Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

Martin Larocca1,2, Piotr Czarnik2, Kunal Sharma3,2, Gopikrishnan Muraleedharan2, Patrick J. Coles2, and M. Cerezo4,5

1Departamento de Física “J. J. Giambiagi” and IFIBA, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA USA
4Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
5Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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


Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of $\mathfrak{g}$.

In this work, we provide a novel framework for diagnosing the presence or absence of Barren Plateaus (BPs) in variational quantum algorithms and quantum machine learning models. Our work leverages tools from quantum control theory to connect the scaling of the cost-function gradients with the dimension of the so-called dynamical Lie algebra (DLA), the Lie closure of the generators of the parametrized quantum circuit. Our results greatly improve our understanding of the BP phenomenon, allowing us to predict their happening in a wide range of scenarios that were not covered by previous literature. Taken together, this work provides novel strategies for an active trainability-aware design of quantum neural network architectures, and showcases the importance of the DLA in variational quantum computing.

► BibTeX data

► References

[1] Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science, pages 124–134. Ieee, 1994. 10.1109/​SFCS.1994.365700. URL https:/​/​ieeexplore.ieee.org/​document/​365700.

[2] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103 (15): 150502, 2009. 10.1103/​PhysRevLett.103.150502. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.103.150502.

[3] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating hamiltonian dynamics with a truncated taylor series. Physical Review Letters, 114 (9): 090502, 2015. 10.1103/​PhysRevLett.114.090502. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.114.090502.

[4] Iulia M Georgescu, Sahel Ashhab, and Franco Nori. Quantum simulation. Reviews of Modern Physics, 86 (1): 153, 2014. 10.1103/​RevModPhys.86.153. URL https:/​/​journals.aps.org/​rmp/​abstract/​10.1103/​RevModPhys.86.153.

[5] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. 10.22331/​q-2018-08-06-79. URL https:/​/​quantum-journal.org/​papers/​q-2018-08-06-79/​.

[6] 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.

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

[8] Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost. Near-term quantum algorithms for linear systems of equations. arXiv preprint arXiv:1909.07344, 2019. URL https:/​/​arxiv.org/​abs/​1909.07344.

[9] Xiaosi Xu, Jinzhao Sun, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational algorithms for linear algebra. Science Bulletin, 66 (21): 2181–2188, 2021. 10.1016/​j.scib.2021.06.023. URL https:/​/​www.sciencedirect.com/​science/​article/​pii/​S2095927321004631.

[10] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1): 1–6, 2019. 10.1038/​s41534-019-0187-2. URL https:/​/​www.nature.com/​articles/​s41534-019-0187-2.

[11] Harper R Grimsley, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications, 10 (1): 1–9, 2019. 10.1038/​s41467-019-10988-2. URL https:/​/​www.nature.com/​articles/​s41467-019-10988-2.

[12] 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. 10.1038/​s41534-020-00302-0. URL https:/​/​www.nature.com/​articles/​s41534-020-00302-0.

[13] Benjamin Commeau, M. Cerezo, Zoë Holmes, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. Variational hamiltonian diagonalization for dynamical quantum simulation. arXiv preprint arXiv:2009.02559, 2020. URL https:/​/​arxiv.org/​abs/​2009.02559.

[14] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. Long-time simulations with high fidelity on quantum hardware. arXiv preprint arXiv:2102.04313, 2021. URL https:/​/​arxiv.org/​abs/​2102.04313.

[15] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P Orth. Adaptive variational quantum dynamics simulations. PRX Quantum, 2 (3): 030307, 2021. 10.1103/​PRXQuantum.2.030307. URL https:/​/​journals.aps.org/​prxquantum/​abstract/​10.1103/​PRXQuantum.2.030307.

[16] Suguru Endo, Jinzhao Sun, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational quantum simulation of general processes. Physical Review Letters, 125 (1): 010501, 2020. 10.1103/​PhysRevLett.125.010501. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.125.010501.

[17] Jonathan Wei Zhong Lau, Kishor Bharti, Tobias Haug, and Leong Chuan Kwek. Quantum assisted simulation of time dependent hamiltonians. arXiv preprint arXiv:2101.07677, 2021. URL https:/​/​arxiv.org/​abs/​2101.07677.

[18] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 1–7, 2014. doi.org/​10.1038/​ncomms5213. URL https:/​/​www.nature.com/​articles/​ncomms5213#citeas.

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

[20] 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. 10.1007/​978-94-015-8330-5_4. URL https:/​/​iopscience.iop.org/​article/​10.1088/​1367-2630/​18/​2/​023023.

[21] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T Sornborger, and Patrick J Coles. Quantum-assisted quantum compiling. Quantum, 3: 140, 2019. 10.22331/​q-2019-05-13-140. URL https:/​/​quantum-journal.org/​papers/​q-2019-05-13-140/​.

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

[23] Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J Coles. Variational quantum state diagonalization. npj Quantum Information, 5 (1): 1–10, 2019. 10.1038/​s41534-019-0167-6. URL https:/​/​www.nature.com/​articles/​s41534-019-0167-6.

[24] Andrew Arrasmith, Lukasz Cincio, Andrew T Sornborger, Wojciech H Zurek, and Patrick J Coles. Variational consistent histories as a hybrid algorithm for quantum foundations. Nature Communications, 10 (1): 1–7, 2019. 10.1038/​s41467-019-11417-0. URL https:/​/​www.nature.com/​articles/​s41467-019-11417-0.

