An Adaptive Optimizer for Measurement-Frugal Variational Algorithms

Jonas M. Kübler1,2, Andrew Arrasmith1, Lukasz Cincio1, and Patrick J. Coles1

1Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
2Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 Tübingen, Germany.

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


Variational hybrid quantum-classical algorithms (VHQCAs) have the potential to be useful in the era of near-term quantum computing. However, recently there has been concern regarding the number of measurements needed for convergence of VHQCAs. Here, we address this concern by investigating the classical optimizer in VHQCAs. We introduce a novel optimizer called individual Coupled Adaptive Number of Shots (iCANS). This adaptive optimizer frugally selects the number of measurements (i.e., number of shots) both for a given iteration and for a given partial derivative in a stochastic gradient descent. We numerically simulate the performance of iCANS for the variational quantum eigensolver and for variational quantum compiling, with and without noise. In all cases, and especially in the noisy case, iCANS tends to out-perform state-of-the-art optimizers for VHQCAs. We therefore believe this adaptive optimizer will be useful for realistic VHQCA implementations, where the number of measurements is limited.

One of the major technological questions of our time is whether near-term quantum computers will have practical applications. A promising strategy to address this question is the paradigm of variational quantum algorithms, which combine classical optimization with evaluation of objective functions on a quantum device. These algorithms make the most of the limited resources of the comparatively small (hundreds of qubits), noisy quantum devices that will likely be available in the coming years. This potential makes variational algorithms arguably the best candidates for demonstrating the first advantage in computational speed or cost for quantum computers.

However, there is a legitimate concern over whether or not the number of times a quantum state must be prepared and measured (i.e. the number of “shots” taken on the quantum device) in order for the variational algorithm to converge will be prohibitive. This and related questions have sparked a recent interest in researching which classical optimizer should be used in variational algorithms.

In this work, we propose a shot-frugal optimization strategy for variational algorithms that dynamically adjusts the number of shots expended (and thus the precision) for each update step in a stochastic gradient descent procedure that we name iCANS (individual Coupled Adaptive Number of Shots). Allocating measurement resources separately for each component of the gradient estimates, iCANS takes advantage of very inexpensive update steps requiring few shots early in the optimization and smoothly increases the number of shots used in order to achieve a high precision optimization result. We present comparisons between iCANS and other optimizers that have been discussed for the context of variational algorithms and find that iCANS often performs better than these other methods.

► BibTeX data

► References

[1] J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).

[2] 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).

[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).

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

[5] P. D. Johnson, J. Romero, J. Olson, Y. Cao, and A. Aspuru-Guzik, QVECTOR: an algorithm for device-tailored quantum error correction, arXiv:1711.02249 (2017).

[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] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, Variational quantum state diagonalization, npj Quantum Information 5, 57 (2019).

[8] 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, 3438 (2019).

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

[10] T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, Variational quantum algorithms for discovering hamiltonian spectra, Physical Review A 99, 062304 (2019).

[11] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019).

[12] Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Physical Review X 7, 021050 (2017).

[13] C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. Joshi, P. Jurcevic, C. Muschik, P. Silvi, R. Blatt, C. Roos, et al., Self-verifying variational quantum simulation of lattice models, Nature 569, 355 (2019).

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

[15] T. Jones and S. C. Benjamin, Quantum compilation and circuit optimisation via energy dissipation, arXiv:1811.03147 (2018).

[16] K. Heya, Y. Suzuki, Y. Nakamura, and K. Fujii, Variational quantum gate optimization, arXiv:1810.12745 (2018).

[17] S. Endo, Y. Li, S. Benjamin, and X. Yuan, Variational quantum simulation of general processes, arXiv:1812.08778 (2018).

[18] K. Sharma, S. Khatri, M. Cerezo, and P. Coles, Noise resilience of variational quantum compiling, New Journal of Physics (2020), 10.1088/​1367-2630/​ab784c.

