewer qubits compared to the QAOA. We highlight the numerical instabilities generated by encoding the problem into the variational ansatz and propose a classical optimization procedure to find the ground-state of the ansatz in less iterations with a better or similar objective. Additionally, we propose a method to embed the linear interpolation of the MaxCut problem on a quantum device. Furthermore, we compare classical optimizers for this variational ansatz on quadratic unconstrained binary optimization and graph partitioning problems.