[25] M. Cerezo, Alexander Poremba, Lukasz Cincio, and Patrick J Coles. Variational quantum fidelity estimation. Quantum, 4: 248, 2020. 10.22331/​q-2020-03-26-248. URL https:/​/​quantum-journal.org/​papers/​q-2020-03-26-248/​.

[26] Y. Li and S. C. Benjamin. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X, 7: 021050, Jun 2017. 10.1103/​PhysRevX.7.021050. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.7.021050.

[27] Kentaro Heya, Ken M Nakanishi, Kosuke Mitarai, and Keisuke Fujii. Subspace variational quantum simulator. arXiv preprint arXiv:1904.08566, 2019. URL https:/​/​arxiv.org/​abs/​1904.08566.

[28] Kishor Bharti and Tobias Haug. Quantum-assisted simulator. Physical Review A, 104 (4): 042418, 2021. 10.1103/​PhysRevA.104.042418. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.104.042418.

[29] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J Coles. Variational quantum state eigensolver. npj Quantum Information, 8 (1): 1–11, 2022. 10.1038/​s41534-022-00611-6. URL https:/​/​doi.org/​10.1038/​s41534-022-00611-6.

[30] Jacob L Beckey, M. Cerezo, Akira Sone, and Patrick J Coles. Variational quantum algorithm for estimating the quantum Fisher information. Physical Review Research, 4 (1): 013083, 2022. 10.1103/​PhysRevResearch.4.013083. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.4.013083.

[31] Lennart Bittel and Martin Kliesch. Training variational quantum algorithms is np-hard. Phys. Rev. Lett., 127: 120502, Sep 2021. 10.1103/​PhysRevLett.127.120502. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.127.120502.

[32] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning. Physical Review A, 98 (3): 032309, 2018. 10.1103/​PhysRevA.98.032309. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.98.032309.

[33] 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.

[34] 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/​.

[35] 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/​.

[36] 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. URL https:/​/​arxiv.org/​abs/​2004.06252.

[37] 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): 1–6, 2018. 10.1038/​s41467-018-07090-4. URL https:/​/​www.nature.com/​articles/​s41467-018-07090-4.

[38] 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.

[39] 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.

[40] 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.

[41] 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.

[42] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J Coles. Effect of barren plateaus on gradient-free optimization. Quantum, 5: 558, 2021. 10.22331/​q-2021-10-05-558. URL https:/​/​quantum-journal.org/​papers/​q-2021-10-05-558/​.

[43] 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.

[44] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. Entanglement-induced barren plateaus. PRX Quantum, 2 (4): 040316, 2021. 10.1103/​PRXQuantum.2.040316. URL https:/​/​journals.aps.org/​prxquantum/​abstract/​10.1103/​PRXQuantum.2.040316.

[45] Taylor L Patti, Khadijeh Najafi, Xun Gao, and Susanne F Yelin. Entanglement devised barren plateau mitigation. Physical Review Research, 3 (3): 033090, 2021. 10.1103/​PhysRevResearch.3.033090. URL https:/​/​par.nsf.gov/​servlets/​purl/​10328786.

[46] 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.

[47] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J Coles. Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum, 3: 010313, Jan 2022. 10.1103/​PRXQuantum.3.010313. URL https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.3.010313.

[48] Andrew Arrasmith, Zoë Holmes, Marco Cerezo, and Patrick J Coles. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. Quantum Science and Technology, 7 (4): 045015, 2022. 10.1088/​2058-9565/​ac7d06. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​ac7d06.

[49] 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, 2021c. 10.1038/​s41467-021-21728-w. URL https:/​/​www.nature.com/​articles/​s41467-021-21728-w.

[50] AV Uvarov and Jacob D Biamonte. On barren plateaus and cost function locality in variational quantum algorithms. Journal of Physics A: Mathematical and Theoretical, 54 (24): 245301, 2021. 10.1088/​1751-8121/​abfac7. URL https:/​/​doi.org/​10.1088/​1751-8121/​abfac7.

[51] Tyler Volkoff and Patrick J Coles. Large gradients via correlation in random parameterized quantum circuits. Quantum Science and Technology, 6 (2): 025008, 2021. 10.1088/​2058-9565/​abd89. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​abd891.

[52] Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural networks via classical neural networks. arXiv preprint arXiv:1907.05415, 2019. URL https:/​/​arxiv.org/​abs/​1907.05415.

[53] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum, 3: 214, 2019. 10.22331/​q-2019-12-09-214. URL https:/​/​quantum-journal.org/​papers/​q-2019-12-09-214/​.

[54] Andrea Skolik, Jarrod R McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib. Layerwise learning for quantum neural networks. Quantum Machine Intelligence, 3 (1): 1–11, 2021. 10.1007/​s42484-020-00036-4. URL https:/​/​doi.org/​10.1007/​s42484-020-00036-4.

[55] M Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio. A semi-agnostic ansatz with variable structure for quantum machine learning. arXiv preprint arXiv:2103.06712, 2021. URL https:/​/​arxiv.org/​abs/​2103.06712.