[19] J. Carolan, M. Mosheni, J. P. Olson, M. Prabhu, C. Chen, D. Bunandar, N. C. Harris, F. N. Wong, M. Hochberg, S. Lloyd, et al., Variational quantum unsampling on a quantum photonic processor, arXiv:1904.10463 (2019).

[20] N. Yoshioka, Y. O. Nakagawa, K. Mitarai, and K. Fujii, Variational quantum algorithm for non-equilirium steady states, arXiv:1908.09836 (2019).

[21] C. Bravo-Prieto, LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver: A hybrid algorithm for linear systems, arXiv:1909.05820 (2019).

[22] X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational algorithms for linear algebra, arXiv:1909.03898 (2019).

[23] S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Information 5, 1 (2019).

[24] C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time, arXiv:1910.04292 (2019).

[25] D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).

[26] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. Sawaya, et al., Quantum chemistry in the age of quantum computing, Chemical reviews (2018), 10.1021/​acs.chemrev.8b00803.

[27] S. McArdle, S. Endo, A. Aspuru-Guzik, S. Benjamin, and X. Yuan, Quantum computational chemistry, arXiv:1808.10402 (2018).

[28] A. Jena, S. Genin, and M. Mosca, Pauli partitioning with respect to gate sets, arXiv:1907.07859 (2019).

[29] A. F. Izmaylov, T.-C. Yen, R. A. Lang, and V. Verteletskyi, Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method, arXiv:1907.09040 (2019).

[30] T.-C. Yen, V. Verteletskyi, and A. F. Izmaylov, Measuring all compatible operators in one series of a single-qubit measurements using unitary transformations, arXiv:1907.09386 (2019).

[31] P. Gokhale, O. Angiuli, Y. Ding, K. Gui, T. Tomesh, M. Suchara, M. Martonosi, and F. T. Chong, Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families, arXiv:1907.13623 (2019).

[32] O. Crawford, B. van Straaten, D. Wang, T. Parks, E. Campbell, and S. Brierley, Efficient quantum measurement of pauli operators, arXiv:1908.06942 (2019).

[33] P. Gokhale and F. T. Chong, $o(n^3)$ measurement cost for variational quantum eigensolver on molecular hamiltonians, arXiv:1908.11857 (2019).

[34] W. J. Huggins, J. McClean, N. Rubin, Z. Jiang, N. Wiebe, K. B. Whaley, and R. Babbush, Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers, arXiv:1907.13117 (2019).

[35] 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:1907.05415 (2019).

[36] M. Wilson, S. Stromswold, F. Wudarski, S. Hadfield, N. M. Tubman, and E. Rieffel, Optimizing quantum heuristics with meta-learning, arXiv:1908.03185 (2019).

[37] K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for quantum-classical hybrid algorithms, arXiv:1903.12166 (2019).

[38] R. M. Parrish, J. T. Iosue, A. Ozaeta, and P. L. McMahon, A Jacobi diagonalization and Anderson acceleration algorithm for variational quantum algorithm parameter optimization, arXiv:1904.03206 (2019).

[39] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, arXiv:1909.02108 (2019).

[40] L. Balles, J. Romero, and P. Hennig, in Proceedings of the Thirty-Third Conference on Uncertainty in Artificial Intelligence (UAI) (2017) pp. 410–419.

[41] G. A., Qiskit: An Open-source Framework for Quantum Computing, (2019).

[42] D. P. Kingma and J. Ba, in Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2015).

[43] J. C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE transactions on automatic control 37, 332 (1992).

[44] Y. LeCun, Y. Bengio, and G. Hinton, Deep learning, Nature 521, 436 (2015).

[45] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).

[46] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).

[47] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, and N. Killoran, Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv:1811.04968 (2018).

[48] A. Harrow and J. Napp, Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms, arXiv:1901.05374 (2019).

[49] G. G. Guerreschi and M. Smelyanskiy, Practical optimization for hybrid quantum-classical algorithms, arXiv:1701.01450 (2017).

