Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2305.03857 · npj Quantum Information

Alignment between initial state and mixer improves QAOA performance for constrained optimization

Zichang He, Ruslan Shaydulin, Shouvanik Chakrabarti, Dylan Herman, Changhao Li, Yue Sun, Marco Pistoia

Published Nov 27, 2023• 48 citations
arXivHTMLPDF

Abstract

Abstract Quantum alternating operator ansatz (QAOA) has a strong connection to the adiabatic algorithm, which it can approximate with sufficient depth. However, it is unclear to what extent the lessons from the adiabatic regime apply to QAOA as executed in practice with small to moderate depth. In this paper, we demonstrate that the intuition from the adiabatic algorithm applies to the task of choosing the QAOA initial state. Specifically, we observe that the best performance is obtained when the initial state of QAOA is set to be the ground state of the mixing Hamiltonian, as required by the adiabatic algorithm. We provide numerical evidence using the examples of constrained portfolio optimization problems with both low (p ≤ 3) and high (p = 100) QAOA depth. Additionally, we successfully apply QAOA with XY mixer to portfolio optimization on a trapped-ion quantum processor using 32 qubits and discuss our findings in near-term experiments.

DOI
10.1038/s41534-023-00787-5

Related papers

XY vs X mixer in Quantum Alternating Operator Ansatz for optimization problems with constraints

Zhihui Wang, Nicholas C. Rubin, Eleanor Gilbert Rieffel

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.

Bulletin of the American Physical Society•Mar 8, 2018•0 citations

Constrained Portfolio Optimization via Quantum Approximate Optimization Algorithm (QAOA) with XY-Mixers and Trotterized Initialization: A Hybrid Approach for Direct Indexing

Javier Mancilla, Theodoros D. Bouloumis, Frederic Goguikian

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.

quant-pharXiv:2602.14827•Feb 17, 2026•0 citations

XY mixers: Analytical and numerical results for the quantum alternating operator ansatz

Zhihui Wang, Nicholas C. Rubin, Jason Dominy +1 more

The quantum alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using $XY$ Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the $XY$ model, we demonstrate that, for an integer variable admitting $\ensuremath{\kappa}$ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\ensuremath{\kappa})$. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using $XY$ mixers is borne out numerically, demonstrating a significant improvement over the general $X$ mixer, and moreover the generalized $W$ state yields better performance than easier-to-generate classical initial states when $XY$ mixers are used.

arXiv:1904.09314•Physical Review A•Jan 14, 2020•189 citations

Quantum Alternating Operator Ansatz (QAOA) beyond low depth with gradually changing unitaries

Vladimir Kremenetski, Anuj Apte, Tad Hogg +2 more

The Quantum Approximate Optimization Algorithm and its generalization to Quantum Alternating Operator Ansatz (QAOA) is a promising approach for applying quantum computers to challenging problems such as combinatorial optimization and computational chemistry. In this paper, we study the underlying mechanisms governing the behavior of QAOA circuits beyond shallow depth in the practically relevant setting of gradually varying unitaries. We use the discrete adiabatic theorem, which complements and generalizes the insights obtained from the continuous-time adiabatic theorem primarily considered in prior work. Our analysis explains some general properties that are conspicuously depicted in the recently introduced QAOA performance diagrams. For parameter sequences derived from continuous schedules (e.g. linear ramps), these diagrams capture the algorithm's performance over different parameter sizes and circuit depths. Surprisingly, they have been observed to be qualitatively similar across different performance metrics and application domains. Our analysis explains this behavior as well as entails some unexpected results, such as connections between the eigenstates of the cost and mixer QAOA Hamiltonians changing based on parameter size and the possibility of reducing circuit depth without sacrificing performance.

arXiv:2305.04455•May 8, 2023•6 citations

Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry

Vladimir Kremenetski, Tad Hogg, Stuart Hadfield +2 more

Determining Hamiltonian ground states and energies is a challenging task with many possible approaches on quantum computers. While variational quantum eigensolvers are popular approaches for near term hardware, adiabatic state preparation is an alternative that does not require noisy optimization of parameters. Beyond adiabatic schedules, QAOA is an important method for optimization problems. In this work we modify QAOA to apply to finding ground states of molecules and empirically evaluate the modified algorithm on several molecules. This modification applies physical insights used in classical approximations to construct suitable QAOA operators and initial state. We find robust qualitative behavior for QAOA as a function of the number of steps and size of the parameters, and demonstrate this behavior also occurs in standard QAOA applied to combinatorial search. To this end we introduce QAOA phase diagrams that capture its performance and properties in various limits. In particular we show a region in which non-adiabatic schedules perform better than the adiabatic limit while employing lower quantum circuit depth. We further provide evidence our results and insights also apply to QAOA applications beyond chemistry.

