Effect of barren plateaus on gradient-free optimization

Andrew Arrasmith1, M. Cerezo1,2, Piotr Czarnik1, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

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Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

Quantum machine learning and variational quantum algorithms provide the potential for near-term, practical applications of quantum computers. However, these techniques require the use of a classical optimization loop that can, in some cases, prove prohibitive. In particular, some of these optimization landscapes present barren plateaus, where the gradient of the cost function is exponentially suppressed except within an exponentially small region. As a result, optimization on a barren plateau landscape requires computational resources that scale exponentially with the number of qubits used.

Perhaps because of the mention of gradients in the definition of a barren plateau, a number of researchers have postulated that a gradient free optimizer might be able to beat the resource scaling. This thought turns out to be incorrect. Here we provide a proof that no gradient free strategy (including models that build surrogate gradients) can escape this resource scaling in barren plateau landscapes.

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► References

[1] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics 1, 19 (2021a).

[2] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum (nisq) algorithms, arXiv preprint arXiv:2101.08448 (2021).

[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, 1 (2014).

[4] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18, 023023 (2016).

[5] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv preprint arXiv:1411.4028 (2014).

[6] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).

[7] S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, Quantum-assisted quantum compiling, Quantum 3, 140 (2019).

[8] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, Variational quantum state diagonalization, npj Quantum Information 5, 1 (2019).

[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 (2019).

[10] M. Cerezo, A. Poremba, L. Cincio, and P. J. Coles, Variational quantum fidelity estimation, Quantum 4, 248 (2020a).

[11] C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time, npj Quantum Information 6, 1 (2020).

[12] K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, Noise resilience of variational quantum compiling, New Journal of Physics 22, 043006 (2020a).

[13] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. Coles, Variational quantum linear solver, arXiv preprint arXiv:1909.05820 (2019).

[14] M. Cerezo, K. Sharma, A. Arrasmith, and P. J. Coles, Variational quantum state eigensolver, arXiv preprint arXiv:2004.01372 (2020b).

[15] M. Schuld, I. Sinayskiy, and F. Petruccione, The quest for a quantum neural network, Quantum Information Processing 13, 2567 (2014).

[16] I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nature Physics 15, 1273 (2019).

[17] K. Beer, D. Bondarenko, T. Farrelly, T. J. Osborne, R. Salzmann, D. Scheiermann, and R. Wolf, Training deep quantum neural networks, Nature Communications 11, 808 (2020).

[18] G. Verdon, J. Pye, and M. Broughton, A universal training algorithm for quantum deep learning, arXiv preprint arXiv:1806.09729 (2018).

[19] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications 9, 1 (2018).

[20] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nature Communications 12, 1791 (2021b).

[21] K. Sharma, M. Cerezo, L. Cincio, and P. J. Coles, Trainability of dissipative perceptron-based quantum neural networks, arXiv preprint arXiv:2005.12458 (2020b).

[22] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, arXiv preprint arXiv:2007.14384 (2020).

[23] M. Cerezo and P. J. Coles, Higher order derivatives of quantum neural networks with barren plateaus, Quantum Science and Technology 6, 035006 (2021).

[24] Z. Holmes, A. Arrasmith, B. Yan, P. J. Coles, A. Albrecht, and A. T. Sornborger, Barren plateaus preclude learning scramblers, Physical Review Letters 126, 190501 (2021a).

[25] A. Pesah, M. Cerezo, S. Wang, T. Volkoff, A. T. Sornborger, and P. J. Coles, Absence of barren plateaus in quantum convolutional neural networks, arXiv preprint arXiv:2011.02966 (2020).

[26] K. Zhang, M.-H. Hsieh, L. Liu, and D. Tao, Toward trainability of quantum neural networks, arXiv preprint arXiv:2011.06258 (2020).

[27] A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, The power of quantum neural networks, Nature Computational Science 1, 403 (2021).

[28] C. O. Marrero, M. Kieferová, and N. Wiebe, Entanglement induced barren plateaus, arXiv preprint arXiv:2010.15968 (2020).