[56] Alicia B Magann, Christian Arenz, Matthew D Grace, Tak-San Ho, Robert L Kosut, Jarrod R McClean, Herschel A Rabitz, and Mohan Sarovar. From pulses to circuits and back again: A quantum optimal control perspective on variational quantum algorithms. PRX Quantum, 2 (1): 010101, 2021. https:/​/​doi.org/​10.1103/​PRXQuantum.2.010101. URL https:/​/​journals.aps.org/​prxquantum/​abstract/​10.1103/​PRXQuantum.2.010101.

[57] Stuart Hadfield, Zhihui Wang, Bryan O'Gorman, Eleanor G Rieffel, Davide Venturelli, and Rupak Biswas. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms, 12 (2): 34, 2019. 10.3390/​a12020034. URL https:/​/​www.mdpi.com/​1999-4893/​12/​2/​34.

[58] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92: 042303, Oct 2015. 10.1103/​PhysRevA.92.042303. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.92.042303.

[59] 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/​abstract/​10.1103/​PRXQuantum.1.020319.

[60] Linghua Zhu, Ho Lun Tang, George S Barron, Nicholas J Mayhall, Edwin Barnes, and Sophia E Economou. Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer. Physical Review Research, 4 (3): 033029, 2022. 10.1103/​PhysRevResearch.4.033029. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.4.033029.

[61] Alexandre Choquette, Agustin Di Paolo, Panagiotis Kl Barkoutsos, David Sénéchal, Ivano Tavernelli, and Alexandre Blais. Quantum-optimal-control-inspired ansatz for variational quantum algorithms. Physical Review Research, 3 (2): 023092, 2021. 10.1103/​PhysRevResearch.3.023092. URL https:/​/​journals.aps.org/​prresearch/​abstract/​10.1103/​PhysRevResearch.3.023092.

[62] Supanut Thanasilp, Samson Wang, Nhat A Nghiem, Patrick J. Coles, and M. Cerezo. Subtleties in the trainability of quantum machine learning models. arXiv preprint arXiv:2110.14753, 2021. URL https:/​/​arxiv.org/​abs/​2110.14753.

[63] D. D'Alessandro. Introduction to Quantum Control and Dynamics. Chapman & Hall/​CRC Applied Mathematics & Nonlinear Science. Taylor & Francis, 2007. ISBN 9781584888840. URL https:/​/​books.google.sm/​books?id=HbMYmAEACAAJ.

[64] Sukin Sim, Peter D Johnson, and Alán Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quantum Technologies, 2 (12): 1900070, 2019. 10.1002/​qute.201900070. URL https:/​/​onlinelibrary.wiley.com/​doi/​full/​10.1002/​qute.201900070.

[65] Carlton M Caves. Quantum error correction and reversible operations. Journal of Superconductivity, 12 (6): 707–718, 1999. 10.1023/​A:1007720606911. URL https:/​/​link.springer.com/​article/​10.1023/​A:1007720606911.

[66] P Rungta, WJ Munro, K Nemoto, P Deuar, Gerard J Milburn, and CM Caves. Qudit entanglement. In Directions in Quantum Optics, pages 149–164. Springer, 2001. 10.1007/​3-540-40894-0_14. URL https:/​/​link.springer.com/​chapter/​10.1007.

[67] Nicholas Hunter-Jones. Unitary designs from statistical mechanics in random quantum circuits. arXiv preprint arXiv:1905.12053, 2019. URL https:/​/​arxiv.org/​abs/​1905.12053.

[68] Yoshifumi Nakata, Masato Koashi, and Mio Murao. Generating a state t-design by diagonal quantum circuits. New Journal of Physics, 16 (5): 053043, 2014. 10.1088/​1367-2630/​16/​5/​053043. URL https:/​/​iopscience.iop.org/​article/​10.1088/​1367-2630/​16/​5/​053043.

[69] Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, and Claudio Chamon. Optimizing variational quantum algorithms using pontryagin’s minimum principle. Physical Review X, 7 (2): 021027, 2017. 10.1103/​PhysRevX.7.021027. URL https:/​/​journals.aps.org/​prx/​abstract/​10.1103/​PhysRevX.7.021027.

[70] Oinam Romesh Meitei, Bryan T Gard, George S Barron, David P Pappas, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-vqe. arXiv preprint arXiv:2008.04302, 2020. URL https:/​/​arxiv.org/​abs/​2008.04302.

[71] Juneseo Lee, Alicia B Magann, Herschel A Rabitz, and Christian Arenz. Progress toward favorable landscapes in quantum combinatorial optimization. Physical Review A, 104 (3): 032401, 2021. 10.1103/​PhysRevA.104.032401. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.104.032401.

[72] Jun Li, Xiaodong Yang, Xinhua Peng, and Chang-Pu Sun. Hybrid quantum-classical approach to quantum optimal control. Physical Review Letters, 118 (15): 150503, 2017. 10.1103/​PhysRevLett.118.150503. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.118.150503.

[73] Viswanath Ramakrishna and Herschel Rabitz. Relation between quantum computing and quantum controllability. Physical Review A, 54 (2): 1715, 1996. 10.1103/​PhysRevA.54.1715. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.54.1715.

[74] Seth Lloyd. Quantum approximate optimization is computationally universal. arXiv preprint arXiv:1812.11075, 2018. URL https:/​/​arxiv.org/​abs/​1812.11075.

