Comparing QAOA Implementations of Penalty Function and XY-Mixer for Constrained Optimization: Application to the Circular Maximum Diversity Problem on IBM Quantum Computer
N. Rhodes, Sandy Spiers, Hòa T. Bùi, Maiyuren Srikumar, S. S. Kulkarni, Ryan C. Loxton, M. Egan
We compare two different implementations of Quantum Alternating Operator Ansatz (QAOA) for constrained optimization: the penalty function and the XY-mixer. The penalty function method converts a constrained problem to an unconstrained problem by penalizing constraint violations, whereas the XY-mixer uses two-qubit operators to maintain feasibility. We use a special case of the maximum diversity problem (MDP) to demonstrate the effectiveness of the XY-mixer. The MDP aims at selecting a subset of elements from a larger set to maximize diversity between the selected elements. The special case deals with instances in which elements are arranged in a circular structure. The circular MDP instances are challenging to solve using classical methods due to solution space symmetry but are well-suited for the topology of IBM quantum devices. We introduce a grouping and an approximation mechanism facilitating XY-mixer implementation on IBM quantum computers. Experiments with different-sized instances show that the XY-mixer generates more and better feasible solutions than the penalty function approach and increases the probability of finding good solutions.
We compare two different implementations of Quantum Alternating Operator Ansatz (QAOA) for constrained optimization: the penalty function and the XY-mixer. The penalty function method converts a constrained problem to an unconstrained problem by penalizing constraint violations, whereas the XY-mixer uses two-qubit operators to maintain feasibility. We use a special case of the maximum diversity problem (MDP) to demonstrate the effectiveness of the XY-mixer. The MDP aims at selecting a subset of elements from a larger set to maximize diversity between the selected elements. The special case deals with instances in which elements are arranged in a circular structure. The circular MDP instances are challenging to solve using classical methods due to solution space symmetry but are well-suited for the topology of IBM quantum devices. We introduce a grouping and an approximation mechanism facilitating XY-mixer implementation on IBM quantum computers. Experiments with different-sized instances show that the XY-mixer generates more and better feasible solutions than the penalty function approach and increases the probability of finding good solutions.
Quantum Approximate Optimization Algorithm, further generalized as Quantum Alternating Operator Ansatz (QAOA), is a family of algorithms for combinatorial optimization problems. It is a leading candidate to run on emerging universal quantum computers to gain insight into quantum heuristics. In constrained optimization, penalties are often introduced so that the ground state of the cost Hamiltonian encodes the solution (a standard practice in quantum annealing). An alternative is to choose a mixing Hamiltonian such that the constraint corresponds to a constant of motion and the quantum evolution stays in the feasible subspace. Better performance of the algorithm is speculated due to a much smaller search space. We consider problems with a constant Hamming weight as the constraint. We also compare different methods of generating the generalized W-state, which serves as a natural initial state for the Hamming-weight constraint. Using graph-coloring as an example, we compare the performance of using XY model as a mixer that preserves the Hamming weight with the performance of adding a penalty term in the cost Hamiltonian.
Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by 5.30% on ER random graphs and 5.41% on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only 15.30% and 25.18% of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.
ABSTRACT Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA‐QAOA+. Unlike QAOA+, AMA‐QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA‐QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA‐QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by on Erdős–Rényi random graphs and on 3‐regular graphs. Moreover, AMA‐QAOA+ significantly reduces the CNOT gate consumption, requiring only and of the CNOT gates used by QAOA+ on Erdős–Rényi and 3‐regular random graphs, respectively. These results demonstrate that AMA‐QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource‐efficient quantum circuits.
This paper proposes a novel combination of constraint encoding methods for the Quantum Approximate Optimization Ansatz (QAOA). Real-world optimization problems typically consist of multiple types of constraints. To solve these optimization problems with quantum methods, normally, all constraints are added as quadratic penalty terms to the objective, which expands the search space and increases problem complexity. This work proposes a general workflow that extracts and encodes specific constraints directly into the circuit of QAOA: One-hot constraints are enforced throughXY-mixers that restrict the search space to the feasible sub-space naturally. Inequality constraints are implemented through oracle-based Indicator Functions (IF). This paper focuses on the numerical benchmarks of the combined approach for solving the Multi-Knapsack (MKS) and the Prosumer Problem (PP), a modification of the MKS in the domain of electricity optimization. To this end, we introduce computational techniques that efficiently simulate the two presented constraint architectures. SinceXY-mixers restrict the search space, specific state vector entries are always zero and can be omitted from the simulation, saving valuable memory and computing resources. We benchmark the combined method against the established QUBO formulation, yielding a better solution quality and probability of sampling the optimal solution. Despite more complex circuits, the time-to-solution is more than an order of magnitude faster compared to the baseline methods and exhibits more favorable scaling properties.
Realizing the potential of near-term quantum computers to solve industry-relevant constrained-optimization problems is a promising path to quantum advantage. In this work, we consider the extractive summarization constrained-optimization problem and demonstrate the largest-to-date execution of a quantum optimization algorithm that natively preserves constraints on quantum hardware. We report results with the Quantum Alternating Operator Ansatz algorithm with a Hamming-weight-preserving XY mixer (XY-QAOA) on trapped-ion quantum computer. We successfully execute XY-QAOA circuits that restrict the quantum evolution to the in-constraint subspace, using up to 20 qubits and a two-qubit gate depth of up to 159. We demonstrate the necessity of directly encoding the constraints into the quantum circuit by showing the trade-off between the in-constraint probability and the quality of the solution that is implicit if unconstrained quantum optimization methods are used. We show that this trade-off makes choosing good parameters difficult in general. We compare XY-QAOA to the Layer Variational Quantum Eigensolver algorithm, which has a highly expressive constant-depth circuit, and the Quantum Approximate Optimization Algorithm. We discuss the respective trade-offs of the algorithms and implications for their execution on near-term quantum hardware.
XY mixer are a type of mixing Hamiltonians used in the quantum alternating operator ansatzs (QAOA) framework for solving combinatorial optimization problems where the feasible subspace consists of states with the same Hamming weight. In this paper, we propose an extension of the XY-mixer ansatzes to solve optimization problems that do not adhere to this specific subspace structure. For problems, such as maximal independent set, one can employ positive semidefinite programming and a greedy algorithm to obtain upper and lower bounds for the optimal solution, then utilize the XY-mixer ansatzes in the separated subspaces between the upper and lower bounds to address the problem at hand. Additionally, we find suitable counterdiabatic (CD) driving terms that complement the XY-mixer ansatzes. These driving terms accelerate the QAOA evolution towards the target state and also confine the evolution within the subspace. By combining XY-mixer ansatzes with counterdiabatic driving (XY-CD mixers), we present a generalized QAOA-based scheme for finding higher-quality approximate solutions to combinatorial problems.