Julien Gacon1,2, Christa Zoufal1,3, Giuseppe Carleo2, and Stefan Woerner1
1IBM Quantum, IBM Research – Zurich, CH-8803 Rüschlikon, Switzerland
2Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Institute for Theoretical Physics, ETH Zurich, CH-8092 Zürich, Switzerland
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Abstract
The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with $d$ parameters, however, is computationally expensive and generally requires $mathcal{O}(d^2)$ function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.
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► References
[1] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated Quantum Computation of Molecular Energies. Science, 309 (5741): 1704–1707, September 2005. 10.1126/science.1113479.
https://doi.org/10.1126/science.1113479
[2] Alberto Peruzzo et al. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5: 4213, July 2014. 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213
[3] Mari Carmen Bañuls et al. Simulating lattice gauge theories within quantum technologies. European Physical Journal D, 74 (8): 165, August 2020. 10.1140/epjd/e2020-100571-8.
https://doi.org/10.1140/epjd/e2020-100571-8
[4] Alejandro Perdomo-Ortiz, Neil Dickson, Marshall Drew-Brook, Geordie Rose, and Alán Aspuru-Guzik. Finding low-energy conformations of lattice protein models by quantum annealing. Scientific Reports, 2: 571, August 2012. 10.1038/srep00571.
https://doi.org/10.1038/srep00571
[5] Mark Fingerhuth, Tomáš Babej, and Christopher Ing. A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding. arXiv, October 2018. URL https://arxiv.org/abs/1810.13411.
arXiv:1810.13411
[6] Anton Robert, Panagiotis Kl. Barkoutsos, Stefan Woerner, and Ivano Tavernelli. Resource-efficient quantum algorithm for protein folding. npj Quantum Information, 7 (1): 38, February 2021. ISSN 2056-6387. 10.1038/s41534-021-00368-4.
https://doi.org/10.1038/s41534-021-00368-4
[7] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimization Algorithm. arXiv, November 2014. URL https://arxiv.org/abs/1411.4028.
arXiv:1411.4028
[8] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea. Grover Adaptive Search for Constrained Polynomial Binary Optimization. arXiv, December 2019. URL https://arxiv.org/abs/1912.04088. 10.22331/q-2021-04-08-428.
https://doi.org/10.22331/q-2021-04-08-428
arXiv:1912.04088
[9] Lee Braine, Daniel J. Egger, Jennifer Glick, and Stefan Woerner. Quantum Algorithms for Mixed Binary Optimization applied to Transaction Settlement. arXiv, October 2019. URL https://arxiv.org/abs/1910.05788. 10.1109/TQE.2021.3063635.
https://doi.org/10.1109/TQE.2021.3063635
arXiv:1910.05788
[10] J. Gacon, C. Zoufal, and S. Woerner. Quantum-enhanced simulation-based optimization. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 47–55, 2020. 10.1109/QCE49297.2020.00017.
https://doi.org/10.1109/QCE49297.2020.00017
[11] D. J. Egger et al. Quantum computing for finance: State-of-the-art and future prospects. IEEE Transactions on Quantum Engineering, 1: 1–24, 2020. 10.1109/TQE.2020.3030314.
https://doi.org/10.1109/TQE.2020.3030314
[12] J. S. Otterbach et al. Unsupervised Machine Learning on a Hybrid Quantum Computer. arXiv, December 2017. URL https://arxiv.org/abs/1712.05771.
arXiv:1712.05771
[13] Vojtěch Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature, 567 (7747): 209–212, March 2019. 10.1038/s41586-019-0980-2.
https://doi.org/10.1038/s41586-019-0980-2
[14] Maria Schuld. Quantum machine learning models are kernel methods. arXiv, January 2021. URL https://arxiv.org/abs/2101.11020.
arXiv:2101.11020
[15] Nikolaj Moll et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3 (3): 030503, July 2018. 10.1088/2058-9565/aab822.
https://doi.org/10.1088/2058-9565/aab822
[16] Sam McArdle et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1), Sep 2019. ISSN 2056-6387. 10.1038/s41534-019-0187-2.
https://doi.org/10.1038/s41534-019-0187-2
[17] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, October 2019. ISSN 2521-327X. 10.22331/q-2019-10-07-191.
https://doi.org/10.22331/q-2019-10-07-191
[18] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner. Variational quantum boltzmann machines. Quantum Machine Intelligence, 3: 7, 2020. ISSN 2524-4914. 10.1007/s42484-020-00033-7.
