Quantum Control Modelling, Methods, and Applications

Nahid Binandeh Dehaghani; Fernando Lobo Pereira; Antonio Pedro Aguiar

doi:10.21467/exr.2.1.5037

Authors

Nahid Binandeh Dehaghani SYSTEC - Research Center for Systems and Technologies, FEUP - Faculty of Engineering, Porto University, Rua Dr. Roberto Frias sn, i219, 4200-465 Porto, Portugal https://orcid.org/0000-0002-0370-7744
Fernando Lobo Pereira SYSTEC - Research Center for Systems and Technologies, FEUP - Faculty of Engineering, Porto University, Rua Dr. Roberto Frias sn, i219, 4200-465 Porto, Portugal https://orcid.org/0000-0002-9602-2452
Antonio Pedro Aguiar SYSTEC - Research Center for Systems and Technologies, FEUP - Faculty of Engineering, Porto University, Portugal

DOI:

https://doi.org/10.21467/exr.2.1.5037

Abstract

This review concerns quantum control results and methods that, over the years, have been used in the various operations involving quantum systems. Most of these methods have been originally developed outside the context of quantum physics, and, then, adapted to take into account the specificities of the various quantum physical platforms. Quantum control consists in designing adequate control signals required to efficiently manipulate systems conforming the laws of quantum mechanics in order to ensure the associated desired behaviours and performances. This work attempts to provide a thorough and self-contained introduction and review of the various quantum control theories and their applications. It encompasses issues spanning quantum control modelling, problem formulation, concepts of controllability, as well as a selection of the main control theories. Given the vastness of the field, we tried our best to be as concise as possible, and, for the details, the reader is pointed out to a profusion of references. The contents of the review are organized in the three major classes of control problems - open-loop control, closed-loop learning control, and feedback control - and, for each one of them, we present the main developments in quantum control theory. Finally, concerning the importance of attaining robustness and reliability due to inherent fragility of quantum systems, methods for quantum robust control are also surveyed.

Keywords:

Quantum Systems, Quantum Control Modelling, Quantum Control Methods

Downloads

Download data is not yet available.

References

J. Madronero et al., ‘Quantum chaos, transport, and control—in quantum optics’, Advances In Atomic, Molecular, and Optical Physics, vol. 53, pp. 33–73, 2006.

J. Miguel-Sánchez et al., ‘Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry--Perot cavity’, New Journal of Physics, vol. 15, no. 4, p. 045002, 2013.

E. Bauch et al., ‘Ultralong dephasing times in solid-state spin ensembles via quantum control’, Physical Review X, vol. 8, no. 3, p. 031025, 2018.

J. Alonso, F. M. Leupold, B. C. Keitch, and J. P. Home, ‘Quantum control of the motional states of trapped ions through fast switching of trapping potentials’, New Journal of Physics, vol. 15, no. 2, p. 023001, 2013.

S. Amri, R. Corgier, D. Sugny, E. M. Rasel, N. Gaaloul, and E. Charron, ‘Optimal control of the transport of Bose-Einstein condensates with atom chips’, Scientific reports, vol. 9, no. 1, pp. 1–11, 2019.

G.-Q. Liu, X. Feng, N. Wang, Q. Li, and R.-B. Liu, ‘Coherent quantum control of nitrogen-vacancy center spins near 1000 kelvin’, Nature communications, vol. 10, no. 1, pp. 1–8, 2019.

S. Gamble, ‘Quantum computing: What it is, why we want it, and how we’re trying to get it’, in Frontiers of Engineering: Reports on Leading-Edge Engineering from the 2018 Symposium, 2019.

L. Zyga, ‘Physicists find quantum coherence and quantum entanglement are two sides of the same coin’, Phys. org, 2015.

V. Ramakrishna and H. Rabitz, ‘Relation between quantum computing and quantum controllability’, Physical Review A, vol. 54, no. 2, p. 1715, 1996.