[29] T. L. Patti, K. Najafi, X. Gao, and S. F. Yelin, Entanglement devised barren plateau mitigation, Physical Review Research 3, 033090 (2021).

[30] A. Uvarov and J. D. Biamonte, On barren plateaus and cost function locality in variational quantum algorithms, Journal of Physics A: Mathematical and Theoretical 54, 245301 (2021).

[31] Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, arXiv preprint arXiv:2101.02138 (2021b).

[32] Y. Du, M.-H. Hsieh, T. Liu, S. You, and D. Tao, On the learnability of quantum neural networks, arXiv preprint arXiv:2007.12369 (2020).

[33] A. Arrasmith, Z. Holmes, M. Cerezo, and P. J. Coles, Equivalence of quantum barren plateaus to cost concentration and narrow gorges, arXiv preprint arXiv:2104.05868 (2021).

[34] G. Verdon, M. Broughton, J. R. McClean, K. J. Sung, R. Babbush, Z. Jiang, H. Neven, and M. Mohseni, Learning to learn with quantum neural networks via classical neural networks, arXiv preprint arXiv:1907.05415 (2019).

[35] T. Volkoff and P. J. Coles, Large gradients via correlation in random parameterized quantum circuits, Quantum Science and Technology 6, 025008 (2021).

[36] A. Skolik, J. R. McClean, M. Mohseni, P. van der Smagt, and M. Leib, Layerwise learning for quantum neural networks, Quantum Machine Intelligence 3, 1 (2021).

[37] E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).

[38] E. Campos, A. Nasrallah, and J. Biamonte, Abrupt transitions in variational quantum circuit training, Physical Review A 103, 032607 (2021).

[39] E. Farhi and H. Neven, Classification with quantum neural networks on near term processors, arXiv preprint arXiv:1802.06002 (2018).

[40] N. Killoran, T. R. Bromley, J. M. Arrazola, M. Schuld, N. Quesada, and S. Lloyd, Continuous-variable quantum neural networks, Physical Review Research 1, 033063 (2019).

[41] J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles, An adaptive optimizer for measurement-frugal variational algorithms, Quantum 4, 263 (2020).

[42] R. Sweke, F. Wilde, J. J. Meyer, M. Schuld, P. K. Fährmann, B. Meynard-Piganeau, and J. Eisert, Stochastic gradient descent for hybrid quantum-classical optimization, Quantum 4, 314 (2020).

[43] A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles, Operator sampling for shot-frugal optimization in variational algorithms, arXiv preprint arXiv:2004.06252 (2020).

[44] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).

[45] J. A. Nelder and R. Mead, A simplex method for function minimization, The computer journal 7, 308 (1965).

[46] R. R. Barton and J. S. Ivey Jr, Modifications of the nelder-mead simplex method for stochastic simulation response optimization, 1991 Winter Simulation Conference Proceedings (1991), 10.1109/​WSC.1991.185709.

[47] X. Huang, Robust simplex algorithm for online optimization, Physical Review Accelerators and Beams 21, 104601 (2018).

[48] M. J. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives, The computer journal 7, 155 (1964).

[49] R. P. Brent, Algorithms for minimization without derivatives (Courier Corporation, 2013).

[50] M. J. Powell, in Advances in optimization and numerical analysis (Springer, 1994) pp. 51–67.

[51] A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Communications in Mathematical Physics 291, 257 (2009).

[52] F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Communications in Mathematical Physics 346, 397 (2016).

[53] A. Harrow and S. Mehraban, Approximate unitary $ t $-designs by short random quantum circuits using nearest-neighbor and long-range gates, arXiv preprint arXiv:1809.06957 (2018).

[54] J. Močkus, in Optimization techniques IFIP technical conference (Springer, 1975) pp. 400–404.

[55] A. Anand, M. Degroote, and A. Aspuru-Guzik, Natural evolutionary strategies for variational quantum computation, Machine Learning: Science and Technology (2021), 10.1088/​2632-2153/​abf3ac.

