Parameter concentrations in quantum approximate optimization
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The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. In QAOA, a quantum circuit is trained---by repeatedly adjusting circuit parameters---to solve a problem instance. Several recent findings have reported parameter concentration effects in QAOA and their presence has become one of folklore: while empirically observed, the concentrations have not been defined and analytical approaches remain scarce, focusing on limiting system sizes while neglecting parameter scaling as system size increases. We found that optimal QAOA circuit parameters concentrate as an inverse polynomial in the problem size, providing an optimistic result for improving circuit training. Our results are analytically demonstrated for QAOA at $p=1,2$ (corresponding to 2 and 4 tunable parameters, respectively). The technique is also applicable for higher depths wherein the concentration effect is cross verified numerically. Parameter concentrations allow for training on a fraction $w
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Classes of problems exhibit pronounced parameter concentration of optimal or near-optimal angles.
“On the other hand, there is increasing evidence that classes of problems exhibit pronounced parameter concentration of the optimal or near optimal angles [19], [51], [1], [6], [61], [17], [48], [41]”
From High-Round QAOA for MAX $k$-SAT on Trapped Ion NISQ Devices (2023) · cites this paper as [1]
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Problem-size-independent angles for a Grover-driven quantum approximate optimization algorithm
Headley, D., & Wilhelm, F. K. (2023) · Physical Review A · doi:10.1103/physreva.107.012412 · Cited by 10 · see citing works
Quantum approximate optimization algorithm applied to the binary perceptron
Torta, P., Mbeng, G. B., Baldassi, C., Zecchina, R., & Santoro, G. E. (2023) · Physical review. B./Physical review. B · doi:10.1103/physrevb.107.094202 · Cited by 10 · see citing works
Robustness of variational quantum algorithms against stochastic parameter perturbation
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Self-citationProgress towards Analytically Optimal Angles in Quantum Approximate Optimisation
Rabinovich, D., Sengupta, R., Campos, E., Akshay, V., & Biamonte, J. (2022) · Mathematics · doi:10.3390/math10152601 · Cited by 9 · see citing works
Self-citationIon-native variational ansatz for quantum approximate optimization
Rabinovich, D., Adhikary, S., Campos, E., Akshay, V., Anikin, E. V., Sengupta, R., Lakhmanskaya, O., Lakhmanskiy, K., & Biamonte, J. (2022) · Physical Review A · doi:10.1103/physreva.106.032418 · Cited by 8 · see citing works
Self-citationNear-term quantum algorithm for solving the MaxCut problem with fewer quantum resources
Xiumei, Z., Li, Y., Li, J., Wang, S., Wang, S., Qin, S., & Gao, F. (2024) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2024.129951 · Cited by 8 · see citing works
Exponential qubit reduction in optimization for financial transaction settlement
Huber, E. X., Tan, B. Y. L., Griffin, P., & Angelakis, D. G. (2024) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-024-00262-w · Cited by 9 · see citing works
Yang, Y. (2025) · Frontiers in Physics · doi:10.3389/fphy.2025.1551209 · Cited by 7 · see citing works
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Self-citationEfficient quantum circuits based on the quantum natural gradient
Roy, A., Erramilli, S., & Konik, R. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043083 · Cited by 6 · see citing works
Solving Higher Order Binary Optimization Problems on NISQ Devices: Experiments and Limitations
Gilbert, V., Rodriguez, J., Louise, S., & Sirdey, R. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-36030-5_18 · Cited by 5 · see citing works
Parallel circuit implementation of variational quantum algorithms
Cattelan, M., Yarkoni, S., & Lechner, W. (2025) · npj Quantum Information · doi:10.1038/s41534-025-00982-6 · Cited by 8 · see citing works
Constant Depth Code Deformations in the Parity Architecture
Messinger, A., Fellner, M., & Lechner, W. (2023) · IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.00022 · Cited by 5 · see citing works
Fuzzy Clustering for QAOA Complexity Reduction
Acampora, G., Chiatto, A., & Vitiello, A. (2023) · IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) · doi:10.1109/fuzz52849.2023.10309767 · Cited by 4 · see citing works
A Method to Compute QAOA Fixed Angles
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Universal resources for quantum approximate optimization algorithm and quantum annealing
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Depth scaling of unstructured search via quantum approximate optimization
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Self-citationA Monte Carlo Tree Search approach to QAOA: finding a needle in the haystack
Agirre, A., van Nieuwenburg, E., & Wauters, M. M. (2025) · New Journal of Physics · doi:10.1088/1367-2630/adc765 · Cited by 3 · see citing works
Hybrid GRU-CNN bilinear parameters initialization for quantum approximate optimization algorithm
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Heuristic Time Complexity of NISQ Shortest-Vector-Problem Solvers
Prokop, M., & Wallden, P. (2025) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2025.3620104 · Cited by 2 · see citing works
Efficient large-scale quantum optimization via counterdiabatic ansatz
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Hierarchical QAOA circuit design framework for distributed quantum computing
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Pelofske, E., Rams, M. M., Bärtschi, A., Czarnik, P., Braccia, P., Cincio, Ł., & Eidenbenz, S. (2026) · Physical Review Research · doi:10.1103/p2lg-z4kn · Cited by 1 · see citing works
Efficient online quantum circuit learning with no upfront training
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Digital Zero-Noise Extrapolation with Quantum Circuit Unoptimization
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Variational Quantum Algorithm Parameter Tuning with Estimation of Distribution Algorithms
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Fuzzy Clustering-based Transferability of QAOA Parameters for Solving Vertex Cover Problems
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Entropic property of randomized QAOA circuits
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Exponential Qubit Reduction in Optimization for Financial Transaction SettlementPreprint
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Noise-Aware Machine Learning for Accelerating QAOA in Noisy Intermediate-Scale Quantum Computing
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Genetic Algorithm-based QAOA Training improved with Fuzzy Clustering
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CVAE-Guided QAOA Parameter Initialization and Dynamic Injection
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Sampling-Based Quantum Optimization Algorithm with Quantum Relaxation
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Approximating Ground States of Quantum Hamiltonians with Snapshot-QAOA
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Quantum optimization for the Physician Scheduling Problem
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Performance evaluation and control improvements for solving optimization problems on NISQ platforms
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Implementation of the Quantum Approximate Optimization Algorithm for Max-Cut
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Role of overparametrization in quantum approximate optimization
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Self-citationAnalytical Dimensions and Performance Evaluation of the Quantum Approximate Optimization Algorithm
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- Authors
- Akshay, V., Rabinovich, D., Campos, E., & Biamonte, J.
- Affiliations
- Skolkovo Institute of Science and Technology (RU)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2021-07-07
- Volume
- 104
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 2469-9926
- 2469-9934
- OpenAlex
- W3139223064
- Cited by
- 103
- References
- 47
- Citations by year
- 2026: 10 · 2025: 25 · 2024: 23 · 2023: 25 · 2022: 17 · 2021: 3
- FWCI
- 6.01
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University) (cc-by)
- Funders
- Leading Research Center on Quantum Computing (014/20)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Metaheuristic Optimization Algorithms Research
- Keywords
- quantum approximate optimization algorithm, variational quantum algorithms, circuit depth
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