F. Albertini and D. D’Alessandro, ‘Notions of controllability for quantum mechanical systems’, in Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No. 01CH37228), 2001, vol. 2, pp. 1589–1594.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, ‘Whither the future of controlling quantum phenomena?’, Science, vol. 288, no. 5467, pp. 824–828, 2000.

W. S. Warren, H. Rabitz, and M. Dahleh, ‘Coherent control of quantum dynamics: the dream is alive’, Science, vol. 259, no. 5101, pp. 1581–1589, 1993.

R. J. Gordon, L. Zhu, and T. Seideman, ‘Coherent control of chemical reactions’, Accounts of Chemical Research, vol. 32, no. 12, pp. 1007–1016, 1999.

N. B. Dehaghani and F. L. Pereira, ‘Optimal Control of Quantum Systems by Pontryagin Maximum Principle’, U. Porto Journal of Engineering, vol. 8, no. 2, pp. 194–201, 2022.

N. B. Dehaghani and F. L. Pereira, ‘High Fidelity Quantum State Transfer by Pontryagin Maximum Principle’, arXiv preprint arXiv:2203. 04361, 2022.

N. B. Dehaghani and F. L. Pereira, ‘A Quantum Optimal Control Problem with State Constrained Preserving Coherence’, arXiv preprint arXiv:2203. 15727, 2022.

K. W. M. Tibbetts et al., ‘Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations’, Physical Review A, vol. 86, no. 6, p. 062309, 2012.

J. P. Palao and R. Kosloff, ‘Optimal control theory for unitary transformations’, Physical Review A, vol. 68, no. 6, p. 062308, 2003.

S. Cong, Control of quantum systems: theory and methods. John Wiley & Sons, 2014.

T. Untidt et al., ‘Design of NMR pulse experiments with optimum sensitivity: coherence-order-selective transfer in I2S and I3S spin systems’, Molecular Physics, vol. 95, no. 5, pp. 787–796, 1998.

J.-W. Wu, C.-W. Li, T.-J. Tarn, and J. Zhang, ‘Optimal bang-bang control for SU (1, 1) coherent states’, Physical Review A, vol. 76, no. 5, p. 053403, 2007.

U. Boscain and B. Piccoli, Optimal syntheses for control systems on 2-D manifolds, vol. 43. Springer Science & Business Media, 2003.

S. Mancini and H. M. Wiseman, ‘Optimal control of entanglement via quantum feedback’, Physical Review A, vol. 75, no. 1, p. 012330, 2007.

J. M. Geremia and H. Rabitz, ‘Optimal identification of Hamiltonian information by closed-loop laser control of quantum systems’, Physical review letters, vol. 89, no. 26, p. 263902, 2002.

B.-S. Chen, W.-H. Chen, F. Hsu, and W. Zhang, ‘Optimal tracking control design of quantum systems via tensor formal power series method’, The Open Automation and Control Systems Journal, vol. 1, no. 1, 2008.

F. Ticozzi and L. Viola, ‘Analysis and synthesis of attractive quantum Markovian dynamics’, Automatica, vol. 45, no. 9, pp. 2002–2009, 2009.

D. Dong, C. Zhang, H. Rabitz, A. Pechen, and T.-J. Tarn, ‘Incoherent control of locally controllable quantum systems’, The Journal of Chemical Physics, vol. 129, no. 15, p. 154103, 2008.

L. Roa, A. Delgado, M. L. L. de Guevara, and A. B. Klimov, ‘Measurement-driven quantum evolution’, Physical Review A, vol. 73, no. 1, p. 012322, 2006.

K. Zhdanov, ‘An improvement of D-MORPH method for finding quantum optimal control’, arXiv preprint arXiv:1709. 05364, 2017.

M. Tsubouchi and T. Momose, ‘Rovibrational wave-packet manipulation using shaped midinfrared femtosecond pulses toward quantum computation: Optimization of pulse shape by a genetic algorithm’, Physical Review A, vol. 77, no. 5, p. 052326, 2008.

