Quantum approximate optimization algorithm (QAOA) has attracted much attention as an algorithm that has the potential to efficiently solve combinatorial optimization problems. Among them, a fermionic QAOA (FQAOA) for solving constrained optimization problems has been developed [Yoshioka, Sasada, Nakano, and Fujii, Phys. Rev. Research vol. 5, 023071, 2023]. In this algorithm, the constraints are essentially imposed as fermion number conservation at arbitrary approximation level. We take the portfolio optimization problem as an application example and propose a new driver Hamiltonian on an one-dimensional cyclic lattice. Our FQAOA with the new driver Hamiltonian reduce the number of gate operations in quantum circuits. Experiments on a trapped-ion quantum computer using 16 qubits on Amazon Braket demonstrates that the proposed driver Hamiltonian effectively suppresses noise effects compared to the previous FQAOA.
Quantum computers are expected to accelerate solving combinatorial optimization problems, including algorithms such as Grover adaptive search and quantum approximate optimization algorithm (QAOA). However, many combinatorial optimization problems involve constraints which, when imposed as soft constraints in the cost function, can negatively impact the performance of the optimization algorithm. In this paper, we propose fermionic quantum approximate optimization algorithm (FQAOA) for solving combinatorial optimization problems with constraints. Specifically, FQAOA tackles the constrains issue by using fermion particle number preservation to intrinsically impose them throughout QAOA. We provide a systematic guideline for designing the driver Hamiltonian for a given problem Hamiltonian with constraints. The initial state can be chosen to be a superposition of states satisfying the constraint and the ground state of the driver Hamiltonian. This property is important since FQAOA reduced to quantum adiabatic computation in the large limit of circuit depth
p and improved performance, even for shallow circuits with optimizing the parameters starting from the fixed-angle determined by Trotterized quantum adiabatic evolution. We perform an extensive numerical simulation and demonstrate that proposed FQAOA provides substantial performance advantages against existing approaches in portfolio optimization problems. Furthermore, the Hamiltonian design guideline is useful not only for QAOA but also Grover adaptive search and quantum phase estimation to solve combinatorial optimization problems with constraints. Since software tools for fermionic systems have been developed in quantum computational chemistry both for noisy intermediate-scale quantum computers and fault-tolerant quantum computers, FQAOA allows us to apply these tools for constrained combinatorial optimization problems.
Quantum Approximation Optimization Algorithms (QAOA) have been actively developed, among which Fermionic QAOA (FQAOA) has been successfully applied to financial portfolio optimization problems. We improve FQAOA and apply it to the optimization of electricity demand portfolios aiming to procure a target amount of electricity with minimum risk. Our new algorithm, FQAOA-SCLFM, allows approximate integration of constraints on the target amount of power by utilizing self-consistent local field modulation (SCLFM) in a driver Hamiltonian. We demonstrate that this approach performs better than the currently widely used XY-QAOA and the previous FQAOA in all instances subjected to this study.
Farhi et al. recently proposed a class of quantum algorithms, the quantum approximate optimization algorithm (QAOA), for approximately solving combinatorial optimization problems (E. Farhi et al., arXiv:1411.4028; arXiv:1412.6062; arXiv:1602.07674). A level-
p QAOA circuit consists of
p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The
2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As
p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The success of the QAOA approach will depend, in part, on finding effective parameter-setting strategies. Here we analytically and numerically study parameter setting for the QAOA applied to MaxCut. For the level-1 QAOA, we derive an analytical expression for a general graph. In principle, expressions for higher
p could be derived, but the number of terms quickly becomes prohibitive. For a special case of MaxCut, the ``ring of disagrees,'' or the one-dimensional antiferromagnetic ring, we provide an analysis for an arbitrarily high level. Using a fermionic representation, the evolution of the system under the QAOA translates into quantum control of an ensemble of independent spins. This treatment enables us to obtain analytical expressions for the performance of the QAOA for any
p. It also greatly simplifies the numerical search for the optimal values of the parameters. By exploring symmetries, we identify a lower-dimensional submanifold of interest; the search effort can be accordingly reduced. This analysis also explains an observed symmetry in the optimal parameter values. Further, we numerically investigate the parameter landscape and show that it is a simple one in the sense of having no local optima.
Combinatorial optimization is a promising area for achieving quantum speedup. The quantum approximate optimization algorithm (QAOA) is designed to search for low-energy states of the Ising model, which correspond to near-optimal solutions of combinatorial optimization problems (COPs). However, effectively dealing with constraints of COPs remains a significant challenge. Existing methods, such as tailoring mixing operators, are typically limited to specific constraint types, like one-hot constraints. To address these limitations, we introduce a method for engineering a compressed space that represents the feasible solution space with fewer qubits than the original. Our approach includes a scalable technique for determining the unitary transformation between the compressed and original spaces on gate-based quantum computers. We then propose compressed space QAOA, which seeks near-optimal solutions within this reduced space, while utilizing the Ising model formulated in the original Hilbert space. Experimental results on a quantum simulator demonstrate the effectiveness of our method in solving various constrained COPs.
Constrained combinatorial optimization with strict linear constraints underpins applications in drug discovery, power grids, logistics, and finance, yet remains computationally demanding for classical algorithms, especially at large scales. The Quantum Approximate Optimization Algorithm (QAOA) offers a promising quantum framework, but conventional penalty-based formulations distort optimization landscapes and demand deep circuits, undermining scalability on near-term hardware. In this work, we introduce Hamming Weight Operators, a new class of constraint-aware operators that confine quantum evolution strictly within the feasible subspace. Building on this idea, we develop Adaptive Hamming Weight Operator QAOA, which dynamically selects the most effective operators to construct shallow, problem-tailored circuits. We validate our approach on benchmark tasks from both finance and high-energy physics, specifically portfolio optimization and two-jet clustering with energy balance. Across these problems, our method inherently satisfies all constraints by construction, converges faster, and achieves higher Approximation Ratios than penalty-based QAOA, while requiring roughly half as many gates. By embedding constraint-aware operators into an adaptive variational framework, our approach establishes a scalable and hardware-efficient pathway for solving practical constrained optimization problems on near-term quantum devices.
Abstract We present a detailed study of portfolio optimization using different versions of the quantum approximate optimization algorithm (QAOA). For a given list of assets, the portfolio optimization problem is formulated as quadratic binary optimization constrained on the number of assets contained in the portfolio. QAOA has been suggested as a possible candidate for solving this problem (and similar combinatorial optimization problems) more efficiently than classical computers in the case of a sufficiently large number of assets. However, the practical implementation of this algorithm requires a careful consideration of several technical issues, not all of which are discussed in the present literature. The present article intends to fill this gap and thereby provides the reader with a useful guide for applying QAOA to the portfolio optimization problem (and similar problems). In particular, we will discuss several possible choices of the variational form and of different classical algorithms for finding the corresponding optimized parameters. Viewing at the application of QAOA on error-prone NISQ hardware, we also analyse the influence of statistical sampling errors (due to a finite number of shots) and gate and readout errors (due to imperfect quantum hardware). Finally, we define a criterion for distinguishing between ‘easy’ and ‘hard’ instances of the portfolio optimization problem.