[50] J. S. Bergstra, R. Bardenet, Y. Bengio, and B. Kégl, in Advances in Neural Information Processing Systems 24 (2011) pp. 2546–2554.

[51] L. Liu, H. Jiang, P. He, W. Chen, X. Liu, J. Gao, and J. Han, On the variance of the adaptive learning rate and beyond, arXiv:1908.03265 (2019).

[52] J. C. Spall, Implementation of the simultaneous perturbation algorithm for stochastic optimization, IEEE Transactions on aerospace and electronic systems 34, 817 (1998).

[53] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242 (2017).

[54] 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).

[55] R. P. Brent, Algorithms for Minimization Without Derivatives (Dover Publications, 2013).

[56] D. G. Anderson, Iterative procedures for nonlinear integral equations, Journal of the ACM (JACM) 12, 547 (1965).

[57] P. Pulay, Improved scf convergence acceleration, Journal of Computational Chemistry 3, 556 (1982).

[58] IBM Q 16 Melbourne backend specification, https:/​/​​Qiskit/​ibmq-device-information/​tree/​master/​backends/​melbourne/​V1 (2018).

[59] R. Sweke, F. Wilde, J. Meyer, M. Schuld, P. K. Fährmann, B. Meynard-Piganeau, and J. Eisert, Stochastic gradient descent for hybrid quantum-classical optimization, arXiv:1910.01155 (2019).

Cited by

[1] Yutaka Shikano, Hiroshi C. Watanabe, Ken M. Nakanishi, and Yu-ya Ohnishi, "Post-Hartree–Fock method in quantum chemistry for quantum computer", The European Physical Journal Special Topics 230 4, 1037 (2021).

[2] 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).

[3] Ruhan Wang, Philip Richerme, and Fan Chen, "A hybrid quantum–classical neural network for learning transferable visual representation", Quantum Science and Technology 8 4, 045021 (2023).

[4] Chenfeng Cao, Hiroshi Yano, and Yuya O. Nakagawa, "Accelerated variational quantum eigensolver with joint Bell measurement", Physical Review Research 6 1, 013205 (2024).

[5] Philip Mocz and Aaron Szasz, "Toward Cosmological Simulations of Dark Matter on Quantum Computers", The Astrophysical Journal 910 1, 29 (2021).

[6] 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).

[7] 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).

[8] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum Linear Solver", Quantum 7, 1188 (2023).

[9] Ryan Shaffer, Lucas Kocia, and Mohan Sarovar, "Surrogate-based optimization for variational quantum algorithms", Physical Review A 107 3, 032415 (2023).

[10] 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).

[11] 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).

[12] Guoming Wang, Dax Enshan Koh, Peter D. Johnson, and Yudong Cao, "Minimizing Estimation Runtime on Noisy Quantum Computers", PRX Quantum 2 1, 010346 (2021).

[13] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational quantum state eigensolver", npj Quantum Information 8 1, 113 (2022).

[14] David Fitzek, Robert S. Jonsson, Werner Dobrautz, and Christian Schäfer, "Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes", Quantum 8, 1313 (2024).

[15] Maria Schuld and Nathan Killoran, "Is Quantum Advantage the Right Goal for Quantum Machine Learning?", PRX Quantum 3 3, 030101 (2022).

[16] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[17] Brian Coyle, Maxwell Henderson, Justin Chan Jin Le, Niraj Kumar, Marco Paini, and Elham Kashefi, "Quantum versus classical generative modelling in finance", Quantum Science and Technology 6 2, 024013 (2021).

[18] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, "Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information", Quantum 5, 567 (2021).

[19] Patrick Selig, Niall Murphy, Ashwin Sundareswaran R, David Redmond, and Simon Caton, 2021 International Conference on Rebooting Computing (ICRC) 24 (2021) ISBN:978-1-6654-2332-8.