R. Storn and K. Price, ‘Differential evolution--a simple and efficient heuristic for global optimization over continuous spaces’, Journal of global optimization, vol. 11, no. 4, pp. 341–359, 1997.

S. Yang, M. Wang, and Others, ‘A quantum particle swarm optimization’, in Proceedings of the 2004 congress on evolutionary computation (IEEE Cat. No. 04TH8753), 2004, vol. 1, pp. 320–324.

R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction. MIT press, 2018.

D. Dong, C. Chen, H. Li, and T.-J. Tarn, ‘Quantum reinforcement learning’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 38, no. 5, pp. 1207–1220, 2008.

R. S. Judson and H. Rabitz, ‘Teaching lasers to control molecules’, Physical review letters, vol. 68, no. 10, p. 1500, 1992.

C. Brif, R. Chakrabarti, and H. Rabitz, ‘Control of quantum phenomena: past, present and future’, New Journal of Physics, vol. 12, no. 7, p. 075008, 2010.

Y. Sun, C. Wu, Z. Zhu, and C. Chen, ‘Comparison of learning methods for landscape control of open quantum systems’, in Proceeding of the 11th World Congress on Intelligent Control and Automation, 2014, pp. 1241–1246.

X. Wang and S. G. Schirmer, ‘Analysis of Lyapunov method for control of quantum states’, IEEE Transactions on Automatic control, vol. 55, no. 10, pp. 2259–2270, 2010.

S. Cong and F. Meng, ‘A survey of quantum lyapunov control methods’, The Scientific World Journal, vol. 2013, 2013.

W. Cui, Y. Pan, and D. Dong, ‘Variable structure control of quantum systems’, in 2015 54th IEEE Conference on Decision and Control (CDC), 2015, pp. 6441–6446.

D. Dong and I. R. Petersen, ‘Variable structure control of uncontrollable quantum systems’, IFAC Proceedings Volumes, vol. 42, no. 6, pp. 237–242, 2009.

D. Dong and I. R. Petersen, ‘Controllability of quantum systems with switching control’, International Journal of Control, vol. 84, no. 1, pp. 37–46, 2011.

L. M. Johansen, ‘Quantum theory of successive projective measurements’, Physical Review A, vol. 76, no. 1, p. 012119, 2007.

A. N. Korotkov and D. V. Averin, ‘Continuous weak measurement of quantum coherent oscillations’, Physical Review B, vol. 64, no. 16, p. 165310, 2001.

H. M. Wiseman, ‘Quantum theory of continuous feedback’, Physical Review A, vol. 49, no. 3, p. 2133, 1994.

R. Ruskov and A. N. Korotkov, ‘Quantum feedback control of a solid-state qubit’, Physical Review B, vol. 66, no. 4, p. 041401, 2002.

C. Ahn, A. C. Doherty, and A. J. Landahl, ‘Continuous quantum error correction via quantum feedback control’, Physical Review A, vol. 65, no. 4, p. 042301, 2002.

J. Combes and K. Jacobs, ‘Rapid state reduction of quantum systems using feedback control’, Physical review letters, vol. 96, no. 1, p. 010504, 2006.

C. Hill and J. Ralph, ‘Weak measurement and control of entanglement generation’, Physical Review A, vol. 77, no. 1, p. 014305, 2008.

A. M. Brańczyk, P. E. Mendonça, A. Gilchrist, A. C. Doherty, and S. D. Bartlett, ‘Quantum control of a single qubit’, Physical Review A, vol. 75, no. 1, p. 012329, 2007.

M. R. James, H. I. Nurdin, and I. R. Petersen, ‘$H^infty $ Control of Linear Quantum Stochastic Systems’, IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1787–1803, 2008.

