11.4 C
New York

Non-trivial symmetries in quantum landscapes and their resilience to quantum noise


Enrico Fontana1,2,3, M. Cerezo1,4, Andrew Arrasmith1, Ivan Rungger5, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
3National Physical Laboratory, Teddington TW11 0LW, UK
4Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
5National Physical Laboratory, Teddington, UK

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


Very little is known about the cost landscape for parametrized Quantum Circuits (PQCs). Nevertheless, PQCs are employed in Quantum Neural Networks and Variational Quantum Algorithms, which may allow for near-term quantum advantage. Such applications require good optimizers to train PQCs. Recent works have focused on quantum-aware optimizers specifically tailored for PQCs. However, ignorance of the cost landscape could hinder progress towards such optimizers. In this work, we analytically prove two results for PQCs: (1) We find an exponentially large symmetry in PQCs, yielding an exponentially large degeneracy of the minima in the cost landscape. Alternatively, this can be cast as an exponential reduction in the volume of relevant hyperparameter space. (2) We study the resilience of the symmetries under noise, and show that while it is conserved under unital noise, non-unital channels can break these symmetries and lift the degeneracy of minima, leading to multiple new local minima. Based on these results, we introduce an optimization method called Symmetry-based Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs. Our numerical simulations show that SYMH improves the overall optimizer performance in the presence of non-unital noise at a level comparable to current hardware. Overall, this work derives large-scale circuit symmetries from local gate transformations, and uses them to construct a noise-aware optimization method.

In this work, we study the cost landscape for parametrized quantum circuits (PQCs), which are employed in quantum neural networks and variational quantum algorithms. We unravel the presence of an exponentially large symmetry in the PQCs landscape, yielding an exponentially large degeneracy of the cost function minima. We then study the resilience of these symmetries under quantum noise, and show that while they are conserved under unital noise, non-unital channels can break these symmetries and lift the degeneracy of minima. Based on these results, we introduce an optimization method called Symmetry-based Minima Hopping (SYMH), which exploits the underlying symmetries in PQCs.

► BibTeX data

► References

[1] J. Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, 2018. 10.22331/​q-2018-08-06-79.

[2] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics, 3 (1): 625–644, 2021a. 10.1038/​s42254-021-00348-9. URL https:/​/​www.nature.com/​articles/​s42254-021-00348-9.

[3] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5: 4213, 2014. 10.1038/​ncomms5213. URL https:/​/​www.nature.com/​articles/​ncomms5213.

[4] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, 2016. 10.1088/​1367-2630/​18/​2/​023023. URL https:/​/​iopscience.iop.org/​article/​10.1088/​1367-2630/​18/​2/​023023/​meta.

[5] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014. 10.48550/​arXiv.1411.4028. URL https:/​/​arxiv.org/​abs/​1411.4028.

[6] J. Romero, J. P. Olson, and A. Aspuru-Guzik. Quantum autoencoders for efficient compression of quantum data. Quantum Science and Technology, 2 (4): 045001, Dec 2017. 10.1088/​2058-9565/​aa8072. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​aa8072.

[7] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger, and Patrick J. Coles. Quantum-assisted quantum compiling. Quantum, 3: 140, May 2019. ISSN 2521-327X. 10.22331/​q-2019-05-13-140. URL https:/​/​doi.org/​10.22331/​q-2019-05-13-140.

[8] R. LaRose, A. Tikku, É. O’Neel-Judy, L. Cincio, and P. J. Coles. Variational quantum state diagonalization. npj Quantum Information, 5: 1–10, 2018. 10.1038/​s41534-019-0167-6. URL https:/​/​www.nature.com/​articles/​s41534-019-0167-6.

[9] A. Arrasmith, L. Cincio, A. T. Sornborger, W. H. Zurek, and P. J. Coles. Variational consistent histories as a hybrid algorithm for quantum foundations. Nature communications, 10 (1): 3438, 2019. 10.1038/​s41467-019-11417-0. URL https:/​/​www.nature.com/​articles/​s41467-019-11417-0.

[10] M. Cerezo, Alexander Poremba, Lukasz Cincio, and Patrick J Coles. Variational quantum fidelity estimation. Quantum, 4: 248, 2020a. 10.22331/​q-2020-03-26-248.

