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

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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.

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► 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] Tobias Haug, Kishor Bharti, and M. S. Kim, "Capacity and quantum geometry of parametrized quantum circuits", arXiv:2102.01659.

[2] Kamal Choudhary, "Quantum Computation for Predicting Solids-state Material Properties", arXiv:2102.11452.

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

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