H. I. Nurdin, M. R. James, and I. R. Petersen, ‘Coherent quantum LQG control’, Automatica, vol. 45, no. 8, pp. 1837–1846, 2009.

A. J. Shaiju, I. R. Petersen, and M. R. James, ‘Guaranteed cost LQG control of uncertain linear stochastic quantum systems’, in 2007 American Control Conference, 2007, pp. 2118–2123.

D. Dong and I. R. Petersen, ‘Sliding mode control of quantum systems’, New Journal of Physics, vol. 11, no. 10, p. 105033, 2009.

F. Chen, R. Hou, B. Jiang, and G. Tao, ‘Study on fast terminal sliding mode control for a helicopter via quantum information technique and nonlinear fault observer’, International Journal of Innovative Computing, Information and Control, vol. 9, no. 8, pp. 3437–3447, 2013.

D. Dong and I. R. Petersen, ‘Sliding mode control of two-level quantum systems’, Automatica, vol. 48, no. 5, pp. 725–735, 2012.

D. Dong, I. R. Petersen, and H. Rabitz, ‘Sampled-data design for robust control of a single qubit’, IEEE Transactions on Automatic Control, vol. 58, no. 10, pp. 2654–2659, 2013.

N. Yamamoto and L. Bouten, ‘Quantum risk-sensitive estimation and robustness’, IEEE Transactions on Automatic Control, vol. 54, no. 1, pp. 92–107, 2009.

J.-S. Li and N. Khaneja, ‘Ensemble control of Bloch equations’, IEEE Transactions on Automatic Control, vol. 54, no. 3, pp. 528–536, 2009.

R. L. Kosut, M. D. Grace, and C. Brif, ‘Robust control of quantum gates via sequential convex programming’, Physical Review A, vol. 88, no. 5, p. 052326, 2013.

W. Yang, Z.-Y. Wang, and R.-B. Liu, ‘Preserving qubit coherence by dynamical decoupling’, Frontiers of Physics in China, vol. 6, no. 1, pp. 2–14, 2011.

J. R. West, D. A. Lidar, B. H. Fong, and M. F. Gyure, ‘High fidelity quantum gates via dynamical decoupling’, Physical review letters, vol. 105, no. 23, p. 230503, 2010.

P. A. M. Dirac, The principles of quantum mechanics. Oxford university press, 1981.

M. A. Nielsen and I. Chuang, ‘Quantum computation and quantum information’. American Association of Physics Teachers, 2002.

H.-P. Breuer, F. Petruccione, and Others, The theory of open quantum systems. Oxford University Press on Demand, 2002.

R. Alicki and K. Lendi, Quantum dynamical semigroups and applications, vol. 717. Springer, 2007.

R. Van Handel, J. K. Stockton, and H. Mabuchi, ‘Feedback control of quantum state reduction’, IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 768–780, 2005.

J. Zhang, R.-B. Wu, Y.-X. Liu, C.-W. Li, and T.-J. Tarn, ‘Quantum coherent nonlinear feedback with applications to quantum optics on chip’, IEEE transactions on automatic control, vol. 57, no. 8, pp. 1997–2008, 2012.

S. Attal, ‘Quantum noises’, in Open Quantum Systems II, Springer, 2006, pp. 79–147.

J. W. Clark, D. G. Lucarelli, and T.-J. Tarn, ‘Control of quantum systems’, International Journal of Modern Physics B, vol. 17, no. 28, pp. 5397–5411, 2003.

S. G. Schirmer, H. Fu, and A. I. Solomon, ‘Complete controllability of quantum systems’, Physical Review A, vol. 63, no. 6, p. 063410, 2001.

F. Albertini and D. D’Alessandro, ‘Notions of controllability for bilinear multilevel quantum systems’, IEEE Transactions on Automatic Control, vol. 48, no. 8, pp. 1399–1403, 2003.

L. Pinna, ‘On the controllability of the quantum dynamics of closed and open systems’, Université Paris-Saclay (ComUE), 2018.