[11] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. Variational fast forwarding for quantum simulation beyond the coherence time. npj Quantum Information, 6 (1): 1–10, 2020. URL 10.1038/​s41534-020-00302-0.

[12] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick Coles. Variational quantum linear solver. arXiv preprint arXiv:1909.05820, 2019. 10.48550/​arXiv.1909.05820. URL https:/​/​arxiv.org/​abs/​1909.05820.

[13] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J Coles. Variational quantum state eigensolver. arXiv preprint arXiv:2004.01372, 2020b. 10.48550/​arXiv.2004.01372. URL https:/​/​arxiv.org/​abs/​2004.01372.

[14] Ivan Rungger, Nathan Fitzpatrick, Honxiang Chen, CH Alderete, Harriett Apel, Alexander Cowtan, Andrew Patterson, D Munoz Ramo, Yingyue Zhu, Nhung Hong Nguyen, et al. Dynamical mean field theory algorithm and experiment on quantum computers. arXiv preprint arXiv:1910.04735, 2019. 10.48550/​arXiv.1910.04735. URL https:/​/​arxiv.org/​abs/​1910.04735.

[15] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. The quest for a quantum neural network. Quantum Information Processing, 13 (11): 2567–2586, 2014. 10.1007/​s11128-014-0809-8. URL https:/​/​link.springer.com/​article/​10.1007/​s11128-014-0809-8.

[16] Iris Cong, Soonwon Choi, and Mikhail D Lukin. Quantum convolutional neural networks. Nature Physics, 15 (12): 1273–1278, 2019. 10.1038/​s41567-019-0648-8. URL https:/​/​www.nature.com/​articles/​s41567-019-0648-8.

[17] Kerstin Beer, Dmytro Bondarenko, Terry Farrelly, Tobias J Osborne, Robert Salzmann, Daniel Scheiermann, and Ramona Wolf. Training deep quantum neural networks. Nature Communications, 11 (1): 1–6, 2020. 10.1038/​s41467-020-14454-2. URL https:/​/​www.nature.com/​articles/​s41467-020-14454-2.

[18] Guillaume Verdon, Jason Pye, and Michael Broughton. A universal training algorithm for quantum deep learning. arXiv preprint arXiv:1806.09729, 2018. 10.48550/​arXiv.1806.09729. URL https:/​/​arxiv.org/​abs/​1806.09729.

[19] Andrew Patterson, Hongxiang Chen, Leonard Wossnig, Simone Severini, Dan Browne, and Ivan Rungger. Quantum state discrimination using noisy quantum neural networks. Physical Review Research, 3 (1): 013063, 2021. 10.1103/​PhysRevResearch.3.013063. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.3.013063.

[20] Patrick Huembeli and Alexandre Dauphin. Characterizing the loss landscape of variational quantum circuits. Quantum Science and Technology, 6 (2): 025011, 2021. 10.1088/​2058-9565/​abdbc9. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​abdbc9.

[21] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. Quantum circuit learning. Phys. Rev. A, 98 (3): 032309, 2018. 10.1103/​PhysRevA.98.032309. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.98.032309.

[22] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99 (3): 032331, 2019. 10.1103/​PhysRevA.99.032331. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.99.032331.

[23] Kosuke Mitarai and Keisuke Fujii. Methodology for replacing indirect measurements with direct measurements. Physical Review Research, 1 (1): 013006, 2019. 10.1103/​PhysRevResearch.1.013006. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.1.013006.

[24] M. Cerezo and Patrick J Coles. Higher order derivatives of quantum neural networks with barren plateaus. Quantum Science and Technology, 6 (2): 035006, 2021. 10.1088/​2058-9565/​abf51a. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​abf51a.

[25] Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A, 103: 012405, Jan 2021. 10.1103/​PhysRevA.103.012405. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.103.012405.

[26] Jonas M Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J Coles. An adaptive optimizer for measurement-frugal variational algorithms. Quantum, 4: 263, 2020. 10.22331/​q-2020-05-11-263. URL https:/​/​quantum-journal.org/​papers/​q-2020-05-11-263/​.