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[1] 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).

[2] Stefano Markidis, "On physics-informed neural networks for quantum computers", Frontiers in Applied Mathematics and Statistics 8, 1036711 (2022).

[3] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).

[4] 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).

[5] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee, "Quantum machine learning: from physics to software engineering", Advances in Physics: X 8 1, 2165452 (2023).

[6] Junyu Liu, Zexi Lin, and Liang Jiang, "Laziness, barren plateau, and noises in machine learning", Machine Learning: Science and Technology 5 1, 015058 (2024).

[7] 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).

[8] Lucas Friedrich and Jonas Maziero, "Evolution strategies: application in hybrid quantum-classical neural networks", Quantum Information Processing 22 3, 132 (2023).

[9] 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).

[10] Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto, "Understanding quantum machine learning also requires rethinking generalization", Nature Communications 15 1, 2277 (2024).

[11] 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).

[12] Hiroshi Ohno, "Grover’s search with learning oracle for constrained binary optimization problems", Quantum Machine Intelligence 6 1, 12 (2024).

[13] 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, 1791 (2021).

[14] V. A. Zaytsev, M. E. Groshev, I. A. Maltsev, A. V. Durova, and V. M. Shabaev, "Calculation of the moscovium ground‐state energy by quantum algorithms", International Journal of Quantum Chemistry 124 1, e27232 (2024).

[15] 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).

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

[17] Fabio Zoratti, Giacomo De Palma, Bobak Kiani, Quynh T. Nguyen, Milad Marvian, Seth Lloyd, and Vittorio Giovannetti, "Improving the speed of variational quantum algorithms for quantum error correction", Physical Review A 108 2, 022611 (2023).

[18] Jonathan Wei Zhong Lau, Kian Hwee Lim, Harshank Shrotriya, and Leong Chuan Kwek, "NISQ computing: where are we and where do we go?", AAPPS Bulletin 32 1, 27 (2022).

[19] Tyler Volkoff, Zoë Holmes, and Andrew Sornborger, "Universal Compiling and (No-)Free-Lunch Theorems for Continuous-Variable Quantum Learning", PRX Quantum 2 4, 040327 (2021).

[20] Cenk Tüysüz, Giuseppe Clemente, Arianna Crippa, Tobias Hartung, Stefan Kühn, and Karl Jansen, "Classical splitting of parametrized quantum circuits", Quantum Machine Intelligence 5 2, 34 (2023).

[21] Giuseppe Buonaiuto, Francesco Gargiulo, Giuseppe De Pietro, Massimo Esposito, and Marco Pota, "The effects of quantum hardware properties on the performances of variational quantum learning algorithms", Quantum Machine Intelligence 6 1, 9 (2024).

[22] Han Qi, Lei Wang, Hongsheng Zhu, Abdullah Gani, and Changqing Gong, "The barren plateaus of quantum neural networks: review, taxonomy and trends", Quantum Information Processing 22 12, 435 (2023).

[23] Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, "Fock state-enhanced expressivity of quantum machine learning models", EPJ Quantum Technology 9 1, 16 (2022).

[24] David A. Herrera-Martí, "Policy Gradient Approach to Compilation of Variational Quantum Circuits", Quantum 6, 797 (2022).

[25] Raphael César de Souza Pimenta and Anibal Thiago Bezerra, "Revisiting semiconductor bulk hamiltonians using quantum computers", Physica Scripta 98 4, 045804 (2023).

[26] Christopher K. Long, Kieran Dalton, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur, and Normann Mertig, "Layering and subpool exploration for adaptive variational quantum eigensolvers: Reducing circuit depth, runtime, and susceptibility to noise", Physical Review A 109 4, 042413 (2024).

[27] Hannes Leipold, Federico M. Spedalieri, and Eleanor Rieffel, "Tailored Quantum Alternating Operator Ansätzes for Circuit Fault Diagnostics", Algorithms 15 10, 356 (2022).

[28] G. Paradezhenko, A. Pervishko, and D. Yudin, "Quantum-assisted Open-pit Optimization", JETP Letters (2024).