C. Altafini, ‘Controllability of quantum mechanical systems by root space decomposition of su (N)’, Journal of Mathematical Physics, vol. 43, no. 5, pp. 2051–2062, 2002.

G. Turinici and H. Rabitz, ‘Wavefunction controllability for finite-dimensional bilinear quantum systems’, Journal of Physics A: Mathematical and General, vol. 36, no. 10, p. 2565, 2003.

R. Diestel, ‘Graph theory 3rd ed’, Graduate texts in mathematics, vol. 173, 2005.

S. G. Schirmer, A. I. Solomon, and J. V. Leahy, ‘Degrees of controllability for quantum systems and application to atomic systems’, Journal of Physics A: Mathematical and General, vol. 35, no. 18, p. 4125, 2002.

D. Dong and I. R. Petersen, ‘Quantum control theory and applications: a survey’, IET Control Theory & Applications, vol. 4, no. 12, pp. 2651–2671, 2010.

D. d’Alessandro, Introduction to quantum control and dynamics. CRC press, 2007.

J. Werschnik and E. K. U. Gross, ‘Quantum optimal control theory’, Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 40, no. 18, p. R175, 2007.

M. R. James, ‘Optimal Quantum Control Theory’, Annual Review of Control, Robotics, and Autonomous Systems, vol. 4, pp. 343–367, 2021.

H. O. Fattorini, Infinite dimensional optimization and control theory, vol. 54. Cambridge University Press, 1999.

W. Zhu and H. Rabitz, ‘A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator’, The Journal of Chemical Physics, vol. 109, no. 2, pp. 385–391, 1998.

Y. Maday and G. Turinici, ‘New formulations of monotonically convergent quantum control algorithms’, The Journal of Chemical Physics, vol. 118, no. 18, pp. 8191–8196, 2003.

O. V. Morzhin and A. N. Pechen, ‘Krotov method for optimal control of closed quantum systems’, Russian Mathematical Surveys, vol. 74, no. 5, p. 851, 2019.

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, ‘Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms’, Journal of magnetic resonance, vol. 172, no. 2, pp. 296–305, 2005.

F. K. Wilhelm, S. Kirchhoff, S. Machnes, N. Wittler, and D. Sugny, ‘An introduction into optimal control for quantum technologies’, arXiv preprint arXiv:2003. 10132, 2020.

J. Sørensen, M. O. Aranburu, T. Heinzel, and J. F. Sherson, ‘Quantum optimal control in a chopped basis: Applications in control of Bose-Einstein condensates’, Physical Review A, vol. 98, no. 2, p. 022119, 2018.

C. Altafini and F. Ticozzi, ‘Modeling and control of quantum systems: an introduction’, IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 1898–1917, 2012.

A. Sinatra, F. Castelli, L. A. Lugiato, P. Grangier, and J. P. Poizat, ‘Effective two-level model versus three-level model’, Quantum and Semiclassical Optics: Journal of the European Optical Society Part B, vol. 7, no. 3, p. 405, 1995.

I. Kuprov and C. T. Rodgers, ‘Derivatives of spin dynamics simulations’, The Journal of chemical physics, vol. 131, no. 23, p. 234108, 2009.

R. Fisher, ‘Optimal control of multi-level quantum systems’, Technische Universität München, 2010.

H. Yuan and N. Khaneja, ‘Time optimal control of coupled qubits under nonstationary interactions’, Physical Review A, vol. 72, no. 4, p. 040301, 2005.

D. D’alessandro and M. Dahleh, ‘Optimal control of two-level quantum systems’, IEEE Transactions on Automatic Control, vol. 46, no. 6, pp. 866–876, 2001.

Y. Lou and S. Cong, ‘State transfer control of quantum systems on the Bloch sphere’, Journal of Systems Science and Complexity, vol. 24, no. 3, p. 506, 2011.