[75] Mauro ES Morales, JD Biamonte, and Zoltán Zimborás. On the universality of the quantum approximate optimization algorithm. Quantum Information Processing, 19 (9): 1–26, 2020. 10.1007/​s11128-020-02748-9. URL https:/​/​link.springer.com/​article/​10.1007/​s11128-020-02748-9.

[76] V Akshay, H Philathong, Mauro ES Morales, and Jacob D Biamonte. Reachability deficits in quantum approximate optimization. Physical Review Letters, 124 (9): 090504, 2020. 10.1103/​PhysRevLett.124.090504. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.124.090504.

[77] Robert Zeier and Thomas Schulte-Herbrüggen. Symmetry principles in quantum systems theory. Journal of mathematical physics, 52 (11): 113510, 2011. https:/​/​doi.org/​10.1063/​1.3657939. URL https:/​/​aip.scitation.org/​doi/​pdf/​10.1063/​1.3657939.

[78] Thomas Polack, Haim Suchowski, and David J Tannor. Uncontrollable quantum systems: A classification scheme based on lie subalgebras. Physical Review A, 79 (5): 053403, 2009. https:/​/​doi.org/​10.1103/​PhysRevA.79.053403. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.79.053403.

[79] Leonardo Banchi, Daniel Burgarth, and Michael J Kastoryano. Driven quantum dynamics: will it blend? Physical Review X, 7 (4): 041015, 2017. 10.1103/​PhysRevX.7.041015. URL https:/​/​journals.aps.org/​prx/​abstract/​10.1103/​PhysRevX.7.041015.

[80] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, Sep 2017. ISSN 1476-4687. 10.1038/​nature23879. URL https:/​/​doi.org/​10.1038/​nature23879.

[81] Aram W Harrow and Richard A Low. Random quantum circuits are approximate 2-designs. Communications in Mathematical Physics, 291 (1): 257–302, 2009. 10.1007/​s00220-009-0873-6. URL https:/​/​link.springer.com/​article/​10.1007.

[82] Fernando GSL Brandao, Aram W Harrow, and Michał Horodecki. Local random quantum circuits are approximate polynomial-designs. Communications in Mathematical Physics, 346 (2): 397–434, 2016. 10.1007/​s00220-016-2706-8. URL https:/​/​link.springer.com/​article/​10.1007.

[83] Aram Harrow and Saeed Mehraban. Approximate unitary $ t $-designs by short random quantum circuits using nearest-neighbor and long-range gates. arXiv preprint arXiv:1809.06957, 2018. URL https:/​/​arxiv.org/​abs/​1809.06957.

[84] Andrew Lucas. Ising formulations of many np problems. Frontiers in Physics, 2: 5, 2014. 10.3389/​fphy.2014.00005. URL https:/​/​www.frontiersin.org/​articles/​10.3389/​fphy.2014.00005/​full.

[85] Michael Streif and Martin Leib. Training the quantum approximate optimization algorithm without access to a quantum processing unit. Quantum Science and Technology, 5 (3): 034008, 2020. 10.1088/​2058-9565/​ab8c2b. URL https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​ab8c2b.

[86] M. Cerezo, Raúl Rossignoli, N Canosa, and E Ríos. Factorization and criticality in finite $xxz$ systems of arbitrary spin. Physical Review Letters, 119 (22): 220605, 2017. 10.1103/​PhysRevLett.119.220605. URL https:/​/​journals.aps.org/​prl/​abstract/​10.1103/​PhysRevLett.119.220605.

[87] Xiaoting Wang, Daniel Burgarth, and S Schirmer. Subspace controllability of spin-1 2 chains with symmetries. Physical Review A, 94 (5): 052319, 2016. https:/​/​doi.org/​10.1103/​PhysRevA.94.052319. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.94.052319.

[88] Benoı̂t Collins and Piotr Śniady. Integration with respect to the haar measure on unitary, orthogonal and symplectic group. Communications in Mathematical Physics, 264 (3): 773–795, 2006. 10.1007/​s00220-006-1554-3. URL https:/​/​link.springer.com/​article/​10.1007.

[89] PM Poggi and Diego Ariel Wisniacki. Optimal control of many-body quantum dynamics: Chaos and complexity. Physical Review A, 94 (3): 033406, 2016. 10.1103/​PhysRevA.94.033406. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.94.033406.

[90] Martín Larocca and Diego Wisniacki. Krylov-subspace approach for the efficient control of quantum many-body dynamics. Physical Review A, 103 (2): 023107, 2021. 10.1103/​PhysRevA.103.023107. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.103.023107.

[91] P Erdos and A Renyi. On random graphs i. Publ. math. debrecen, 6 (290-297): 18, 1959. URL http:/​/​snap.stanford.edu/​class/​cs224w-readings/​erdos59random.pdf.

[92] Christian Arenz and Herschel Rabitz. Drawing together control landscape and tomography principles. Physical Review A, 102 (4): 042207, 2020. 10.1103/​PhysRevA.102.042207. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.102.042207.

[93] Zbigniew Puchala and Jaroslaw Adam Miszczak. Symbolic integration with respect to the haar measure on the unitary groups. Bulletin of the Polish Academy of Sciences Technical Sciences, 65 (1): 21–27, 2017. 10.1515/​bpasts-2017-0003. URL http:/​/​journals.pan.pl/​dlibra/​publication/​121307/​edition/​105697/​content.

[94] 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.

