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The Japanese Society for Artificial Intelligence

Application to optimization problem with multiple constraints using quantum approximation optimization algorithm (QAOA)

Koudai Shiba

Published Apr 1, 2020• 0 citations
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The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm that offers the potential to handle combinatorial optimization problems. Introducing constraints in such combinatorial optimization problems poses a significant challenge in the extensions of QAOA to support relevant larger-scale problems. This paper introduces a quantum machine learning approach to learn the mixer Hamiltonian required to hard constrain the search subspace. We show that this method can be used for encoding any general form of constraints. One can directly plug the learnt unitary into the QAOA framework using an adaptable ansatz. This procedure gives the flexibility to control the depth of the circuit at the cost of the accuracy of enforcing the constraint, thus having immediate application in the Noisy Intermediate Scale Quantum (NISQ) era. We also develop an intuitive metric that uses Wasserstein distance to assess the performance of general approximate optimization algorithms with/without constraints. Finally, using this metric, we evaluate the performance of the proposed algorithm.

arXiv:2105.06770•May 14, 2021•4 citations
When tackling binary optimization problems using quantum algorithms, the conventional Ising representation and Quantum Approximate Optimization Algorithm (QAOA) encounter difficulties in efficiently handling errors for large-scale problems involving multiple constraints. To address these challenges, this paper presents a hybrid framework that combines the use of standard Ising Hamiltonians to solve a subset of the constraints, while employing non-Ising formulations to represent and address the remaining constraints. The resolution of these non-Ising constraints is achieved through either penalty dephasing or the quantum Zeno effect. This innovative approach leads to a collection of quantum circuits with adaptable structures, depending on the chosen representation for each constraint. Furthermore, this paper introduces a novel technique that utilizes the quantum Zeno effect by frequently measuring the constraint flag, enabling the resolution of any optimization constraint. Theoretical properties of these algorithms are discussed, and their performance in addressing practical aircraft loading problems is highly promising, showcasing significant potential for a wide range of industrial applications.
Abstract Constrained optimization problems are ubiquitous in science and industry. Quantum algorithms have shown promise in solving optimization problems, yet none of the current algorithms can effectively handle arbitrary constraints. We introduce a technique that uses quantum Zeno dynamics to solve optimization problems with multiple arbitrary constraints, including inequalities. We show that the dynamics of quantum optimization can be efficiently restricted to the in-constraint subspace on a fault-tolerant quantum computer via repeated projective measurements, requiring only a small number of auxiliary qubits and no post-selection. Our technique has broad applicability, which we demonstrate by incorporating it into the quantum approximate optimization algorithm (QAOA) and variational quantum circuits for optimization. We evaluate our method numerically on portfolio optimization problems with multiple realistic constraints and observe better solution quality and higher in-constraint probability than state-of-the-art techniques. We implement a proof-of-concept demonstration of our method on the Quantinuum H1-2 quantum processor.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm that offers the potential to handle combinatorial optimization problems. Introducing constraints in such combinatorial optimization problems poses a significant challenge in the extensions of QAOA to support relevant larger-scale problems. This paper introduces a quantum machine learning approach to learn the mixer Hamiltonian required to hard constrain the search subspace. We show that this method can be used for encoding any general form of constraints. One can directly plug the learnt unitary into the QAOA framework using an adaptable ansatz. This procedure gives the flexibility to control the depth of the circuit at the cost of the accuracy of enforcing the constraint, thus having immediate application in the Noisy Intermediate Scale Quantum (NISQ) era. We also develop an intuitive metric that uses Wasserstein distance to assess the performance of general approximate optimization algorithms with/without constraints. Finally, using this metric, we evaluate the performance of the proposed algorithm.