[20] Adam Glos, Aleksandra Krawiec, and Zoltán Zimborás, "Space-efficient binary optimization for variational quantum computing", npj Quantum Information 8 1, 39 (2022).

[21] 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).

[22] Juliane Muller, Wim Lavrijsen, Costin Iancu, and Wibe de Jong, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 215 (2022) ISBN:978-1-6654-9113-6.

[23] Chen Zhao and Xiao-Shan Gao, "Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus", Quantum 5, 466 (2021).

[24] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles, "Effect of barren plateaus on gradient-free optimization", Quantum 5, 558 (2021).

[25] 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).

[26] Maiyuren Srikumar, Charles D Hill, and Lloyd C L Hollenberg, "Clustering and enhanced classification using a hybrid quantum autoencoder", Quantum Science and Technology 7 1, 015020 (2022).

[27] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond, "Quantum machine learning on near-term quantum devices: Current state of supervised and unsupervised techniques for real-world applications", Physical Review Applied 21 6, 067001 (2024).

[28] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", Physical Review A 103 5, 052416 (2021).

[29] Arul Rhik Mazumder, Anuvab Sen, and Udayon Sen, Lecture Notes in Networks and Systems 1018, 651 (2024) ISBN:978-3-031-62268-7.

[30] Maria Schuld and Francesco Petruccione, Encyclopedia of Machine Learning and Data Science 1 (2023) ISBN:978-1-4899-7502-7.

[31] Ijaz Ahamed Mohammad, Matej Pivoluska, and Martin Plesch, "Meta-optimization of resources on quantum computers", Scientific Reports 14 1, 10312 (2024).

[32] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, "Universal noise-precision relations in variational quantum algorithms", Physical Review Research 5 2, 023025 (2023).

[33] Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J. Coles, "Non-trivial symmetries in quantum landscapes and their resilience to quantum noise", Quantum 6, 804 (2022).

[34] Andrew Jena, Scott N. Genin, and Michele Mosca, "Optimization of variational-quantum-eigensolver measurement by partitioning Pauli operators using multiqubit Clifford gates on noisy intermediate-scale quantum hardware", Physical Review A 106 4, 042443 (2022).

[35] Tirthak Patel, Daniel Silver, and Devesh Tiwari, 2022 Design, Automation & Test in Europe Conference & Exhibition (DATE) 334 (2022) ISBN:978-3-9819263-6-1.

[36] Igor O. Sokolov, Panagiotis Kl. Barkoutsos, Lukas Moeller, Philippe Suchsland, Guglielmo Mazzola, and Ivano Tavernelli, "Microcanonical and finite-temperature ab initio molecular dynamics simulations on quantum computers", Physical Review Research 3 1, 013125 (2021).

[37] Fumiyoshi Kobayashi, Kosuke Mitarai, and Keisuke Fujii, "Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers", Physical Review A 105 5, 052415 (2022).

[38] Tyson Jones and Simon C. Benjamin, "Robust quantum compilation and circuit optimisation via energy minimisation", Quantum 6, 628 (2022).

[39] 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).

[40] Frédéric Sauvage and Florian Mintert, "Optimal Quantum Control with Poor Statistics", PRX Quantum 1 2, 020322 (2020).

[41] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, "Estimating the gradient and higher-order derivatives on quantum hardware", Physical Review A 103 1, 012405 (2021).

[42] Jonathan Foldager, Arthur Pesah, and Lars Kai Hansen, "Noise-assisted variational quantum thermalization", Scientific Reports 12 1, 3862 (2022).

[43] 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).

[44] Wim Lavrijsen, Ana Tudor, Juliane Muller, Costin Iancu, and Wibe de Jong, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 267 (2020) ISBN:978-1-7281-8969-7.

[45] Tyler Volkoff and Patrick J Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

[46] Sharu Theresa Jose and Osvaldo Simeone, "Error-Mitigation-Aided Optimization of Parameterized Quantum Circuits: Convergence Analysis", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[47] Koustubh Phalak and Swaroop Ghosh, "Shot Optimization in Quantum Machine Learning Architectures to Accelerate Training", IEEE Access 11, 41514 (2023).