[29] Zidu Liu, L.-M. Duan, and Dong-Ling Deng, "Solving quantum master equations with deep quantum neural networks", Physical Review Research 4 1, 013097 (2022).

[30] Brian Coyle, Mina Doosti, Elham Kashefi, and Niraj Kumar, "Progress toward practical quantum cryptanalysis by variational quantum cloning", Physical Review A 105 4, 042604 (2022).

[31] Muhammad Kashif and Saif Al-Kuwari, "The unified effect of data encoding, ansatz expressibility and entanglement on the trainability of HQNNs", International Journal of Parallel, Emergent and Distributed Systems 38 5, 362 (2023).

[32] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).

[33] Lucas Friedrich and Jonas Maziero, "Quantum neural network cost function concentration dependency on the parametrization expressivity", Scientific Reports 13 1, 9978 (2023).

[34] Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, Yong-Jian Han, and Guo-Ping Guo, "Mitigating barren plateaus with transfer-learning-inspired parameter initializations", New Journal of Physics 25 1, 013039 (2023).

[35] Ze‐Tong Li, Fan‐Xu Meng, Han Zeng, Zhai‐Rui Gong, Zai‐Chen Zhang, and Xu‐Tao Yu, "A Gradient‐Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz", Advanced Quantum Technologies 6 5, 2200130 (2023).

[36] Raoul Heese, Patricia Bickert, and Astrid Elisa Niederle, "Representation of binary classification trees with binary features by quantum circuits", Quantum 6, 676 (2022).

[37] S. Shin, Y. S. Teo, and H. Jeong, "Exponential data encoding for quantum supervised learning", Physical Review A 107 1, 012422 (2023).

[38] Pierre Decoodt, Tan Jun Liang, Soham Bopardikar, Hemavathi Santhanam, Alfaxad Eyembe, Begonya Garcia-Zapirain, and Daniel Sierra-Sosa, "Hybrid Classical–Quantum Transfer Learning for Cardiomegaly Detection in Chest X-rays", Journal of Imaging 9 7, 128 (2023).

[39] Korbinian Kottmann, Friederike Metz, Joana Fraxanet, and Niccolò Baldelli, "Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer", Physical Review Research 3 4, 043184 (2021).

[40] Francesco Scala, Stefano Mangini, Chiara Macchiavello, Daniele Bajoni, and Dario Gerace, 2022 International Joint Conference on Neural Networks (IJCNN) 1 (2022) ISBN:978-1-7281-8671-9.

[41] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?", Quantum 8, 1287 (2024).

[42] Matthias Möller, Computational Methods in Applied Sciences 58, 357 (2023) ISBN:978-3-031-29081-7.

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

[44] Weiyuan Gong, Dong Yuan, Weikang Li, and Dong-Ling Deng, "Enhancing quantum adversarial robustness by randomized encodings", Physical Review Research 6 2, 023020 (2024).

[45] Oriel Kiss, Michele Grossi, Pavel Lougovski, Federico Sanchez, Sofia Vallecorsa, and Thomas Papenbrock, "Quantum computing of the Li6 nucleus via ordered unitary coupled clusters", Physical Review C 106 3, 034325 (2022).

[46] Cristian L. Cortes and Stephen K. Gray, "Quantum Krylov subspace algorithms for ground- and excited-state energy estimation", Physical Review A 105 2, 022417 (2022).

[47] Y. S. Teo, "Optimized numerical gradient and Hessian estimation for variational quantum algorithms", Physical Review A 107 4, 042421 (2023).

[48] Xin-Yu Chen, Pan Gao, Chu-Dan Qiu, Ya-Nan Lu, Fan Yang, Yuanyuan Zhao, Hang Li, Jiang Zhang, Shijie Wei, Tonghao Xing, Xin-Yu Pan, Dong Ruan, Feihao Zhang, Keren Li, and Guilu Long, "A noise-robust quantum dynamics learning protocol based on Choi–Jamiolkowski isomorphism: theory and experiment", New Journal of Physics 26 3, 033023 (2024).