[27] Ken M Nakanishi, Keisuke Fujii, and Synge Todo. Sequential minimal optimization for quantum-classical hybrid algorithms. Physical Review Research, 2 (4): 043158, 2020a. URL 10.1103/​PhysRevResearch.2.043158.

[28] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, 9 (1): 4812, 2018. 10.1038/​s41467-018-07090-4. URL https:/​/​www.nature.com/​articles/​s41467-018-07090-4.

[29] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications, 12 (1): 1–12, 2021b. 10.1038/​s41467-021-21728-w. URL https:/​/​www.nature.com/​articles/​s41467-021-21728-w.

[30] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J Coles. Trainability of dissipative perceptron-based quantum neural networks. Physical Review Letters, 128 (18): 180505, 2022. 10.1103/​PhysRevLett.128.180505.

[31] Zoë Holmes, Andrew Arrasmith, Bin Yan, Patrick J. Coles, Andreas Albrecht, and Andrew T Sornborger. Barren plateaus preclude learning scramblers. Physical Review Letters, 126 (19): 190501, 2021. 10.1103/​PhysRevLett.126.190501. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.126.190501.

[32] Arthur Pesah, M. Cerezo, Samson Wang, Tyler Volkoff, Andrew T Sornborger, and Patrick J Coles. Absence of barren plateaus in quantum convolutional neural networks. Physical Review X, 11 (4): 041011, 2021. 10.1103/​PhysRevX.11.041011. URL https:/​/​journals.aps.org/​prx/​abstract/​10.1103/​PhysRevX.11.041011.

[33] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. Entanglement-induced barren plateaus. PRX Quantum, 2 (4): 040316, 2021. 10.1103/​PRXQuantum.2.040316.

[34] Kathleen E Hamilton, Tyler Kharazi, Titus Morris, Alexander J McCaskey, Ryan S Bennink, and Raphael C Pooser. Scalable quantum processor noise characterization. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 430–440. IEEE, 2020. 10.1109/​QCE49297.2020.00060. URL https:/​/​ieeexplore.ieee.org/​abstract/​document/​9259938.

[35] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications, 12 (1): 1–11, 2021. 10.1038/​s41467-021-27045-6. URL https:/​/​www.nature.com/​articles/​s41467-021-27045-6.

[36] Kunal Sharma, Sumeet Khatri, M. Cerezo, and Patrick J Coles. Noise resilience of variational quantum compiling. New Journal of Physics, 22 (4): 043006, 2020. 10.1088/​1367-2630/​ab784c. URL https:/​/​iopscience.iop.org/​article/​10.1088/​1367-2630/​ab784c.

[37] Enrico Fontana, Nathan Fitzpatrick, David Muñoz Ramo, Ross Duncan, and Ivan Rungger. Evaluating the noise resilience of variational quantum algorithms. Physical Review A, 104 (2): 022403, 2021. 10.1103/​PhysRevA.104.022403. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.104.022403.

[38] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum natural gradient. Quantum, 4: 269, 2020. 10.22331/​q-2020-05-25-269. URL https:/​/​quantum-journal.org/​papers/​q-2020-05-25-269/​.

[39] Bálint Koczor and Simon C Benjamin. Quantum natural gradient generalised to non-unitary circuits. arXiv preprint arXiv:1912.08660, 2019. 10.48550/​arXiv.1912.08660. URL https:/​/​arxiv.org/​abs/​1912.08660.

[40] Ken M Nakanishi, Keisuke Fujii, and Synge Todo. Sequential minimal optimization for quantum-classical hybrid algorithms. Physical Review Research, 2 (4): 043158, 2020b. 10.1103/​PhysRevResearch.2.043158. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.2.043158.

[41] Andrew Arrasmith, Lukasz Cincio, Rolando D Somma, and Patrick J Coles. Operator sampling for shot-frugal optimization in variational algorithms. arXiv preprint arXiv:2004.06252, 2020. 10.48550/​arXiv.2004.06252. URL https:/​/​arxiv.org/​abs/​2004.06252.