[48] 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).

[49] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, "Fermionic Partial Tomography via Classical Shadows", Physical Review Letters 127 11, 110504 (2021).

[50] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley, "Efficient quantum measurement of Pauli operators in the presence of finite sampling error", Quantum 5, 385 (2021).

[51] Sukin Sim, Jonathan Romero, Jérôme F Gonthier, and Alexander A Kunitsa, "Adaptive pruning-based optimization of parameterized quantum circuits", Quantum Science and Technology 6 2, 025019 (2021).

[52] H. Chen, L. Wossnig, S. Severini, H. Neven, and M. Mohseni, "Universal discriminative quantum neural networks", Quantum Machine Intelligence 3 1, 1 (2021).

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

[54] Wenyang Qian, Robert A. M. Basili, Mary Mehrnoosh Eshaghian-Wilner, Ashfaq Khokhar, Glenn Luecke, and James P. Vary, "Comparative Study of Variations in Quantum Approximate Optimization Algorithms for the Traveling Salesman Problem", Entropy 25 8, 1238 (2023).

[55] Bálint Koczor and Simon C. Benjamin, "Quantum analytic descent", Physical Review Research 4 2, 023017 (2022).

[56] 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).

[57] Aram W. Harrow and John C. Napp, "Low-Depth Gradient Measurements Can Improve Convergence in Variational Hybrid Quantum-Classical Algorithms", Physical Review Letters 126 14, 140502 (2021).

[58] 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).

[59] Matthew T. Scoggins and Armin Rahmani, "Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: Application to the XXZ model on the square lattice", Physical Review Research 3 4, 043165 (2021).

[60] Wenyang Qian, Robert Basili, Soham Pal, Glenn Luecke, and James P. Vary, "Solving hadron structures using the basis light-front quantization approach on quantum computers", Physical Review Research 4 4, 043193 (2022).

[61] 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).

[62] Lucas Slattery, Benjamin Villalonga, and Bryan K. Clark, "Unitary block optimization for variational quantum algorithms", Physical Review Research 4 2, 023072 (2022).

[63] Ruhee D’Cunha, Matthew Otten, Matthew R. Hermes, Laura Gagliardi, and Stephen K. Gray, "State Preparation in Quantum Algorithms for Fragment-Based Quantum Chemistry", Journal of Chemical Theory and Computation 20 8, 3121 (2024).

[64] 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).

[65] Yang Qian, Yuxuan Du, and Dacheng Tao, "Shuffle-QUDIO: accelerate distributed VQE with trainability enhancement and measurement reduction", Quantum Machine Intelligence 6 1, 32 (2024).

[66] Peter Demetriou, Conrad J. Haupt, and Ken J. Nixon, 2021 IEEE Computer Society Annual Symposium on VLSI (ISVLSI) 422 (2021) ISBN:978-1-6654-3946-6.

[67] Mario Motta and Julia E. Rice, "Emerging quantum computing algorithms for quantum chemistry", WIREs Computational Molecular Science 12 3, e1580 (2022).

[68] Yifei Huang, Yuguo Shao, Weiluo Ren, Jinzhao Sun, and Dingshun Lv, "Efficient Quantum Imaginary Time Evolution by Drifting Real-Time Evolution: An Approach with Low Gate and Measurement Complexity", Journal of Chemical Theory and Computation 19 13, 3868 (2023).

[69] Cristina Cîrstoiu, Zoë 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, 82 (2020).

[70] Johannes Weidenfeller, Lucia C. Valor, Julien Gacon, Caroline Tornow, Luciano Bello, Stefan Woerner, and Daniel J. Egger, "Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware", Quantum 6, 870 (2022).