[95] Christian Kokail, Christine Maier, Rick van Bijnen, Tiff Brydges, Manoj K Joshi, Petar Jurcevic, Christine A Muschik, Pietro Silvi, Rainer Blatt, Christian F Roos, et al. Self-verifying variational quantum simulation of lattice models. Nature, 569 (7756): 355–360, 2019. 10.1038/​s41586-019-1177-4. URL https:/​/​www.nature.com/​articles/​s41586-019-1177-4.

[96] 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.

[97] Nikolay V Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M Anisimov, Andrew T Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A Dub. Correlation-informed permutation of qubits for reducing ansatz depth in vqe. PRX Quantum, 2 (2): 020337, 2021. 10.1103/​PRXQuantum.2.020337. URL https:/​/​journals.aps.org/​prxquantum/​abstract/​10.1103/​PRXQuantum.2.020337.

[98] Bobak Toussi Kiani, Seth Lloyd, and Reevu Maity. Learning unitaries by gradient descent. arXiv preprint arXiv:2001.11897, 2020. URL https:/​/​arxiv.org/​abs/​2001.11897.

[99] Zhihui Wang, Nicholas C Rubin, Jason M Dominy, and Eleanor G Rieffel. $XY$ mixers: Analytical and numerical results for the quantum alternating operator ansatz. Physical Review A, 101 (1): 012320, 2020. 10.1103/​PhysRevA.101.012320. URL https:/​/​journals.aps.org/​pra/​abstract/​10.1103/​PhysRevA.101.012320.

[100] Andreas Bärtschi and Stephan Eidenbenz. Grover mixers for qaoa: Shifting complexity from mixer design to state preparation. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 72–82. IEEE, 2020. 10.1109/​QCE49297.2020.00020. URL https:/​/​www.computer.org/​csdl/​proceedings-article/​qce/​2020/​896900a072/​1p2VnUCmpYA.

[101] Wen Wei Ho and Timothy H. Hsieh. Efficient variational simulation of non-trivial quantum states. SciPost Phys., 6: 29, 2019. 10.21468/​SciPostPhys.6.3.029. URL https:/​/​scipost.org/​10.21468/​SciPostPhys.6.3.029.

[102] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. Strategies for solving the fermi-hubbard model on near-term quantum computers. Physical Review B, 102 (23): 235122, 2020. 10.1103/​PhysRevB.102.235122. URL https:/​/​journals.aps.org/​prb/​abstract/​10.1103/​PhysRevB.102.235122.

[103] Chen Zhao and Xiao-Shan Gao. Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. Quantum, 5: 466, June 2021. ISSN 2521-327X. 10.22331/​q-2021-06-04-466. URL https:/​/​doi.org/​10.22331/​q-2021-06-04-466.

[104] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao. Toward trainability of quantum neural networks. arXiv preprint arXiv:2011.06258, 2020. URL https:/​/​arxiv.org/​abs/​2011.06258.

[105] Frederic Sauvage, Sukin Sim, Alexander A Kunitsa, William A Simon, Marta Mauri, and Alejandro Perdomo-Ortiz. Flip: A flexible initializer for arbitrarily-sized parametrized quantum circuits. arXiv preprint arXiv:2103.08572, 2021. URL https:/​/​arxiv.org/​abs/​2103.08572.

[106] Yidong Liao, Min-Hsiu Hsieh, and Chris Ferrie. Quantum optimization for training quantum neural networks. arXiv preprint arXiv:2103.17047, 2021. URL https:/​/​arxiv.org/​abs/​2103.17047.

[107] Raj Chakrabarti and Herschel Rabitz. Quantum control landscapes. International Reviews in Physical Chemistry, 26 (4): 671–735, 2007. 10.1080/​01442350701633300. URL https:/​/​www.tandfonline.com/​doi/​abs/​10.1080/​01442350701633300.

[108] Martín Larocca, Pablo M Poggi, and Diego A Wisniacki. Quantum control landscape for a two-level system near the quantum speed limit. Journal of Physics A: Mathematical and Theoretical, 51 (38): 385305, aug 2018. 10.1088/​1751-8121/​aad657. URL https:/​/​doi.org/​10.1088/​1751-8121/​aad657.

[109] Martín Larocca, Esteban Calzetta, and Diego A. Wisniacki. Exploiting landscape geometry to enhance quantum optimal control. Physical Review A, 101: 023410, Feb 2020. 10.1103/​PhysRevA.101.023410. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.101.023410.

[110] Winton G. Brown and Lorenza Viola. Convergence rates for arbitrary statistical moments of random quantum circuits. Phys. Rev. Lett., 104: 250501, Jun 2010. 10.1103/​PhysRevLett.104.250501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.104.250501.

[111] Domenico D’Alessandro and Jonas T Hartwig. Dynamical decomposition of bilinear control systems subject to symmetries. Journal of Dynamical and Control Systems, 27 (1): 1–30, 2021. https:/​/​doi.org/​10.1007/​s10883-020-09488-0.

Cited by

[1] Sergey V. Grebnev, Maxim A. Gavreev, Evgeniy O. Kiktenko, Anton P. Guglya, Albert R. Efimov, and Aleksey K. Fedorov, "Pitfalls of the Sublinear QAOA-Based Factorization Algorithm", IEEE Access 11, 134760 (2023).

[2] Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz", Physical Review A 106 6, L060401 (2022).

