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Bulletin of the American Physical Society

Investigations of the Quantum Alternating Operator Ansatz

Eleanor Gilbert Rieffel, Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, Davide Venturelli

Published Mar 8, 2018• 0 citations
OpenAlex

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Quantum alternating operator ansatz with PSO optimizer for portfolio optimization problem

Wenjie Liu, Yue Ma, Yuchen Gu +2 more

Recently, Quantum Approximate Optimization Algorithm (QAOA) and Quantum Alternating Operator ansatz (QAOAz) are utilized to solve the Mean Variance (MV) model for the portfolio optimization problem (QAOAz-MV), which shows performance advantages over classical algorithms on this huge search space. For the more complex and comprehensive risk parity (RP) model, a novel QAOAz solution (QAOAz-RP) is proposed. We begin by defining the RP model for the portfolio optimization problem. Next, we detail the QAOAz algorithm process, where the problem Hamiltonian with the ZZZZ term is derived, the corresponding quantum circuit is ingeniously constructed using the parity check method, and the whole quantum circuit containing the ring XY-mixer is given. Finally, to improve the optimization performance of QAOAz, a Particle Swarm Optimization (PSO) optimizer is introduced to tune the parameters of the quantum circuits, which is applicable to both QAOAz-MV and QAOAz-RP. The experiment conducted on multiple financial markets (e.g. Chinese, U.S., and European) demonstrate that PSO-QAOAz-RP is significantly better for portfolio optimization than ABC-LP, GWO, GA and QAOAz-RP on eight portfolios in all metrics. PSO-QAOAz-MV also has advantage for MV model over the quantum algorithms, including QAOA (improves approximate ratio by 54.79% on average) and QAOAz (improves approximate ratio by 15.38% on average). This study not only provides a breakthrough quantum solution for portfolio optimization, but also provides a reusable technology paradigm for the deep integration of quantum computing and financial engineering. • It is the first QAOAz solution for the RP model of portfolio optimization problem. • The problem Hamiltonian and quantum circuit are derived and ingeniously constructed. • A PSO optimizer is proposed to optimize the parameters of QAOAz’s quantum circuit. • Our QAOAz-RP and PSO-QAOAz-RP outperform state-of-the-art classical algorithms. • Our PSO optimizer is effective on both MV and RP models.

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This study proposes a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problem (CVRP) by integrating the Column Generation (CG) method with the Quantum Alternating Operator Ansatz (QAOAnsatz). The CG method divides the CVRP into the reduced master problem, which finds the best combination of the routes under the route set, and one or more subproblems, which generate the routes that would be beneficial to add to the route set. This method is iteratively refined by adding new routes identified via subproblems and continues until no improving route can be found. We leverage the QAOAnsatz to solve the subproblems. Our algorithm restricts the search space by designing the QAOAnsatz mixer Hamiltonian to enforce one-hot constraints. Moreover, to handle capacity constraints in QAOAnsatz, we employ an Augmented Lagrangian-inspired method that obviates the need for additional slack variables, reducing the required number of qubits. Experimental results on small-scale CVRP instances (up to 6 customers) show that QAOAnsatz converges more quickly to optimal routes than the standard QAOA approach, demonstrating the potential of this hybrid framework in tackling realworld logistical optimization problems on near-term quantum hardware.

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Depth One Quantum Alternating Operator Ansatz as an Approximate Gibbs Distribution Sampler

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This study numerically investigates the thermal sampling properties of QAOA, the Quantum Alternating Operator Ansatz which was generalized from the original Quantum Approximate Optimization Algorithm. Specifically, the ability of QAOA to sample from the Gibbs distribution, equivalently the Boltzmann distribution, defined by a classical Ising model, specifically a fully connected disordered spin glass (Sherrington-Kirkpatrick) model. We focus on two different QAOA mixers; the standard transverse field X mixer, and the Grover mixer. At a QAOA depth of one we examine, for a single full QAOA parameter search space period, the energy landscape, the Shannon entropy landscape of the QAOA probability distribution, and the tradeoff between Boltzmann distribution sampling temperature and error rate (how close to the true Boltzmann distribution is the QAOA distribution). We find that at very high temperatures one-round Grover mixer QAOA can sample from the Boltzmann distribution more accurately than the standard X mixer QAOA at one round. Both X mixer and Grover mixer depth one QAOA can serve as approximate Boltzmann distribution samplers, and how good this approximation is depends heavily on the QAOA angle choice.