[42] Ryan Sweke, Frederik Wilde, Johannes Jakob Meyer, Maria Schuld, Paul K Fährmann, Barthélémy Meynard-Piganeau, and Jens Eisert. Stochastic gradient descent for hybrid quantum-classical optimization. Quantum, 4: 314, 2020. 10.22331/​q-2020-08-31-314. URL https:/​/​quantum-journal.org/​papers/​q-2020-08-31-314/​.

[43] Kevin J Sung, Jiahao Yao, Matthew P Harrigan, Nicholas C Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, and Jarrod R McClean. Using models to improve optimizers for variational quantum algorithms. Quantum Science and Technology, 5 (4): 044008, 2020. 10.1088/​2058-9565/​abb6d9. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​abb6d9.

[44] Wim Lavrijsen, Ana Tudor, Juliane Müller, Costin Iancu, and Wibe de Jong. Classical optimizers for noisy intermediate-scale quantum devices. arXiv preprint arXiv:2004.03004, 2020. 10.1109/​QCE49297.2020.00041. URL https:/​/​arxiv.org/​abs/​2004.03004.

[45] Aram Harrow and John Napp. Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms. arXiv preprint arXiv:1901.05374, 2019. URL 10.1103/​PhysRevLett.126.140502.

[46] 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 (7671): 242, 2017. 10.1038/​nature23879. URL https:/​/​www.nature.com/​articles/​nature23879.

[47] S. Hadfield, Z. Wang, B. O’Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms, 12 (2): 34, Feb 2019. ISSN 1999-4893. 10.3390/​a12020034. URL https:/​/​www.mdpi.com/​1999-4893/​12/​2/​34.

[48] Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical reviews, 119 (19): 10856–10915, 2019. 10.1021/​acs.chemrev.8b00803. URL https:/​/​pubs.acs.org/​doi/​10.1021/​acs.chemrev.8b00803.

[49] Rodney J Bartlett and Monika Musiał. Coupled-cluster theory in quantum chemistry. Reviews of Modern Physics, 79 (1): 291, 2007. 10.1103/​RevModPhys.79.291. URL https:/​/​journals.aps.org/​rmp/​abstract/​10.1103/​RevModPhys.79.291.

[50] Joonho Lee, William J Huggins, Martin Head-Gordon, and K Birgitta Whaley. Generalized unitary coupled cluster wave functions for quantum computation. Journal of chemical theory and computation, 15 (1): 311–324, 2018. 10.1021/​acs.jctc.8b01004. URL https:/​/​pubs.acs.org/​doi/​10.1021/​acs.jctc.8b01004.

[51] Bob Coecke and Ross Duncan. Interacting quantum observables: categorical algebra and diagrammatics. New Journal of Physics, 13 (4): 043016, 2011. 10.1088/​1367-2630/​13/​4/​043016. URL https:/​/​iopscience.iop.org/​article/​10.1088/​1367-2630/​13/​4/​043016.

[52] Daniel Stilck França and Raul Garcia-Patron. Limitations of optimization algorithms on noisy quantum devices. Nature Physics, 17 (11): 1221–1227, 2021. 10.1038/​s41567-021-01356-3. URL https:/​/​www.nature.com/​articles/​s41567-021-01356-3.

[53] Bryan T Gard, Linghua Zhu, George S Barron, Nicholas J Mayhall, Sophia E Economou, and Edwin Barnes. Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm. npj Quantum Information, 6 (1): 1–9, 2020. 10.1038/​s41534-019-0240-1. URL https:/​/​www.nature.com/​articles/​s41534-019-0240-1.

[54] Michael Streif, Martin Leib, Filip Wudarski, Eleanor Rieffel, and Zhihui Wang. Quantum algorithms with local particle-number conservation: Noise effects and error correction. Physical Review A, 103 (4): 042412, 2021. 10.1103/​PhysRevA.103.042412. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.103.042412.

[55] F. T. Chong, D. Franklin, and M. Martonosi. Programming languages and compiler design for realistic quantum hardware. Nature, 549 (7671): 180, 2017. 10.1038/​nature23459. URL https:/​/​www.nature.com/​articles/​nature23459.

