Abstract
Parameterized quantum circuits (PQCs) are fundamental to many hybrid quantum-classical algorithms. However, existing structure-based optimizers for the training of PQCs, such as Rotosolve and sequential minimal optimization, rely on heuristic node selection or ignore statistical noise, which limits their robustness and accuracy. To address this issue, we propose an interpolation-based coordinate descent (ICD) method as a unified framework for all structure-based optimizers. ICD approximates the cost function through interpolation, recovers its trigonometric structure, and performs global one-dimensional updates on individual parameters. Unlike previous methods, ICD derives optimal interpolation nodes that minimize statistical errors from measurements. For the common case of r equidistant frequencies, we prove that equidistant nodes with spacing 2π/(2r + 1) jointly minimize the mean squared error of Fourier coefficient estimates, the condition number of the interpolation matrix, and the average variance of the approximated cost function. Numerical experiments confirm the superior robustness and efficiency of ICD over gradient-based methods.
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The data generated and analyzed in this study are available in Zenodo67.
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References
Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).
Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).
Shang, Z.-X., Chen, M.-C., Yuan, X., Lu, C.-Y. & Pan, J.-W. Schrödinger-heisenberg variational quantum algorithms. Phys. Rev. Lett. 131, 060406 (2023).
Yuan, X., Endo, S., Zhao, Q., Li, Y. & Benjamin, S. C. Theory of variational quantum simulation. Quantum 3, 191 (2019).
McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5, 75 (2019).
Endo, S., Sun, J., Li, Y., Benjamin, S. C. & Yuan, X. Variational quantum simulation of general processes. Phys. Rev. Lett. 125, 010501 (2020).
Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028 (2014).
Zhou, L., Wang, S.-T., Choi, S., Pichler, H. & Lukin, M. D. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices. Phys. Rev. X 10, 021067 (2020).
Blekos, K. et al. A review on quantum approximate optimization algorithm and its variants. Phys. Rep. 1068, 1–66 (2024).
Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122, 040504 (2019).
Schuld, M., Bocharov, A., Svore, K. M. & Wiebe, N. Circuit-centric quantum classifiers. Phys. Rev. A 101, 032308 (2020).
Perdomo-Ortiz, A., Benedetti, M., Realpe-Gómez, J. & Biswas, R. Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers. Quantum Sci. Technol. 3, 030502 (2018).
Killoran, N. et al. Continuous-variable quantum neural networks. Phys. Rev. Res. 1, 033063 (2019).
Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol. 4, 043001 (2019).
Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E. & Latorre, J. I. Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020).
Yu, Z., Yao, H., Li, M. & Wang, X. Power and limitations of single-qubit native quantum neural networks. Adv. Neural Inf. Process. Syst. 35, 27810–27823 (2022).
Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).
Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403–409 (2021).
Sweke, R. et al. Stochastic gradient descent for hybrid quantum-classical optimization. Quantum 4, 314 (2020).
Wierichs, D., Izaac, J., Wang, C. & Lin, C. Y.-Y. General parameter-shift rules for quantum gradients. Quantum 6, 677 (2022).
Mari, A., Bromley, T. R. & Killoran, N. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A 103, 012405 (2021).
Ding, Z., Ko, T., Yao, J., Lin, L. & Li, X. Random coordinate descent: a simple alternative for optimizing parameterized quantum circuits. Phys. Rev. Res. 6, 033029 (2024).
Powell, M. J. A direct search optimization method that models the objective and constraint functions by linear interpolation. In Advances in Optimization and Numerical Analysis (eds Gomez, S. & Hennart, J.-P.) 51–67 (Springer, 1994).
Nelder, J. A. & Mead, R. A simplex method for function minimization. Comput. J. 7, 308–313 (1965).
Powell, M. J. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J. 7, 155–162 (1964).
Spall, J. C. Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Autom. Control 45, 1839–1853 (2000).
Pellow-Jarman, A., Sinayskiy, I., Pillay, A. & Petruccione, F. A comparison of various classical optimizers for a variational quantum linear solver. Quantum Inf. Process. 20, 202 (2021).
Lockwood, O. An empirical review of optimization techniques for quantum variational circuits. arXiv preprint https://doi.org/10.48550/arXiv.2202.01389 (2022).
Nocedal, J. & Wright, S. J. Numerical Optimization (Springer, 2006).
Byrd, R. H., Lu, P., Nocedal, J. & Zhu, C. A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995).
Kingma, D. P. & Ba, J. Adam: A method for stochastic optimization. In Proc. International Conference on Learning Representations (ICLR 2015).
Reddi, S. J., Kale, S. & Kumar, S. On the convergence of adam and beyond. In Proc. International Conference on Learning Representations (ICLR 2018).
Stokes, J., Izaac, J., Killoran, N. & Carleo, G. Quantum natural gradient. Quantum 4, 269 (2020).
Crooks, G. E. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv preprint https://doi.org/10.48550/arXiv.1905.13311 (2019).
Kyriienko, O. & Elfving, V. E. Generalized quantum circuit differentiation rules. Phys. Rev. A 104, 052417 (2021).
Hai, V. T. & Ho, L. B. Lagrange interpolation approach for general parameter-shift rule. In Quantum Computing: Circuits, Systems, Automation and Applications (eds Thapliyal, H. & Humble, T.) 1–17 (Springer, 2023).
Markovich, L., Malikis, S., Polla, S. & Tura, J. Parameter shift rule with optimal phase selection. Phys. Rev. A 109, 062429 (2024).
Hoch, F. et al. Variational approach to photonic quantum circuits via the parameter shift rule. Phys. Rev. Res. 7, 023227 (2025).