[71] Ryan Sweke, Frederik Wilde, Johannes Meyer, Maria Schuld, Paul K. Faehrmann, Barthélémy Meynard-Piganeau, and Jens Eisert, "Stochastic gradient descent for hybrid quantum-classical optimization", Quantum 4, 314 (2020).

[72] 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).

[73] Shiro Tamiya and Hayata Yamasaki, "Stochastic gradient line Bayesian optimization for efficient noise-robust optimization of parameterized quantum circuits", npj Quantum Information 8 1, 90 (2022).

[74] 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).

[75] Matt Menickelly, Yunsoo Ha, and Matthew Otten, "Latency considerations for stochastic optimizers in variational quantum algorithms", Quantum 7, 949 (2023).

[76] Tomochika Kurita, Mikio Morita, Hirotaka Oshima, and Shintaro Sato, "Pauli String Partitioning Algorithm with the Ising Model for Simultaneous Measurements", The Journal of Physical Chemistry A 127 4, 1068 (2023).

[77] Chong Hian Chee, Adrian M Mak, Daniel Leykam, Panagiotis Kl Barkoutsos, and Dimitris G Angelakis, "Computing electronic correlation energies using linear depth quantum circuits", Quantum Science and Technology 9 2, 025003 (2024).

[78] Carlos Bravo-Prieto, Diego García-Martín, and José I. Latorre, "Quantum singular value decomposer", Physical Review A 101 6, 062310 (2020).

[79] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", Journal of The Royal Society Interface 19 196, 20220541 (2022).

[80] 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).

[81] Linghua Zhu, Senwei Liang, Chao Yang, and Xiaosong Li, "Optimizing Shot Assignment in Variational Quantum Eigensolver Measurement", Journal of Chemical Theory and Computation 20 6, 2390 (2024).

[82] Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran, "Quantum embeddings for machine learning", arXiv:2001.03622, (2020).

[83] Kunal Sharma, Sumeet Khatri, M. Cerezo, and Patrick J. Coles, "Noise resilience of variational quantum compiling", New Journal of Physics 22 4, 043006 (2020).

[84] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252, (2020).

[85] Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C. Benjamin, "Variational-state quantum metrology", New Journal of Physics 22 8, 083038 (2020).

[86] Marco Cerezo, Alexander Poremba, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum Fidelity Estimation", Quantum 4, 248 (2020).

[87] 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).

[88] Barnaby van Straaten and Bálint Koczor, "Measurement Cost of Metric-Aware Variational Quantum Algorithms", PRX Quantum 2 3, 030324 (2021).

[89] Dan-Bo Zhang and Tao Yin, "Collective optimization for variational quantum eigensolvers", Physical Review A 101 3, 032311 (2020).

[90] Ijaz Ahamed Mohammad, Matej Pivoluska, and Martin Plesch, "Resource-efficient utilization of quantum computers", arXiv:2305.08924, (2023).

[91] David Fitzek, Toheed Ghandriz, Leo Laine, Mats Granath, and Anton Frisk Kockum, "Applying quantum approximate optimization to the heterogeneous vehicle routing problem", arXiv:2110.06799, (2021).

[92] Frederic Sauvage and Florian Mintert, "Optimal quantum control with poor statistics", arXiv:1909.01229, (2019).

[93] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).

[94] Eric Sillekens, Wenting Yi, Daniel Semrau, Alessandro Ottino, Boris Karanov, Sujie Zhou, Kevin Law, Jack Chen, Domanic Lavery, Lidia Galdino, Polina Bayvel, and Robert I. Killey, "Experimental Demonstration of Learned Time-Domain Digital Back-Propagation", arXiv:1912.12197, (2019).

[95] Jeffrey Larson, Matt Menickelly, and Jiahao Shi, "A Novel Noise-Aware Classical Optimizer for Variational Quantum Algorithms", arXiv:2401.10121, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-21 18:43:05) and SAO/NASA ADS (last updated successfully 2024-06-21 18:43:06). The list may be incomplete as not all publishers provide suitable and complete citation data.