[49] 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).

[50] Lucas Friedrich and Jonas Maziero, "Restricting to the chip architecture maintains the quantum neural network accuracy", Quantum Information Processing 23 4, 131 (2024).

[51] Laura Gentini, Alessandro Cuccoli, and Leonardo Banchi, "Variational Adiabatic Gauge Transformation on Real Quantum Hardware for Effective Low-Energy Hamiltonians and Accurate Diagonalization", Physical Review Applied 18 3, 034025 (2022).

[52] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, "Diagnosing Barren Plateaus with Tools from Quantum Optimal Control", Quantum 6, 824 (2022).

[53] Steven Herbert, "Quantum computing for data-centric engineering and science", Data-Centric Engineering 3, e36 (2022).

[54] 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 9, 625 (2021).

[55] Rafael Pereira da Silva, "Hamiltonian Minimization in the NISQ Era", SSRN Electronic Journal (2023).

[56] Zidu Liu, Li-Wei Yu, L.-M. Duan, and Dong-Ling Deng, "Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning", Physical Review Letters 129 27, 270501 (2022).

[57] Silvirianti, Bhaskara Narottama, and Soo Young Shin, "Layerwise Quantum Deep Reinforcement Learning for Joint Optimization of UAV Trajectory and Resource Allocation", IEEE Internet of Things Journal 11 1, 430 (2024).

[58] 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).

[59] 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).

[60] Marvin Bechtold, Johanna Barzen, Frank Leymann, Alexander Mandl, Julian Obst, Felix Truger, and Benjamin Weder, "Investigating the effect of circuit cutting in QAOA for the MaxCut problem on NISQ devices", Quantum Science and Technology 8 4, 045022 (2023).

[61] Lixue Cheng, Yu-Qin Chen, Shi-Xin Zhang, and Shengyu Zhang, "Quantum approximate optimization via learning-based adaptive optimization", Communications Physics 7 1, 83 (2024).

[62] Unpil Baek, Diptarka Hait, James Shee, Oskar Leimkuhler, William J. Huggins, Torin F. Stetina, Martin Head-Gordon, and K. Birgitta Whaley, "Say NO to Optimization: A Nonorthogonal Quantum Eigensolver", PRX Quantum 4 3, 030307 (2023).

[63] Cristian L. Cortes, A. Eugene DePrince, and Stephen K. Gray, "Fast-forwarding quantum simulation with real-time quantum Krylov subspace algorithms", Physical Review A 106 4, 042409 (2022).

[64] Marco Ballarin, Stefano Mangini, Simone Montangero, Chiara Macchiavello, and Riccardo Mengoni, "Entanglement entropy production in Quantum Neural Networks", Quantum 7, 1023 (2023).

[65] Shuo Liu, Shi-Xin Zhang, Shao-Kai Jian, and Hong Yao, "Training variational quantum algorithms with random gate activation", Physical Review Research 5 3, L032040 (2023).

[66] Simone Tibaldi, Davide Vodola, Edoardo Tignone, and Elisa Ercolessi, "Bayesian Optimization for QAOA", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[67] Kun Wang, Zhixin Song, Xuanqiang Zhao, Zihe Wang, and Xin Wang, "Detecting and quantifying entanglement on near-term quantum devices", npj Quantum Information 8 1, 52 (2022).

[68] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Long-time simulations for fixed input states on quantum hardware", npj Quantum Information 8 1, 135 (2022).

[69] 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, 6961 (2021).

[70] Francesco Scala, Andrea Ceschini, Massimo Panella, and Dario Gerace, "A General Approach to Dropout in Quantum Neural Networks", Advanced Quantum Technologies 2300220 (2023).

[71] Xiaozhen Ge, Re-Bing Wu, and Herschel Rabitz, "The optimization landscape of hybrid quantum–classical algorithms: From quantum control to NISQ applications", Annual Reviews in Control 54, 314 (2022).

[72] Eric R. Anschuetz and Bobak T. Kiani, "Quantum variational algorithms are swamped with traps", Nature Communications 13 1, 7760 (2022).