[3] Barkha Singh, S. Indu, and Sudipta Majumdar, 2023 3rd International conference on Artificial Intelligence and Signal Processing (AISP) 1 (2023) ISBN:979-8-3503-2074-9.

[4] V Vijendran, Aritra Das, Dax Enshan Koh, Syed M Assad, and Ping Koy Lam, "An expressive ansatz for low-depth quantum approximate optimisation", Quantum Science and Technology 9 2, 025010 (2024).

[5] Manas Sajjan, Rishabh Gupta, Sumit Suresh Kale, Vinit Singh, Keerthi Kumaran, and Sabre Kais, "Physics-Inspired Quantum Simulation of Resonating Valence Bond States─A Prototypical Template for a Spin-Liquid Ground State", The Journal of Physical Chemistry A 127 41, 8751 (2023).

[6] Nic Ezzell, Elliott M Ball, Aliza U Siddiqui, Mark M Wilde, Andrew T Sornborger, Patrick J Coles, and Zoë Holmes, "Quantum mixed state compiling", Quantum Science and Technology 8 3, 035001 (2023).

[7] Lento Nagano, Alexander Miessen, Tamiya Onodera, Ivano Tavernelli, Francesco Tacchino, and Koji Terashi, "Quantum data learning for quantum simulations in high-energy physics", Physical Review Research 5 4, 043250 (2023).

[8] Carlos Mejuto-Zaera and Alexander F Kemper, "Quantum eigenvector continuation for chemistry applications", Electronic Structure 5 4, 045007 (2023).

[9] Chae-Yeun Park and Nathan Killoran, "Hamiltonian variational ansatz without barren plateaus", Quantum 8, 1239 (2024).

[10] Brody Wrighter and Sonia Lopez Alarcon, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 313 (2023) ISBN:979-8-3503-4323-6.

[11] Pejman Jouzdani, Calvin W. Johnson, Eduardo R. Mucciolo, and Ionel Stetcu, "Alternative approach to quantum imaginary time evolution", Physical Review A 106 6, 062435 (2022).

[12] Ayush Asthana, Chenxu Liu, Oinam Romesh Meitei, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Leakage Reduces Device Coherence Demands for Pulse-Level Molecular Simulations", Physical Review Applied 19 6, 064071 (2023).

[13] Yusen Wu, Bujiao Wu, Jingbo Wang, and Xiao Yuan, "Quantum Phase Recognition via Quantum Kernel Methods", Quantum 7, 981 (2023).

[14] Massimiliano Incudini, Francesco Martini, and Alessandra Di Pierro, "Toward Useful Quantum Kernels", Advanced Quantum Technologies 2300298 (2024).

[15] Louis Schatzki, Martín Larocca, Quynh T. Nguyen, Frédéric Sauvage, and M. Cerezo, "Theoretical guarantees for permutation-equivariant quantum neural networks", npj Quantum Information 10 1, 12 (2024).

[16] Roeland Wiersema and Nathan Killoran, "Optimizing quantum circuits with Riemannian gradient flow", Physical Review A 107 6, 062421 (2023).

[17] Eric R. Anschuetz, Andreas Bauer, Bobak T. Kiani, and Seth Lloyd, "Efficient classical algorithms for simulating symmetric quantum systems", Quantum 7, 1189 (2023).

[18] Alicia B. Magann, Kenneth M. Rudinger, Matthew D. Grace, and Mohan Sarovar, "Lyapunov-control-inspired strategies for quantum combinatorial optimization", Physical Review A 106 6, 062414 (2022).

[19] Alexey Pyrkov, Alex Aliper, Dmitry Bezrukov, Dmitriy Podolskiy, Feng Ren, and Alex Zhavoronkov, "Complexity of life sciences in quantum and AI era", WIREs Computational Molecular Science 14 1, e1701 (2024).

[20] Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "The battle of clean and dirty qubits in the era of partial error correction", Quantum 7, 1060 (2023).

[21] Massimiliano Incudini, Michele Grossi, Andrea Ceschini, Antonio Mandarino, Massimo Panella, Sofia Vallecorsa, and David Windridge, "Resource saving via ensemble techniques for quantum neural networks", Quantum Machine Intelligence 5 2, 39 (2023).

[22] Klée Pollock, Peter P. Orth, and Thomas Iadecola, "Variational microcanonical estimator", Physical Review Research 5 3, 033224 (2023).

[23] Martín Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and Marco Cerezo, "Theory of overparametrization in quantum neural networks", Nature Computational Science 3 6, 542 (2023).

[24] Adriana Meijer - van de Griend and Jukka K. Nurminen, "QuantMark: A Benchmarking API for VQE Algorithms", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[25] Seyed Shakib Vedaie, Archismita Dalal, Eduardo J. Páez, and Barry C. Sanders, "Framework for learning and control in the classical and quantum domains", Annals of Physics 458, 169471 (2023).

[26] Shi-Ju Ran and Gang Su, "Tensor Networks for Interpretable and Efficient Quantum-Inspired Machine Learning", Intelligent Computing 2, 0061 (2023).

[27] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[28] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[29] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio, "A semi-agnostic ansatz with variable structure for variational quantum algorithms", Quantum Machine Intelligence 5 2, 43 (2023).

[30] Andy C. Y. Li, M. Sohaib Alam, Thomas Iadecola, Ammar Jahin, Joshua Job, Doga Murat Kurkcuoglu, Richard Li, Peter P. Orth, A. Barış Özgüler, Gabriel N. Perdue, and Norm M. Tubman, "Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model", Physical Review Research 5 3, 033071 (2023).

