Dimensional Expressivity Analysis of Parametric Quantum Circuits

Lena Funcke1, Tobias Hartung2,3, Karl Jansen4, Stefan Kühn2, and Paolo Stornati4,5

1Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada
2Computation-Based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus
3Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
4NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany
5Institut für Physik, Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
37 pages, 13 figures, journal version


Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.

► BibTeX data

► References

[1] J. Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, 2018.

[2] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd. Quantum machine learning. Nature, 549: 195, 2017.

[3] R. Orús, S. Mugel, and E. Lizaso. Quantum computing for finance: Overview and prospects. Reviews in Physics, 4: 100028, 2019.

[4] A. Montanaro. Quantum algorithms: an overview. npj Quantum Information, 2: 15023, 2016.

[5] F. G. S. L. Brandao and K. M. Svore. Quantum speed-ups for solving semidefinite programs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 415–426, 2017.

[6] M. Troyer and U.-J. Wiese. Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations. Phys. Rev. Lett., 94: 170201, 2005.

[7] F. Arute et al. Quantum supremacy using a programmable superconducting processor. Nature, 574: 505, 2019.

[8] K. Temme, S. Bravyi, and J. M. Gambetta. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett., 119: 180509, 2017.

[9] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549: 242, 2017.

[10] S. Endo, S. C. Benjamin, and Y. Li. Practical quantum error mitigation for near-future applications. Phys. Rev. X, 8: 031027, 2018.

[11] K. Yeter-Aydeniz, R. C. Pooser, and G. Siopsis. Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and lanczos algorithms. npj Quantum Information, 6: 63, 2020.

[12] L. Funcke, T. Hartung, K. Jansen, S. Kühn, P. Stornati, and X. Wang. Measurement error mitigation in quantum computers through classical bit-flip correction. arXiv:2007.03663, 2020.

[13] A. Peruzzo, J. McClean, P. Shadbolt, M. Yung, X. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun., 5: 1, 2014.

[14] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New J. Phys., 18: 023023, 2016.

[15] Y. Wang et al. Quantum simulation of helium hydride cation in a solid-state spin register. ACS Nano, 9: 7769, 2015.

[16] C. Hempel et al. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X, 8: 031022, 2018.

[17] C. Kokail et al. Self-verifying variational quantum simulation of the lattice schwinger model. Nature, 569: 355, 2019.

[18] T. Hartung and K. Jansen. Zeta-regularized vacuum expectation values. J. Math. Phys., 60: 093504, 2019.

[19] K. Jansen and T. Hartung. Zeta-regularized vacuum expectation values from quantum computing simulations. PoS (LATTICE 2019) 363, page 153, 2020.

[20] D. Paulson, L. Dellantonio, J. F. Haase, A. Celi, A. Kan, A. Jena, C. Kokail, R. van Bijnen, K. Jansen, P. Zoller, and C. A. Muschik. Towards simulating 2d effects in lattice gauge theories on a quantum computer. arXiv:2008.09252, 2020.

[21] J. F. Haase, L. Dellantonio, A. Celi, D. Paulson, A. Kan, K. Jansen, and C. A. Muschik. A resource efficient approach for quantum and classical simulations of gauge theories in particle physics. Quantum, 5: 393, 2021.

[22] M. R. Geller. Sampling and scrambling on a chain of superconducting qubits. Physical Review Applied, 10: 024052, 2018.

[23] S. Sim, P. D. Johnson, and A. Aspuru‐Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum‐classical algorithms. Adv. Quantum Technol., 2: 1900070, 2019.

[24] M. Bataille. Quantum circuits of CNOT gates. arXiv:2009.13247, 2020.

[25] S. Sim, J. Romero, J. F. Gonthier, and A. A. Kunitsa. Adaptive pruning-based optimization of parameterized quantum circuits. Quantum Sci. Technol., 6: 025019, 2020.

[26] S. E. Rasmussen, N. J. S. Loft, T. Bækkegaard, M. Kues, and N. T. Zinner. Reducing the amount of single‐qubit rotations in vqe and related algorithms. Adv. Quantum Technol., 3: 2000063, 2020.

[27] T. Hubregtsen, J. Pichlmeier, P. Stecher, and K. Bertels. Evaluation of parameterized quantum circuits: on the relation between classification accuracy, expressibility and entangling capability. arXiv:2003.09887, 2020.

[28] M. Schuld, R. Sweke, and J. J. Meyer. The effect of data encoding on the expressive power of variational quantum machine learning models. arXiv:2008.08605, 2020.

[29] E. Fontana, N. Fitzpatrick, D. Muños Ramo, R. Duncan, and I. Rungger. Evaluating the noise resilience of variational quantum algorithms. arXiv:2011.01125, 2020.