[73] Joe Gibbs, Zoë Holmes, Matthias C. Caro, Nicholas Ezzell, Hsin-Yuan Huang, Lukasz Cincio, Andrew T. Sornborger, and Patrick J. Coles, "Dynamical simulation via quantum machine learning with provable generalization", Physical Review Research 6 1, 013241 (2024).

[74] Ju-Young Ryu, Eyuel Elala, and June-Koo Kevin Rhee, "Quantum Graph Neural Network Models for Materials Search", Materials 16 12, 4300 (2023).

[75] Charles Moussa, Max Hunter Gordon, Michal Baczyk, M Cerezo, Lukasz Cincio, and Patrick J Coles, "Resource frugal optimizer for quantum machine learning", Quantum Science and Technology 8 4, 045019 (2023).

[76] Zeyi Tao, Jindi Wu, Qi Xia, and Qun Li, 2023 IEEE International Conference on Quantum Software (QSW) 76 (2023) ISBN:979-8-3503-0479-4.

[77] Zidu Liu, Pei-Xin Shen, Weikang Li, L-M Duan, and Dong-Ling Deng, "Quantum capsule networks", Quantum Science and Technology 8 1, 015016 (2023).

[78] Ramin Fakhimi and Hamidreza Validi, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[79] Tobias Haug and Kishor Bharti, "Generalized quantum assisted simulator", Quantum Science and Technology 7 4, 045019 (2022).

[80] 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).

[81] J. Gidi, B. Candia, A. D. Muñoz-Moller, A. Rojas, L. Pereira, M. Muñoz, L. Zambrano, and A. Delgado, "Stochastic optimization algorithms for quantum applications", Physical Review A 108 3, 032409 (2023).

[82] Francesco Di Marcantonio, Massimiliano Incudini, Davide Tezza, and Michele Grossi, "Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learning", Quantum Machine Intelligence 5 1, 20 (2023).

[83] Lucas Friedrich and Jonas Maziero, "Avoiding barren plateaus with classical deep neural networks", Physical Review A 106 4, 042433 (2022).

[84] Valentin Heyraud, Zejian Li, Kaelan Donatella, Alexandre Le Boité, and Cristiano Ciuti, "Efficient Estimation of Trainability for Variational Quantum Circuits", PRX Quantum 4 4, 040335 (2023).

[85] Vladimir Vargas-Calderón, Fabio A. González, and Herbert Vinck-Posada, "Optimisation-free density estimation and classification with quantum circuits", Quantum Machine Intelligence 4 2, 16 (2022).

[86] 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).

[87] Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt, "Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm", Physical Review Research 5 2, 023147 (2023).

[88] Hao-Kai Zhang, Shuo Liu, and Shi-Xin Zhang, "Absence of Barren Plateaus in Finite Local-Depth Circuits with Long-Range Entanglement", Physical Review Letters 132 15, 150603 (2024).

[89] Gabriele Agliardi and Enrico Prati, "Optimal Tuning of Quantum Generative Adversarial Networks for Multivariate Distribution Loading", Quantum Reports 4 1, 75 (2022).

[90] Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, and Alioscia Hamma, "Retrieving information from a black hole using quantum machine learning", Physical Review A 106 6, 062434 (2022).

[91] Daniel Faílde, José Daniel Viqueira, Mariamo Mussa Juane, and Andrés Gómez, "Using Differential Evolution to avoid local minima in Variational Quantum Algorithms", Scientific Reports 13 1, 16230 (2023).

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

[93] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[94] 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, (2020).

[95] Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, and Masoud Mohseni, "TensorFlow Quantum: A Software Framework for Quantum Machine Learning", arXiv:2003.02989, (2020).

[96] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, "Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus", PRX Quantum 3 1, 010313 (2022).

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

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

[99] M. Cerezo and Patrick J. Coles, "Higher order derivatives of quantum neural networks with barren plateaus", Quantum Science and Technology 6 3, 035006 (2021).

[100] 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).