arXiv:2108.13056•Aug 30, 2021•14 citations

Constrained Counterdiabatic Quantum Approximate Optimization Algorithm for Portfolio Optimization

Jose Falla, Ilya Safro

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.

quant-pharXiv:2605.06858•May 11, 2026•0 citations
Abstract Quantum alternating operator ansatz (QAOA) has a strong connection to the adiabatic algorithm, which it can approximate with sufficient depth. However, it is unclear to what extent the lessons from the adiabatic regime apply to QAOA as executed in practice with small to moderate depth. In this paper, we demonstrate that the intuition from the adiabatic algorithm applies to the task of choosing the QAOA initial state. Specifically, we observe that the best performance is obtained when the initial state of QAOA is set to be the ground state of the mixing Hamiltonian, as required by the adiabatic algorithm. We provide numerical evidence using the examples of constrained portfolio optimization problems with both low (p ≤ 3) and high (p = 100) QAOA depth. Additionally, we successfully apply QAOA with XY mixer to portfolio optimization on a trapped-ion quantum processor using 32 qubits and discuss our findings in near-term experiments.
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.
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 alternating operator ansatz (QAOA) is a promising gate-model metaheuristic for combinatorial optimization. Applying the algorithm to problems with constraints presents an implementation challenge for near-term quantum resources. This paper explores strategies for enforcing hard constraints by using XYXYXY Hamiltonians as mixing operators (mixers). Despite the complexity of simulating the XYXYXY model, we demonstrate that, for an integer variable admitting $\ensuremath{\kappa}$ discrete values represented through one-hot encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\ensuremath{\kappa})$. We also specify general strategies for implementing QAOA circuits on all-to-all connected hardware graphs and linearly connected hardware graphs inspired by fermionic simulation techniques. Performance is validated on graph-coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using XYXYXY mixers is borne out numerically, demonstrating a significant improvement over the general XXX mixer, and moreover the generalized WWW state yields better performance than easier-to-generate classical initial states when XYXYXY mixers are used.
The Quantum Approximate Optimization Algorithm and its generalization to Quantum Alternating Operator Ansatz (QAOA) is a promising approach for applying quantum computers to challenging problems such as combinatorial optimization and computational chemistry. In this paper, we study the underlying mechanisms governing the behavior of QAOA circuits beyond shallow depth in the practically relevant setting of gradually varying unitaries. We use the discrete adiabatic theorem, which complements and generalizes the insights obtained from the continuous-time adiabatic theorem primarily considered in prior work. Our analysis explains some general properties that are conspicuously depicted in the recently introduced QAOA performance diagrams. For parameter sequences derived from continuous schedules (e.g. linear ramps), these diagrams capture the algorithm's performance over different parameter sizes and circuit depths. Surprisingly, they have been observed to be qualitatively similar across different performance metrics and application domains. Our analysis explains this behavior as well as entails some unexpected results, such as connections between the eigenstates of the cost and mixer QAOA Hamiltonians changing based on parameter size and the possibility of reducing circuit depth without sacrificing performance.
Determining Hamiltonian ground states and energies is a challenging task with many possible approaches on quantum computers. While variational quantum eigensolvers are popular approaches for near term hardware, adiabatic state preparation is an alternative that does not require noisy optimization of parameters. Beyond adiabatic schedules, QAOA is an important method for optimization problems. In this work we modify QAOA to apply to finding ground states of molecules and empirically evaluate the modified algorithm on several molecules. This modification applies physical insights used in classical approximations to construct suitable QAOA operators and initial state. We find robust qualitative behavior for QAOA as a function of the number of steps and size of the parameters, and demonstrate this behavior also occurs in standard QAOA applied to combinatorial search. To this end we introduce QAOA phase diagrams that capture its performance and properties in various limits. In particular we show a region in which non-adiabatic schedules perform better than the adiabatic limit while employing lower quantum circuit depth. We further provide evidence our results and insights also apply to QAOA applications beyond chemistry.
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.