Ostaszewski, M., Grant, E. & Benedetti, M. Structure optimization for parameterized quantum circuits. Quantum 5, 391 (2021).
PennyLane. qml.RotosolveOptimizer https://docs.pennylane.ai/en/stable/code/api/pennylane.RotosolveOptimizer.html (2024).
TensorFlow Quantum. tfq.optimizers.rotosolve_minimize https://www.tensorflow.org/quantum/api_docs/python/tfq/optimizers/rotosolve_minimize (2024).
Vidal, J. G. & Theis, D. O. Calculus on parameterized quantum circuits. arXiv preprint https://doi.org/10.48550/arXiv.1812.06323 (2018).
Parrish, R. M., Iosue, J. T., Ozaeta, A. & McMahon, P. L. A Jacobi diagonalization and Anderson acceleration algorithm for variational quantum algorithm parameter optimization. arXiv preprint https://doi.org/10.48550/arXiv.1904.03206 (2019).
Nakanishi, K. M., Fujii, K. & Todo, S. Sequential minimal optimization for quantum-classical hybrid algorithms. Phys. Rev. Res. 2, 043158 (2020).
Jäger, J., Kaldenbach, T. N., Haas, M. & Schultheis, E. Fast gradient-free optimization of excitations in variational quantum eigensolvers. Commun. Phys. 8, 418 (2025).
Nemkov, N. A., Kiktenko, E. O. & Fedorov, A. K. Fourier expansion in variational quantum algorithms. Phys. Rev. A 108, 032406 (2023).
Fontana, E., Rungger, I., Duncan, R. & Círstoiu, C. Efficient recovery of variational quantum algorithms landscapes using classical signal processing. arXiv preprint https://doi.org/10.48550/arXiv.2208.05958 (2022).
Okumura, S. & Ohzeki, M. Fourier coefficient of parameterized quantum circuits and barren plateau problem. arXiv preprint https://doi.org/10.48550/arXiv.2309.06740 (2023).
Stȩchły, M., Gao, L., Yogendran, B., Fontana, E. & Rudolph, M. Connecting the Hamiltonian structure to the QAOA energy and Fourier landscape structure. arXiv preprint https://doi.org/10.48550/arXiv.2305.13594 (2023).
Boyd, J. P. Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding. J. Eng. Math. 56, 203–219 (2006).
McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, https://doi.org/10.1038/s41467-018-07090-4 (2018).
Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge University Press, 2010).
Shi, H.-J. M., Tu, S., Xu, Y. & Yin, W. A primer on coordinate descent algorithms. arXiv preprint https://doi.org/10.48550/arXiv.1610.00040 (2016).
Horn, R. A. & Johnson, C. R. Matrix Analysis (Cambridge University Press, 2012).
Trefethen, L. N. & Bau, D. Numerical Linear Algebra, Twenty-fifth Anniversary Edition (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2022).
Community, Q. Qiskit: An Open-source Framework For Quantum Computing https://github.com/Qiskit/qiskit (2017).
Wiersema, R. et al. Exploring entanglement and optimization within the Hamiltonian variational ansatz. PRX Quantum 1, 020319 (2020).
Grange, C., Poss, M. & Bourreau, E. An introduction to variational quantum algorithms for combinatorial optimization problems. 4OR 21, 363–403 (2023).
Larocca, M. et al. Barren plateaus in variational quantum computing. Nat. Rev. Phys. 7, 174–189 (2025).
Arrasmith, A., Holmes, Z., Cerezo, M. & Coles, P. J. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. Quantum Sci. Technol. 7, 045015 (2022).
Wiersema, R., Kökcü, E., Kemper, A. F. & Bakalov, B. N. Classification of dynamical lie algebras of 2-local spin systems on linear, circular and fully connected topologies. npj Quantum Inf. 10, 110 (2024).
Schreiber, F. J., Eisert, J. & Meyer, J. J. Classical surrogates for quantum learning models. Phys. Rev. Lett. 131, 100803 (2023).
Xu, Y. & Yin, W. A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion. SIAM J. Imaging Sci. 6, 1758–1789 (2013).
Bittel, L. & Kliesch, M. Training variational quantum algorithms is NP-hard. Phys. Rev. Lett. 127, 120502 (2021).
Warga, J. Minimizing certain convex functions. J. Soc. Ind. Appl. Math. 11, 588–593 (1963).
Lai, Z. OICD_for_VQA, v1.0.0. Zenodo. https://doi.org/10.5281/zenodo.17733224 (2025).
Acknowledgements
This work was supported by the National Natural Science Foundation of China under the grant numbers 12501419, 12288101, and 12331010, and the National Key R&D Program of China under the grant number 2024YFA1012901. D.A. acknowledges the support by the Quantum Science and Technology-National Science and Technology Major Project via Project 2024ZD0301900, and the Fundamental Research Funds for the Central Universities, Peking University. Part of this work was completed while JH was affiliated with UC Berkeley. We would like to express our sincere gratitude to Liyuan Cao, Zhiyan Ding, Tianyou Li, Xiantao Li, Xiufan Li, Lin Lin, Yin Liu, and Zaiwen Wen for their valuable feedback and insightful comments on the manuscript.
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Z.L. conceived the idea and carried out the theoretical analysis. Z.L., T.K. and J.W. performed the numerical simulations. Z.L., J.H. and D.A. analyzed the results. All authors contributed to the preparation of the manuscript.
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Lai, Z., Hu, J., Ko, T. et al. Interpolation-based coordinate descent method for parameterized quantum circuits. Commun Phys (2026). https://doi.org/10.1038/s42005-025-02473-8
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DOI: https://doi.org/10.1038/s42005-025-02473-8