https://doi.org/10.1007/s42484-020-00033-7
[19] Taku Matsui. Quantum statistical mechanics and Feller semigroup. Quantum Probability Communications, 1998. 10.1142/9789812816054_0004.
https://doi.org/10.1142/9789812816054_0004
[20] Masoud Khalkhali and Matilde Marcolli. An Invitation to Noncommutative Geometry. World Scientific, 2008. 10.1142/6422.
https://doi.org/10.1142/6422
[21] J. Eisert, M. Friesdorf, and C. Gogolin. Quantum many-body systems out of equilibrium. Nature Physics, 11 (2), 2015. 10.1038/nphys3215.
https://doi.org/10.1038/nphys3215
[22] Fernando G. S. L. Brandão et al. Quantum SDP Solvers: Large speed-ups, optimality, and applications to quantum learning. arXiv, 2017. URL https://arxiv.org/abs/1710.02581.
arXiv:1710.02581
[23] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko. Quantum Boltzmann Machine. Phys. Rev. X, 8, 2018. 10.1103/PhysRevX.8.021050.
https://doi.org/10.1103/PhysRevX.8.021050
[24] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum natural gradient. Quantum, 4: 269, May 2020. ISSN 2521-327X. 10.22331/q-2020-05-25-269.
https://doi.org/10.22331/q-2020-05-25-269
[25] S. Amari and S. C. Douglas. Why natural gradient? In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP ’98 (Cat. No.98CH36181), volume 2, pages 1213–1216 vol.2, 1998. 10.1109/ICASSP.1998.675489.
https://doi.org/10.1109/ICASSP.1998.675489
[26] J.C. Spall. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control, 37 (3): 332–341, 1992. 10.1109/9.119632.
https://doi.org/10.1109/9.119632
[27] Lingyao Meng and James C. Spall. Efficient computation of the fisher information matrix in the em algorithm. In 2017 51st Annual Conference on Information Sciences and Systems (CISS), pages 1–6, 2017. 10.1109/CISS.2017.7926126.
https://doi.org/10.1109/CISS.2017.7926126
[28] A. Cauchy. Methode generale pour la resolution des systemes d’equations simultanees. C.R. Acad. Sci. Paris, 25: 536–538, 1847. 10.1017/cbo9780511702396.063.
https://doi.org/10.1017/cbo9780511702396.063
[29] J. C. Spall. Accelerated second-order stochastic optimization using only function measurements. In Proceedings of the 36th IEEE Conference on Decision and Control, volume 2, pages 1417–1424 vol.2, December 1997. 10.1109/CDC.1997.657661. ISSN: 0191-2216.
https://doi.org/10.1109/CDC.1997.657661
[30] Yuan Yao, Pierre Cussenot, Alex Vigneron, and Filippo M. Miatto. Natural Gradient Optimization for Optical Quantum Circuits. arXiv, June 2021. URL https://arxiv.org/abs/2106.13660.
arXiv:2106.13660
[31] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99 (3): 032331, March 2019. 10.1103/PhysRevA.99.032331.
https://doi.org/10.1103/PhysRevA.99.032331
[32] Johannes Jakob Meyer. Fisher Information in Noisy Intermediate-Scale Quantum Applications. Quantum, 5: 539, September 2021. ISSN 2521-327X. 10.22331/q-2021-09-09-539.
https://doi.org/10.22331/q-2021-09-09-539
[33] Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A, 103 (1): 012405, Jan 2021. 10.1103/PhysRevA.103.012405.
https://doi.org/10.1103/PhysRevA.103.012405
[34] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Phys. Rev. Lett., 87 (16): 167902, Sep 2001. 10.1103/PhysRevLett.87.167902.
https://doi.org/10.1103/PhysRevLett.87.167902
[35] Lukasz Cincio, Yiğit Subaşı, Andrew T. Sornborger, and Patrick J. Coles. Learning the quantum algorithm for state overlap. arXiv, November 2018. URL http://arxiv.org/abs/1803.04114. 10.1088/1367-2630/aae94a.
https://doi.org/10.1088/1367-2630/aae94a
arXiv:1803.04114
[36] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states. Phys. Rev. A, 99 (5), May 2019. 10.1103/PhysRevA.99.052323.
https://doi.org/10.1103/PhysRevA.99.052323
[37] Kristan Temme, Tobias J. Osborne, Karl Gerd H. Vollbrecht, David Poulin, and Frank Verstraete. Quantum Metropolis Sampling. Nature, 471, 2011. 10.1038/nature09770.