U. Boscain and P. Mason, ‘Time minimal trajectories for two-level quantum systems with drift’, in Proceedings of the 44th IEEE conference on decision and control, 2005, pp. 3188–3193.

U. Boscain and P. Mason, ‘Time minimal trajectories for a spin 1/ 2 particle in a magnetic field’, Journal of Mathematical Physics, vol. 47, no. 6, p. 062101, 2006.

N. Khaneja, R. Brockett, and S. J. Glaser, ‘Time optimal control in spin systems’, Physical Review A, vol. 63, no. 3, p. 032308, 2001.

D. Dong, J. Lam, and T. J. Tarn, ‘Rapid incoherent control of quantum systems based on continuous measurements and reference model’, IET control theory & applications, vol. 3, no. 2, pp. 161–169, 2009.

C. Lin, Y. Ma, and D. Sels, ‘Optimal control for quantum metrology via Pontryagin’s principle’, Physical Review A, vol. 103, no. 5, p. 052607, 2021.

D. D’Alessandro, ‘Optimal evaluation of generalized Euler angles with applications to control’, Automatica, vol. 40, no. 11, pp. 1997–2002, 2004.

C. Altafini, ‘Coherent control of open quantum dynamical systems’, Physical Review A, vol. 70, no. 6, p. 062321, 2004.

I. Oliveira, R. Sarthour Jr, T. Bonagamba, E. Azevedo, and J. C. C. Freitas, NMR quantum information processing. Elsevier, 2011.

R. Romano and D. D’Alessandro, ‘Incoherent control and entanglement for two-dimensional coupled systems’, Physical Review A, vol. 73, no. 2, p. 022323, 2006.

R. Romano and D. D’Alessandro, ‘Environment-mediated control of a quantum system’, Physical review letters, vol. 97, no. 8, p. 080402, 2006.

D. Burgarth and V. Giovannetti, ‘Full control by locally induced relaxation’, Physical review letters, vol. 99, no. 10, p. 100501, 2007.

R. V. Mendes and V. I. Man’ko, ‘Quantum control and the Strocchi map’, Physical Review A, vol. 67, no. 5, p. 053404, 2003.

S. Ashhab and F. Nori, ‘Control-free control: Manipulating a quantum system using only a limited set of measurements’, Physical Review A, vol. 82, no. 6, p. 062103, 2010.

K. Jacobs, ‘Feedback control using only quantum back-action’, New Journal of Physics, vol. 12, no. 4, p. 043005, 2010.

S. Tanaka and N. Yamamoto, ‘Robust adaptive measurement scheme for qubit-state preparation’, Physical Review A, vol. 86, no. 6, p. 062331, 2012.

K. Jacobs, ‘How to project qubits faster using quantum feedback’, Physical Review A, vol. 67, no. 3, p. 030301, 2003.

R. Raussendorf and H. J. Briegel, ‘A one-way quantum computer’, Physical Review Letters, vol. 86, no. 22, p. 5188, 2001.

D. K. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. Pascazio, and K. Yuasa, ‘Exponential rise of dynamical complexity in quantum computing through projections’, Nature communications, vol. 5, no. 1, pp. 1–6, 2014.

S. Hacohen-Gourgy, L. P. García-Pintos, L. S. Martin, J. Dressel, and I. Siddiqi, ‘Incoherent qubit control using the quantum Zeno effect’, Physical review letters, vol. 120, no. 2, p. 020505, 2018.

C. Chen, L.-C. Wang, and Y. Wang, ‘Closed-loop and robust control of quantum systems’, The scientific world journal, vol. 2013, 2013.

D. Dong, ‘Learning control of quantum systems’, arXiv preprint arXiv:2101. 07461, 2021.

R. Chakrabarti and H. Rabitz, ‘Quantum control landscapes’, International Reviews in Physical Chemistry, vol. 26, no. 4, pp. 671–735, 2007.