[31] Nikita Astrakhantsev, Guglielmo Mazzola, Ivano Tavernelli, and Giuseppe Carleo, "Phenomenological theory of variational quantum ground-state preparation", Physical Review Research 5 3, 033225 (2023).

[32] Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev, "Quantum computing for finance", Nature Reviews Physics 5 8, 450 (2023).

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

[34] Bingzhi Zhang, Akira Sone, and Quntao Zhuang, "Quantum computational phase transition in combinatorial problems", npj Quantum Information 8 1, 87 (2022).

[35] Ryan LaRose, Eleanor Rieffel, and Davide Venturelli, "Mixer-phaser Ansätze for quantum optimization with hard constraints", Quantum Machine Intelligence 4 2, 17 (2022).

[36] Charles Moussa, Yash J. Patel, Vedran Dunjko, Thomas Bäck, and Jan N. van Rijn, "Hyperparameter importance and optimization of quantum neural networks across small datasets", Machine Learning (2023).

[37] Jeremy Côté, Frédéric Sauvage, Martín Larocca, Matías Jonsson, Lukasz Cincio, and Tameem Albash, "Diabatic quantum annealing for the frustrated ring model", Quantum Science and Technology 8 4, 045033 (2023).

[38] Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J. Coles, "Generalization in quantum machine learning from few training data", Nature Communications 13 1, 4919 (2022).

[39] Andrea Skolik, Michele Cattelan, Sheir Yarkoni, Thomas Bäck, and Vedran Dunjko, "Equivariant quantum circuits for learning on weighted graphs", npj Quantum Information 9 1, 47 (2023).

[40] Yuxuan Du, Yibo Yang, Dacheng Tao, and Min-Hsiu Hsieh, "Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification", Physical Review Letters 131 14, 140601 (2023).

[41] El Amine Cherrat, Snehal Raj, Iordanis Kerenidis, Abhishek Shekhar, Ben Wood, Jon Dee, Shouvanik Chakrabarti, Richard Chen, Dylan Herman, Shaohan Hu, Pierre Minssen, Ruslan Shaydulin, Yue Sun, Romina Yalovetzky, and Marco Pistoia, "Quantum Deep Hedging", Quantum 7, 1191 (2023).

[42] Frédéric Sauvage, Martín Larocca, Patrick J Coles, and M Cerezo, "Building spatial symmetries into parameterized quantum circuits for faster training", Quantum Science and Technology 9 1, 015029 (2024).

[43] M. R. Perelshtein, A. I. Pakhomchik, Ar. A. Melnikov, M. Podobrii, A. Termanova, I. Kreidich, B. Nuriev, S. Iudin, C. W. Mansell, and V. M. Vinokur, "NISQ-compatible approximate quantum algorithm for unconstrained and constrained discrete optimization", Quantum 7, 1186 (2023).

[44] Massimiliano Incudini, Michele Grossi, Antonio Mandarino, Sofia Vallecorsa, Alessandra Di Pierro, and David Windridge, "The Quantum Path Kernel: A Generalized Neural Tangent Kernel for Deep Quantum Machine Learning", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[45] Adrián Pérez-Salinas, Hao Wang, and Xavier Bonet-Monroig, "Analyzing variational quantum landscapes with information content", npj Quantum Information 10 1, 27 (2024).

[46] Supanut Thanasilp, Samson Wang, Nhat Anh Nghiem, Patrick Coles, and Marco Cerezo, "Subtleties in the trainability of quantum machine learning models", Quantum Machine Intelligence 5 1, 21 (2023).

[47] Junyu Liu, Khadijeh Najafi, Kunal Sharma, Francesco Tacchino, Liang Jiang, and Antonio Mezzacapo, "Analytic Theory for the Dynamics of Wide Quantum Neural Networks", Physical Review Letters 130 15, 150601 (2023).

[48] Lukas Broers and Ludwig Mathey, "Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols", Physical Review Research 6 1, 013076 (2024).

[49] Jonathan Foldager and Bálint Koczor, "Can shallow quantum circuits scramble local noise into global white noise?", Journal of Physics A: Mathematical and Theoretical 57 1, 015306 (2024).

[50] Pranav Chandarana, Narendra N. Hegade, Iraitz Montalban, Enrique Solano, and Xi Chen, "Digitized Counterdiabatic Quantum Algorithm for Protein Folding", Physical Review Applied 20 1, 014024 (2023).

[51] Mara Vizzuso, Gianluca Passarelli, Giovanni Cantele, and Procolo Lucignano, "Convergence of digitized-counterdiabatic QAOA: circuit depth versus free parameters", New Journal of Physics 26 1, 013002 (2024).

[52] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[53] Louis Schatzki, Andrew Arrasmith, Patrick J. Coles, and M. Cerezo, "Entangled Datasets for Quantum Machine Learning", arXiv:2109.03400, (2021).

[54] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, "Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe", arXiv:2205.12110, (2022).

[55] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert, "Exploiting Symmetry in Variational Quantum Machine Learning", PRX Quantum 4 1, 010328 (2023).

[56] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, "Theory of overparametrization in quantum neural networks", arXiv:2109.11676, (2021).