https://doi.org/10.1038/nature09770
[38] Man-Hong Yung and Alán Aspuru-Guzik. A quantum–quantum Metropolis algorithm. Proceedings of the National Academy of Sciences, 109 (3), 2012. 10.1073/pnas.1111758109.
https://doi.org/10.1073/pnas.1111758109
[39] David Poulin and Pawel Wocjan. Sampling from the Thermal Quantum Gibbs State and Evaluating Partition Functions with a Quantum Computer. Phys. Rev. Lett., 103 (22), 2009. 10.1103/PhysRevLett.103.220502.
https://doi.org/10.1103/PhysRevLett.103.220502
[40] Mario Motta and et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16 (2), 2020. 10.1038/s41567-019-0704-4.
https://doi.org/10.1038/s41567-019-0704-4
[41] Fernando G. S. L. Brandão and Michael J. Kastoryano. Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States. Communications in Mathematical Physics, 365 (1), 2019. 10.1007/s00220-018-3150-8.
https://doi.org/10.1007/s00220-018-3150-8
[42] Michael J. Kastoryano and Fernando G. S. L. Brandão. Quantum Gibbs Samplers: The Commuting Case. Communications in Mathematical Physics, 344 (3), 2016. 10.1007/s00220-016-2641-8.
https://doi.org/10.1007/s00220-016-2641-8
[43] Jingxiang Wu and Timothy H. Hsieh. Variational Thermal Quantum Simulation via Thermofield Double States. Phys. Rev. Lett., 123 (22), 2019. 10.1103/PhysRevLett.123.220502.
https://doi.org/10.1103/PhysRevLett.123.220502
[44] Anirban Chowdhury, Guang Hao Low, and Nathan Wiebe. A Variational Quantum Algorithm for Preparing Quantum Gibbs States. arXiv, 2020. URL https://arxiv.org/abs/2002.00055.
arXiv:2002.00055
[45] A.D. McLachlan. A variational solution of the time-dependent Schrödinger equation. Molecular Physics, 8 (1), 1964. 10.1080/00268976400100041.
https://doi.org/10.1080/00268976400100041
[46] Héctor Abraham et al. Qiskit: An open-source framework for quantum computing. 2019. 10.5281/zenodo.2562110.
https://doi.org/10.5281/zenodo.2562110
[47] IBM Quantum, 2021. URL https://quantum-computing.ibm.com/services/docs/services/runtime/.
https://quantum-computing.ibm.com/services/docs/services/runtime/
[48] Sergey Bravyi, Jay M. Gambetta, Antonio Mezzacapo, and Kristan Temme. Tapering off qubits to simulate fermionic hamiltonians. arXiv, 2017. URL https://arxiv.org/abs/1701.08213.
arXiv:1701.08213
[49] Abhinav Kandala et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, September 2017. 10.1038/nature23879.
https://doi.org/10.1038/nature23879
[50] Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567 (7749): 491–495, March 2019. 10.1038/s41586-019-1040-7.
https://doi.org/10.1038/s41586-019-1040-7
[51] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles. An Adaptive Optimizer for Measurement-Frugal Variational Algorithms. Quantum, 4: 263, May 2020. ISSN 2521-327X. 10.22331/q-2020-05-11-263.
https://doi.org/10.22331/q-2020-05-11-263
Cited by
[1] Tobias Haug, Kishor Bharti, and M. S. Kim, “Capacity and quantum geometry of parametrized quantum circuits”, arXiv:2102.01659.
[2] Johannes Jakob Meyer, “Fisher Information in Noisy Intermediate-Scale Quantum Applications”, arXiv:2103.15191.
[3] Tobias Haug and M. S. Kim, “Optimal training of variational quantum algorithms without barren plateaus”, arXiv:2104.14543.
[4] Tobias Haug and M. S. Kim, “Natural parameterized quantum circuit”, arXiv:2107.14063.
[5] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, “Theory of overparametrization in quantum neural networks”, arXiv:2109.11676.
[6] Christa Zoufal, David Sutter, and Stefan Woerner, “Error Bounds for Variational Quantum Time Evolution”, arXiv:2108.00022.
[7] Anna Lopatnikova and Minh-Ngoc Tran, “Quantum Natural Gradient for Variational Bayes”, arXiv:2106.05807.
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On Crossref’s cited-by service no data on citing works was found (last attempt 2021-10-23 12:31:36).
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