We introduce a counterdiabatic (CD) extension of the Quantum Approximate Optimization Algorithm (QAOA) for constrained portfolio optimization. By incorporating approximate adiabatic gauge potentials generated from nested commutators of the Ising-type portfolio problem Hamiltonian and the Hamming weight-preserving XY mixer Hamiltonian into our variational ansatz, the resulting Constrained Counterdiabatic QAOA (CCD-QAOA) achieves improved optimization performance under realistic budget and risk constraints. Benchmarking against standard XY-mixer QAOA, Grover-mixer QAOA, and penalty-based QAOA formulations, our numerical simulations demonstrate that, for a fixed QAOA depth, our CCD-QAOA approach consistently results in better approximation ratios.
We introduce a counterdiabatic (CD) extension of the Quantum Approximate Optimization Algorithm (QAOA) for constrained portfolio optimization. By incorporating approximate adiabatic gauge potentials generated from nested commutators of the Ising-type portfolio problem Hamiltonian and the Hamming weight-preserving XY mixer Hamiltonian into our variational ansatz, the resulting Constrained Counterdiabatic QAOA (CCD-QAOA) achieves improved optimization performance under realistic budget and risk constraints. Benchmarking against standard XY-mixer QAOA, Grover-mixer QAOA, and penalty-based QAOA formulations, our numerical simulations demonstrate that, for a fixed QAOA depth, our CCD-QAOA approach consistently results in better approximation ratios.
Portfolio optimization under strict cardinality constraints is a combinatorial challenge that defies classical convex optimization techniques, particularly in the context of "Direct Indexing" and ESG-constrained mandates. In the Noisy Intermediate-Scale Quantum (NISQ) era, the Quantum Approximate Optimization Algorithm (QAOA) offers a promising hybrid approach. However, standard QAOA implementations utilizing transverse field mixers often fail to strictly enforce hard constraints, necessitating soft penalties that distort the energy landscape. This paper presents a comprehensive analysis of a constraint-preserving QAOA formulation against Simulated Annealing (SA) and Hierarchical Risk Parity (HRP). We implement a specific QAOA ansatz utilizing a Dicke state initialization and an XY-mixer Hamiltonian that strictly preserves the Hamming weight of the solution, ensuring only valid portfolios of size K are explored. Furthermore, we introduce a Trotterized parameter initialization schedule inspired by adiabatic quantum computing to mitigate the "Barren Plateau" problem. Backtesting on a basket of 10 US equities over 2025 reveals that our QAOA approach achieves a Sharpe Ratio of 1.81, significantly outperforming Simulated Annealing (1.31) and HRP (0.98). We further analyze the operational implications of the algorithm's high turnover (76.8%), discussing the trade-offs between theoretical optimality and implementation costs in institutional settings.
The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the p → ∞ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of p large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these "error" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as 1 − C ( p ) ∼ 1 / p μ . We show that transfer of parameters between graphs, and interpolating angles for p + 1 given p are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.
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.
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm proposed for Noisy Intermediate-Scale Quantum (NISQ) devices and is regarded as a promising approach to combinatorial optimization problems, with potential applications in the financial sector. In this study, we apply QAOA to the portfolio optimization problem, which is one of the central challenges in financial engineering. A portfolio consists of a combination of multiple assets, and the portfolio optimization problem aims to determine the optimal asset allocation by balancing expected return and risk. In the context of quantum optimization, portfolio optimization is often formulated using discrete variables. Unlike conventional binary formulations, we consider a ternary portfolio optimization problem that accounts for three states-holding, not holding, and short selling-and compare its performance using different mixer operators. Specifically, we implement QAOA with the standard mixer and several XY Mixers (XY Ring, XY Parity Ring, XY Full, and QAMPA), and conducted simulations using real data based on the German stock index (DAX 30) for portfolios consisting of 5 and 8 assets. Furthermore, we introduce noise based on a depolarizing channel to investigate the behavior of the algorithm in realistic environments. The results show that while XY Mixers exhibit superiority in noiseless settings, their advantage degrades in noisy environments, and the optimal choice of mixer depends on both the number of QAOA depths and the noise strength.
Abstract The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm proposed for Noisy Intermediate-Scale Quantum (NISQ) devices and is regarded as a promising approach to combinatorial optimization problems, with potential applications in the financial sector. In this study, we apply QAOA to the portfolio optimization problem, which is one of the central challenges in financial engineering. A portfolio consists of a combination of multiple assets, and the portfolio optimization problem aims to determine the optimal asset allocation by balancing expected return and risk. In the context of quantum optimization, portfolio optimization is often formulated using discrete variables. Unlike conventional binary formulations, we consider a ternary portfolio optimization problem that accounts for three states—holding, not holding, and short selling—and compare its performance using different mixer operators. Specifically, we implement QAOA with the standard mixer and several XY Mixers (XY Ring, XY Parity Ring, XY Full, and QAMPA), and conducted simulations using real data based on the German stock index (DAX 30) for portfolios consisting of 5 and 8 assets. Furthermore, we introduce noise based on a depolarizing channel to investigate the behavior of the algorithm in realistic environments. The results show that while XY Mixers exhibit superiority in noiseless settings, their advantage degrades in noisy environments, and the optimal choice of mixer depends on both the number of QAOA depths and the noise strength.
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.