[30] B. T. Gard, L. Zhu, G. S. Barron, N. J. Mayhall, S. E. Economou, and E. Barnes. Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm. npj Quantum Information, 6: 10, 2020.

[31] G. S. Barron, B. T. Gard, O. J. Altman, N. J. Mayhall, E. Barnes, and S. E. Economou. Preserving symmetries for variational quantum eigensolvers in the presence of noise. arXiv:2003.00171, 2020.

[32] J. Kim, J. Kim, and D. Rosa. Universal effectiveness of high-depth circuits in variational eigenproblems. arXiv:2010.00157, 2020.

[33] H. Abraham et al. Qiskit: An Open-source Framework for Quantum Computing. Zenodo, 2019.

[34] Qiskit documentation, 2020. https:/​/​qiskit.org/​documentation/​_modules/​qiskit/​circuit/​library/​n_local/​efficient_su2.html. accessed on 2020/​07/​14.

[35] L. Zhao, Z. Zhao, P. Rebentrost, and J. Fitzsimons. Compiling basic linear algebra subroutines for quantum computers. arXiv:1902.10394, 2019.

[36] S. Lloyd, M. Mohseni, and P. Rebentrost. Quantum algorithms for supervised and unsupervised machine learning. arXiv:1307.0411, 2013.

[37] IBM Quantum team. Retrieved from https:/​/​quantum-computing.ibm.com, 2020.

Cited by

[1] Kamal Choudhary, "Quantum computation for predicting electron and phonon properties of solids", Journal of Physics: Condensed Matter 33 38, 385501 (2021).

[2] Alona Sakhnenko, Corey O’Meara, Kumar J. B. Ghosh, Christian B. Mendl, Giorgio Cortiana, and Juan Bernabé-Moreno, "Hybrid classical-quantum autoencoder for anomaly detection", Quantum Machine Intelligence 4 2, 27 (2022).

[3] Joonho Kim and Yaron Oz, "Quantum energy landscape and circuit optimization", Physical Review A 106 5, 052424 (2022).

[4] Benedikt Fauseweh and Jian-Xin Zhu, "Quantum computing Floquet energy spectra", Quantum 7, 1063 (2023).

[5] Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao, "Efficient Measure for the Expressivity of Variational Quantum Algorithms", Physical Review Letters 128 8, 080506 (2022).

[6] David Fitzek, Robert S. Jonsson, Werner Dobrautz, and Christian Schäfer, "Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes", Quantum 8, 1313 (2024).

[7] Fernando Gago-Encinas, Tobias Hartung, Daniel M. Reich, Karl Jansen, and Christiane P. Koch, "Determining the ability for universal quantum computing: Testing controllability via dimensional expressivity", Quantum 7, 1214 (2023).

[8] Xiaoyang Wang, Xu Feng, Tobias Hartung, Karl Jansen, and Paolo Stornati, "Critical behavior of the Ising model by preparing the thermal state on a quantum computer", Physical Review A 108 2, 022612 (2023).

[9] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[10] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics & Astronomy 65 2, 220301 (2022).

[11] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[12] N. Schetakis, D. Aghamalyan, P. Griffin, and M. Boguslavsky, "Review of some existing QML frameworks and novel hybrid classical–quantum neural networks realising binary classification for the noisy datasets", Scientific Reports 12 1, 11927 (2022).

[13] L. Funcke, T. Hartung, K. Jansen, S. Kühn, M. Schneider, P. Stornati, and X. Wang, "Towards quantum simulations in particle physics and beyond on noisy intermediate-scale quantum devices", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 2216, 20210062 (2022).

[14] Emilie Huffman, Miguel García Vera, and Debasish Banerjee, "Toward the real-time evolution of gauge-invariant Z2 and U(1) quantum link models on noisy intermediate-scale quantum hardware with error mitigation", Physical Review D 106 9, 094502 (2022).

[15] Gabriel Matos, Chris N. Self, Zlatko Papić, Konstantinos Meichanetzidis, and Henrik Dreyer, "Characterization of variational quantum algorithms using free fermions", Quantum 7, 966 (2023).

[16] Weikang Li, Zhi-de Lu, and Dong-Ling Deng, "Quantum Neural Network Classifiers: A Tutorial", SciPost Physics Lecture Notes 61 (2022).

[17] Hiroshi C. Watanabe, Rudy Raymond, Yu-Ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, "Optimizing Parameterized Quantum Circuits With Free-Axis Single-Qubit Gates", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[18] Joonho Kim and Yaron Oz, "Entanglement diagnostics for efficient VQA optimization", Journal of Statistical Mechanics: Theory and Experiment 2022 7, 073101 (2022).

[19] Amara Katabarwa, Sukin Sim, Dax Enshan Koh, and Pierre-Luc Dallaire-Demers, "Connecting geometry and performance of two-qubit parameterized quantum circuits", Quantum 6, 782 (2022).