[56] Thomas Häner, Damian S Steiger, Krysta Svore, and Matthias Troyer. A software methodology for compiling quantum programs. Quantum Science and Technology, 3 (2): 020501, 2018. 10.1088/​2058-9565/​aaa5cc. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​aaa5cc.

[57] D. Venturelli, M. Do, E. Rieffel, and J. Frank. Compiling quantum circuits to realistic hardware architectures using temporal planners. Quantum Science and Technology, 3 (2): 025004, 2018. 10.1088/​2058-9565/​aaa331. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​aaa331.

[58] Tyson Jones and Simon C Benjamin. Robust quantum compilation and circuit optimisation via energy minimisation. Quantum, 6: 628, 2022. 10.22331/​q-2022-01-24-628. URL https:/​/​quantum-journal.org/​papers/​q-2022-01-24-628/​.

[59] Kentaro Heya, Yasunari Suzuki, Yasunobu Nakamura, and Keisuke Fujii. Variational quantum gate optimization. arXiv preprint arXiv:1810.12745, 2018. 10.48550/​arXiv.1810.12745. URL https:/​/​arxiv.org/​abs/​1810.12745.

[60] M. J. D. Powell. The BOBYQA algorithm for bound constrained optimization without derivatives. Technical Report, Department of Applied Mathematics and Theoretical Physics, 01 2009. URL https:/​/​www.damtp.cam.ac.uk/​user/​na/​NA_papers/​NA2009_06.pdf.

[61] Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92 (4): 042303, 2015. 10.1103/​PhysRevA.92.042303. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.92.042303.

[62] Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen. Exploring entanglement and optimization within the hamiltonian variational ansatz. PRX Quantum, 1 (2): 020319, 2020. 10.1103/​PRXQuantum.1.020319. URL https:/​/​journals.aps.org/​prxquantum/​pdf/​10.1103/​PRXQuantum.1.020319.

[63] Xuchen You and Xiaodi Wu. Exponentially many local minima in quantum neural networks. In International Conference on Machine Learning, pages 12144–12155. PMLR, 2021. URL https:/​/​proceedings.mlr.press/​v139/​you21c.html.

[64] Hans J Briegel, David E Browne, Wolfgang Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5 (1): 19–26, 2009. 10.1038/​nphys1157. URL https:/​/​www.nature.com/​articles/​nphys1157.

[65] Vincent Danos and Elham Kashefi. Determinism in the one-way model. Physical Review A, 74 (5): 052310, 2006. 10.1103/​PhysRevA.74.052310. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.74.052310.

[66] Scott Kirkpatrick, C Daniel Gelatt, and Mario P Vecchi. Optimization by simulated annealing. science, 220 (4598): 671–680, 1983. 10.1126/​science.220.4598.671. URL https:/​/​www.science.org/​doi/​abs/​10.1126/​science.220.4598.671.

[67] Wagner F Sacco and CREA Oliveira. A new stochastic optimization algorithm based on a particle collision metaheuristic. Proceedings of 6th WCSMO, 2005. URL https:/​/​citeseerx.ist.psu.edu/​viewdoc/​download?doi=

[68] Ana Carolina Rios-Coelho, Wagner F Sacco, and Nélio Henderson. A metropolis algorithm combined with hooke–jeeves local search method applied to global optimization. Applied Mathematics and Computation, 217 (2): 843–853, 2010. 10.1016/​j.amc.2010.06.027. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​S0096300310007125.

[69] Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983, 2016. 10.48550/​arXiv.1608.03983. URL https:/​/​arxiv.org/​abs/​1608.03983.

[70] Oliver Kern, Gernot Alber, and Dima L Shepelyansky. Quantum error correction of coherent errors by randomization. The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics, 32 (1): 153–156, 2005. 10.1140/​epjd/​e2004-00196-9. URL https:/​/​link.springer.com/​article/​10.1140/​epjd/​e2004-00196-9.

[71] Joel J Wallman and Joseph Emerson. Noise tailoring for scalable quantum computation via randomized compiling. Physical Review A, 94 (5): 052325, 2016. URL 10.1103/​PhysRevA.94.052325.