arXiv:2510.10345•Oct 11, 2025•0 citations
Recently, Quantum Approximate Optimization Algorithm (QAOA) and Quantum Alternating Operator ansatz (QAOAz) are utilized to solve the Mean Variance (MV) model for the portfolio optimization problem (QAOAz-MV), which shows performance advantages over classical algorithms on this huge search space. For the more complex and comprehensive risk parity (RP) model, a novel QAOAz solution (QAOAz-RP) is proposed. We begin by defining the RP model for the portfolio optimization problem. Next, we detail the QAOAz algorithm process, where the problem Hamiltonian with the ZZZZ term is derived, the corresponding quantum circuit is ingeniously constructed using the parity check method, and the whole quantum circuit containing the ring XY-mixer is given. Finally, to improve the optimization performance of QAOAz, a Particle Swarm Optimization (PSO) optimizer is introduced to tune the parameters of the quantum circuits, which is applicable to both QAOAz-MV and QAOAz-RP. The experiment conducted on multiple financial markets (e.g. Chinese, U.S., and European) demonstrate that PSO-QAOAz-RP is significantly better for portfolio optimization than ABC-LP, GWO, GA and QAOAz-RP on eight portfolios in all metrics. PSO-QAOAz-MV also has advantage for MV model over the quantum algorithms, including QAOA (improves approximate ratio by 54.79% on average) and QAOAz (improves approximate ratio by 15.38% on average). This study not only provides a breakthrough quantum solution for portfolio optimization, but also provides a reusable technology paradigm for the deep integration of quantum computing and financial engineering. • It is the first QAOAz solution for the RP model of portfolio optimization problem. • The problem Hamiltonian and quantum circuit are derived and ingeniously constructed. • A PSO optimizer is proposed to optimize the parameters of QAOAz’s quantum circuit. • Our QAOAz-RP and PSO-QAOAz-RP outperform state-of-the-art classical algorithms. • Our PSO optimizer is effective on both MV and RP models.
This study proposes a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problem (CVRP) by integrating the Column Generation (CG) method with the Quantum Alternating Operator Ansatz (QAOAnsatz). The CG method divides the CVRP into the reduced master problem, which finds the best combination of the routes under the route set, and one or more subproblems, which generate the routes that would be beneficial to add to the route set. This method is iteratively refined by adding new routes identified via subproblems and continues until no improving route can be found. We leverage the QAOAnsatz to solve the subproblems. Our algorithm restricts the search space by designing the QAOAnsatz mixer Hamiltonian to enforce one-hot constraints. Moreover, to handle capacity constraints in QAOAnsatz, we employ an Augmented Lagrangian-inspired method that obviates the need for additional slack variables, reducing the required number of qubits. Experimental results on small-scale CVRP instances (up to 6 customers) show that QAOAnsatz converges more quickly to optimal routes than the standard QAOA approach, demonstrating the potential of this hybrid framework in tackling realworld logistical optimization problems on near-term quantum hardware.
This study numerically investigates the thermal sampling properties of QAOA, the Quantum Alternating Operator Ansatz which was generalized from the original Quantum Approximate Optimization Algorithm. Specifically, the ability of QAOA to sample from the Gibbs distribution, equivalently the Boltzmann distribution, defined by a classical Ising model, specifically a fully connected disordered spin glass (Sherrington-Kirkpatrick) model. We focus on two different QAOA mixers; the standard transverse field X mixer, and the Grover mixer. At a QAOA depth of one we examine, for a single full QAOA parameter search space period, the energy landscape, the Shannon entropy landscape of the QAOA probability distribution, and the tradeoff between Boltzmann distribution sampling temperature and error rate (how close to the true Boltzmann distribution is the QAOA distribution). We find that at very high temperatures one-round Grover mixer QAOA can sample from the Boltzmann distribution more accurately than the standard X mixer QAOA at one round. Both X mixer and Grover mixer depth one QAOA can serve as approximate Boltzmann distribution samplers, and how good this approximation is depends heavily on the QAOA angle choice.