[20] Hiroshi C. Watanabe, Rudy Raymond, Yu-Ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 100 (2021) ISBN:978-1-6654-1691-7.

[21] Ivana Miháliková, Matej Pivoluska, Martin Plesch, Martin Friák, Daniel Nagaj, and Mojmír Šob, "The Cost of Improving the Precision of the Variational Quantum Eigensolver for Quantum Chemistry", Nanomaterials 12 2, 243 (2022).

[22] Kumar J. B. Ghosh and Sumit Ghosh, "Exploring exotic configurations with anomalous features with deep learning: Application of classical and quantum-classical hybrid anomaly detection", Physical Review B 108 16, 165408 (2023).

[23] Yasar Y. Atas, Jinglei Zhang, Randy Lewis, Amin Jahanpour, Jan F. Haase, and Christine A. Muschik, "SU(2) hadrons on a quantum computer via a variational approach", Nature Communications 12 1, 6499 (2021).

[24] Mahabubul Alam and Swaroop Ghosh, "QNet: A Scalable and Noise-Resilient Quantum Neural Network Architecture for Noisy Intermediate-Scale Quantum Computers", Frontiers in Physics 9, 755139 (2022).

[25] Massimiliano Incudini, Fabio Tarocco, Riccardo Mengoni, Alessandra Di Pierro, and Antonio Mandarino, "Computing graph edit distance on quantum devices", Quantum Machine Intelligence 4 2, 24 (2022).

[26] Tobias Stollenwerk and Stuart Hadfield, "Diagrammatic Analysis for Parameterized Quantum Circuits", Electronic Proceedings in Theoretical Computer Science 394, 262 (2023).

[27] Patrick Selig, Niall Murphy, Ashwin Sundareswaran R, David Redmond, and Simon Caton, 2021 International Conference on Rebooting Computing (ICRC) 24 (2021) ISBN:978-1-6654-2332-8.

[28] Christian W. Bauer, Zohreh Davoudi, A. Baha Balantekin, Tanmoy Bhattacharya, Marcela Carena, Wibe A. de Jong, Patrick Draper, Aida El-Khadra, Nate Gemelke, Masanori Hanada, Dmitri Kharzeev, Henry Lamm, Ying-Ying Li, Junyu Liu, Mikhail Lukin, Yannick Meurice, Christopher Monroe, Benjamin Nachman, Guido Pagano, John Preskill, Enrico Rinaldi, Alessandro Roggero, David I. Santiago, Martin J. Savage, Irfan Siddiqi, George Siopsis, David Van Zanten, Nathan Wiebe, Yukari Yamauchi, Kübra Yeter-Aydeniz, and Silvia Zorzetti, "Quantum Simulation for High-Energy Physics", PRX Quantum 4 2, 027001 (2023).

[29] Tom Weber, Kerstin Borras, Karl Jansen, Dirk Krücker, and Matthias Riebisch, "Construction and volumetric benchmarking of quantum computing noise models", Physica Scripta 99 6, 065106 (2024).

[30] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin, "Modern applications of machine learning in quantum sciences", arXiv:2204.04198, (2022).

[31] Tobias Haug, Kishor Bharti, and M. S. Kim, "Capacity and Quantum Geometry of Parametrized Quantum Circuits", PRX Quantum 2 4, 040309 (2021).

[32] R. R. Ferguson, L. Dellantonio, A. Al Balushi, K. Jansen, W. Dür, and C. A. Muschik, "Measurement-Based Variational Quantum Eigensolver", Physical Review Letters 126 22, 220501 (2021).

[33] Samuel A Wilkinson and Michael J Hartmann, "Evaluating the performance of sigmoid quantum perceptrons in quantum neural networks", arXiv:2208.06198, (2022).

[34] Tobias Stollenwerk and Stuart Hadfield, "Diagrammatic Analysis for Parameterized Quantum Circuits", arXiv:2204.01307, (2022).

[35] Iván Panadero, Yue Ban, Hilario Espinós, Ricardo Puebla, Jorge Casanova, and Erik Torrontegui, "Regressions on quantum neural networks at maximal expressivity", arXiv:2311.06090, (2023).

[36] Hiroshi C. Watanabe, Rudy Raymond, Yu-ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, "Optimizing Parameterized Quantum Circuits with Free-Axis Selection", arXiv:2104.14875, (2021).

[37] Tom Weber, Matthias Riebisch, Kerstin Borras, Karl Jansen, and Dirk Krücker, "Modelling for Quantum Error Mitigation", arXiv:2104.07320, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 17:46:55) and SAO/NASA ADS (last updated successfully 2024-05-21 17:46:56). The list may be incomplete as not all publishers provide suitable and complete citation data.