[72] Osama Moussa, Marcus P da Silva, Colm A Ryan, and Raymond Laflamme. Practical experimental certification of computational quantum gates using a twirling procedure. Physical review letters, 109 (7): 070504, 2012. 10.1103/​PhysRevLett.109.070504. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.109.070504.

[73] Kristan Temme, Sergey Bravyi, and Jay M Gambetta. Error mitigation for short-depth quantum circuits. Physical review letters, 119 (18): 180509, 2017. 10.1103/​PhysRevLett.119.180509. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.119.180509.

[74] Steven T Flammia and Joel J Wallman. Efficient estimation of pauli channels. ACM Transactions on Quantum Computing, 1 (1): 1–32, 2020. 10.1145/​3408039. URL https:/​/​dl.acm.org/​doi/​abs/​10.1145/​3408039.

[75] Ying Li and Simon C Benjamin. Efficient variational quantum simulator incorporating active error minimization. Physical Review X, 7 (2): 021050, 2017. URL 10.1103/​PhysRevX.7.021050.

[76] Suguru Endo, Simon C Benjamin, and Ying Li. Practical quantum error mitigation for near-future applications. Physical Review X, 8 (3): 031027, 2018. 10.1103/​PhysRevX.8.031027. URL https:/​/​journals.aps.org/​prx/​abstract/​10.1103/​PhysRevX.8.031027.

[77] Miroslav Urbanek, Benjamin Nachman, Vincent R Pascuzzi, Andre He, Christian W Bauer, and Wibe A de Jong. Mitigating depolarizing noise on quantum computers with noise-estimation circuits. Physical Review Letters, 127 (27): 270502, 2021. 10.1103/​PhysRevLett.127.270502. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.127.270502.

Cited by

[1] 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”, arXiv:2111.05176.

[2] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Variational Quantum Algorithms”, arXiv:2012.09265.

[3] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin, “Entanglement devised barren plateau mitigation”, Physical Review Research 3 3, 033090 (2021).

[4] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, “Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?”, arXiv:2109.01051.

[5] Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, François Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, and Christopher Eichler, “Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases”, Nature Communications 13, 4144 (2022).

[6] Andrew Arrasmith, Zoë Holmes, M. Cerezo, and Patrick J. Coles, “Equivalence of quantum barren plateaus to cost concentration and narrow gorges”, Quantum Science and Technology 7 4, 045015 (2022).

[7] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, “Theory of overparametrization in quantum neural networks”, arXiv:2109.11676.

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

[9] Dmitry A. Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev, “VQE method: a short survey and recent developments”, Materials Theory 6 1, 2 (2022).

[10] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio, “A semi-agnostic ansatz with variable structure for quantum machine learning”, arXiv:2103.06712.

[11] Enrico Fontana, Nathan Fitzpatrick, David Muñoz Ramo, Ross Duncan, and Ivan Rungger, “Evaluating the noise resilience of variational quantum algorithms”, Physical Review A 104 2, 022403 (2021).

[12] Tobias Stollenwerk and Stuart Hadfield, “Diagrammatic Analysis for Parameterized Quantum Circuits”, arXiv:2204.01307.

[13] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, “Universal noise-precision relations in variational quantum algorithms”, arXiv:2106.03390.

[14] Xiaozhen Ge, Re-Bing Wu, and Herschel Rabitz, “The Optimization Landscape of Hybrid Quantum-Classical Algorithms: from Quantum Control to NISQ Applications”, arXiv:2201.07448.

[15] Joonho Kim and Yaron Oz, “Quantum Energy Landscape and VQA Optimization”, arXiv:2107.10166.

[16] Kun Wang, Zhixin Song, Xuanqiang Zhao, Zihe Wang, and Xin Wang, “Detecting and quantifying entanglement on near-term quantum devices”, npj Quantum Information 8, 52 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2022-09-16 22:19:27). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2022-09-16 22:19:25).

  • Coinsmart. Europe’s Best Bitcoin and Crypto Exchange.Click Here
  • Platoblockchain. Web3 Metaverse Intelligence. Knowledge Amplified. Access Here.
  • Source: https://quantum-journal.org/papers/q-2022-09-15-804/

Latest Intelligence


Latest Intelligence