[57] Martín Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo, "Group-Invariant Quantum Machine Learning", PRX Quantum 3 3, 030341 (2022).

[58] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn, "Avoiding Barren Plateaus Using Classical Shadows", PRX Quantum 3 2, 020365 (2022).

[59] 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, (2021).

[60] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration and untrainability in quantum kernel methods", arXiv:2208.11060, (2022).

[61] Adam Callison and Nicholas Chancellor, "Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond", Physical Review A 106 1, 010101 (2022).

[62] Supanut Thanasilp, Samson Wang, Nhat A. Nghiem, Patrick J. Coles, and M. Cerezo, "Subtleties in the trainability of quantum machine learning models", arXiv:2110.14753, (2021).

[63] James Sud, Stuart Hadfield, Eleanor Rieffel, Norm Tubman, and Tad Hogg, "A Parameter Setting Heuristic for the Quantum Alternating Operator Ansatz", arXiv:2211.09270, (2022).

[64] Andi Gu, Angus Lowe, Pavel A. Dub, Patrick J. Coles, and Andrew Arrasmith, "Adaptive shot allocation for fast convergence in variational quantum algorithms", arXiv:2108.10434, (2021).

[65] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", Quantum 6, 796 (2022).

[66] Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao, "Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits", arXiv:2203.09376, (2022).

[67] Léo Monbroussou, Jonas Landman, Alex B. Grilo, Romain Kukla, and Elham Kashefi, "Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning", arXiv:2309.15547, (2023).

[68] Eric R. Anschuetz and Xun Gao, "Arbitrary Polynomial Separations in Trainable Quantum Machine Learning", arXiv:2402.08606, (2024).

[69] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "Noisy intermediate-scale quantum algorithm for semidefinite programming", Physical Review A 105 5, 052445 (2022).

[70] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and Opportunities in Quantum Machine Learning", arXiv:2303.09491, (2023).

[71] John Napp, "Quantifying the barren plateau phenomenon for a model of unstructured variational ansätze", arXiv:2203.06174, (2022).

[72] Ayush Asthana, Chenxu Liu, Oinam Romesh Meitei, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Minimizing state preparation times in pulse-level variational molecular simulations", arXiv:2203.06818, (2022).

[73] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao, "Toward Trainability of Deep Quantum Neural Networks", arXiv:2112.15002, (2021).

[74] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", Quantum 6, 861 (2022).

[75] Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "The battle of clean and dirty qubits in the era of partial error correction", arXiv:2205.13454, (2022).

[76] L. C. G. Govia, C. Poole, M. Saffman, and H. K. Krovi, "Freedom of the mixer rotation axis improves performance in the quantum approximate optimization algorithm", Physical Review A 104 6, 062428 (2021).

[77] Xinbiao Wang, Junyu Liu, Tongliang Liu, Yong Luo, Yuxuan Du, and Dacheng Tao, "Symmetric Pruning in Quantum Neural Networks", arXiv:2208.14057, (2022).

[78] Xinyu Fei, Lucas T. Brady, Jeffrey Larson, Sven Leyffer, and Siqian Shen, "Switching Time Optimization for Binary Quantum Optimal Control", arXiv:2308.03132, (2023).

[79] Chiara Leadbeater, Louis Sharrock, Brian Coyle, and Marcello Benedetti, "F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits", Entropy 23 10, 1281 (2021).

[80] Enrico Fontana, Ivan Rungger, Ross Duncan, and Cristina Cîrstoiu, "Efficient recovery of variational quantum algorithms landscapes using classical signal processing", arXiv:2208.05958, (2022).

[81] 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, (2022).

[82] Vivek Katial, Kate Smith-Miles, and Charles Hill, "On the Instance Dependence of Optimal Parameters for the Quantum Approximate Optimisation Algorithm: Insights via Instance Space Analysis", arXiv:2401.08142, (2024).

[83] Alistair W. R. Smith, A. J. Paige, and M. S. Kim, "Faster variational quantum algorithms with quantum kernel-based surrogate models", Quantum Science and Technology 8 4, 045016 (2023).

[84] Zeyi Tao, Jindi Wu, Qi Xia, and Qun Li, "LAWS: Look Around and Warm-Start Natural Gradient Descent for Quantum Neural Networks", arXiv:2205.02666, (2022).

[85] Ryan LaRose, Eleanor Rieffel, and Davide Venturelli, "Mixer-Phaser Ansätze for Quantum Optimization with Hard Constraints", arXiv:2107.06651, (2021).

[86] Adrián Pérez-Salinas, Hao Wang, and Xavier Bonet-Monroig, "Analyzing variational quantum landscapes with information content", arXiv:2303.16893, (2023).

[87] Saad Yalouz, Bruno Senjean, Filippo Miatto, and Vedran Dunjko, "Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model", Quantum 5, 572 (2021).

[88] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum Machine Learning for Chemistry and Physics", arXiv:2111.00851, (2021).

[89] Owen Lockwood, "Optimizing Quantum Variational Circuits with Deep Reinforcement Learning", arXiv:2109.03188, (2021).

[90] Yudai Suzuki and Muyuan Li, "Effect of alternating layered ansatzes on trainability of projected quantum kernel", arXiv:2310.00361, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-02 15:31:37) and SAO/NASA ADS (last updated successfully 2024-03-02 15:31:38). The list may be incomplete as not all publishers provide suitable and complete citation data.