[101] 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).

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

[103] 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 the Variational Quantum Eigensolver", PRX Quantum 2 2, 020337 (2021).

[104] 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).

[105] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational quantum algorithms for trace distance and fidelity estimation", Quantum Science and Technology 7 1, 015019 (2022).

[106] Maria Kieferova, Ortiz Marrero Carlos, and Nathan Wiebe, "Quantum Generative Training Using Rényi Divergences", arXiv:2106.09567, (2021).

[107] Stefano Mangini, "Variational quantum algorithms for machine learning: theory and applications", arXiv:2306.09984, (2023).

[108] Lukas Franken, Bogdan Georgiev, Sascha Mücke, Moritz Wolter, Raoul Heese, Christian Bauckhage, and Nico Piatkowski, "Quantum Circuit Evolution on NISQ Devices", arXiv:2012.13453, (2020).

[109] Ali Rad, Alireza Seif, and Norbert M. Linke, "Surviving The Barren Plateau in Variational Quantum Circuits with Bayesian Learning Initialization", arXiv:2203.02464, (2022).

[110] Christa Zoufal, "Generative Quantum Machine Learning", arXiv:2111.12738, (2021).

[111] Hao-Kai Zhang, Chengkai Zhu, Geng Liu, and Xin Wang, "Fundamental limitations on optimization in variational quantum algorithms", arXiv:2205.05056, (2022).

[112] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational Quantum Algorithms for Trace Distance and Fidelity Estimation", arXiv:2012.05768, (2020).

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

[114] Yidong Liao, Min-Hsiu Hsieh, and Chris Ferrie, "Quantum Optimization for Training Quantum Neural Networks", arXiv:2103.17047, (2021).

[115] Yunfei Wang and Junyu Liu, "A comprehensive review of Quantum Machine Learning: from NISQ to Fault Tolerance", arXiv:2401.11351, (2024).

[116] Brian Coyle, "Machine learning applications for noisy intermediate-scale quantum computers", arXiv:2205.09414, (2022).

[117] Lucas Slattery, Ruslan Shaydulin, Shouvanik Chakrabarti, Marco Pistoia, Sami Khairy, and Stefan M. Wild, "Numerical evidence against advantage with quantum fidelity kernels on classical data", Physical Review A 107 6, 062417 (2023).

[118] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[119] Bingzhi Zhang and Quntao Zhuang, "Fast decay of classification error in variational quantum circuits", Quantum Science and Technology 7 3, 035017 (2022).

[120] Akash Kundu, "Reinforcement learning-assisted quantum architecture search for variational quantum algorithms", arXiv:2402.13754, (2024).

[121] Abhinav Anand, Matthias Degroote, and Alán Aspuru-Guzik, "Natural Evolutionary Strategies for Variational Quantum Computation", arXiv:2012.00101, (2020).

[122] 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).

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

[124] Nahum Sá, Ivan S. Oliveira, and Itzhak Roditi, "Towards solving the BCS Hamiltonian gap in Near-Term Quantum Computers", arXiv:2105.14936, (2021).

[125] Elias X. Huber, Benjamin Y. L. Tan, Paul R. Griffin, and Dimitris G. Angelakis, "Exponential Qubit Reduction in Optimization for Financial Transaction Settlement", arXiv:2307.07193, (2023).

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

[127] Adrián Pérez-Salinas, "Algorithmic Strategies for seizing Quantum Computing", arXiv:2112.15175, (2021).

[128] Zidu Liu, Qi Ye, Li-Wei Yu, L. -M. Duan, and Dong-Ling Deng, "Theory on variational high-dimensional tensor networks", arXiv:2303.17452, (2023).

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

[130] Nahum Sá, Ivan S. Oliveira, and Itzhak Roditi, "Towards solving the BCS Hamiltonian gap in near-term quantum computers", Results in Physics 44, 106131 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 04:36:44) and SAO/NASA ADS (last updated successfully 2024-04-19 04:36:46). The list may be incomplete as not all publishers provide suitable and complete citation data.