R.-B. Wu, B. Chu, D. H. Owens, and H. Rabitz, ‘Data-driven gradient algorithm for high-precision quantum control’, Physical Review A, vol. 97, no. 4, p. 042122, 2018.

C.-C. Shu, T.-S. Ho, X. Xing, and H. Rabitz, ‘Frequency domain quantum optimal control under multiple constraints’, Physical Review A, vol. 93, no. 3, p. 033417, 2016.

A. Rothman, T.-S. Ho, and H. Rabitz, ‘Observable-preserving control of quantum dynamics over a family of related systems’, Physical Review A, vol. 72, no. 2, p. 023416, 2005.

M. D. Jegen, M. E. Everett, and A. Schultz, ‘Using homotopy to invert geophysical data’, Geophysics, vol. 66, no. 6, pp. 1749–1760, 2001.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, ‘Optimization by simulated annealing’, science, vol. 220, no. 4598, pp. 671–680, 1983.

M. Dorigo, M. Birattari, and T. Stutzle, ‘Ant colony optimization’, IEEE computational intelligence magazine, vol. 1, no. 4, pp. 28–39, 2006.

P. Moscato and Others, ‘On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms’, Caltech concurrent computation program, C3P Report, vol. 826, p. 1989, 1989.

H. Ma, C. Chen, and D. Dong, ‘Differential evolution with equally-mixed strategies for robust control of open quantum systems’, in 2015 IEEE international conference on systems, man, and cybernetics, 2015, pp. 2055–2060.

M. Pant, H. Zaheer, L. Garcia-Hernandez, A. Abraham, and Others, ‘Differential Evolution: A review of more than two decades of research’, Engineering Applications of Artificial Intelligence, vol. 90, p. 103479, 2020.

J. Kennedy and R. Eberhart, ‘Particle swarm optimization’, in Proceedings of ICNN’95-international conference on neural networks, 1995, vol. 4, pp. 1942–1948.

D. Zeidler, S. Frey, K.-L. Kompa, and M. Motzkus, ‘Evolutionary algorithms and their application to optimal control studies’, Physical Review A, vol. 64, no. 2, p. 023420, 2001.

F. Glover and G. A. Kochenberger, Handbook of metaheuristics. Springer Science & Business Media, 2003.

E. Elbeltagi, T. Hegazy, and D. Grierson, ‘Comparison among five evolutionary-based optimization algorithms’, Advanced engineering informatics, vol. 19, no. 1, pp. 43–53, 2005.

E. Zahedinejad, S. Schirmer, and B. C. Sanders, ‘Evolutionary algorithms for hard quantum control’, Physical Review A, vol. 90, no. 3, p. 032310, 2014.

V. Kachitvichyanukul, ‘Comparison of three evolutionary algorithms: GA, PSO, and DE’, Industrial Engineering and Management Systems, vol. 11, no. 3, pp. 215–223, 2012.

S. Yang, M. Wang, and L. Jiao, ‘A quantum particle swarm optimization’, in Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No. 04TH8753), 2004, vol. 1, pp. 320–324.

J. Sun, W. Xu, and W. Fang, ‘Quantum-behaved particle swarm optimization algorithm with controlled diversity’, in International Conference on Computational Science, 2006, pp. 847–854.

K. V. Price, ‘Differential evolution’, in Handbook of optimization, Springer, 2013, pp. 187–214.

S. Das and P. N. Suganthan, ‘Differential evolution: A survey of the state-of-the-art’, IEEE transactions on evolutionary computation, vol. 15, no. 1, pp. 4–31, 2010.

S. Das, S. S. Mullick, and P. N. Suganthan, ‘Recent advances in differential evolution - an updated survey’, Swarm and Evolutionary Computation, vol. 27, pp. 1–30, 2016.

X. Yang, J. Li, and X. Peng, ‘An improved differential evolution algorithm for learning high-fidelity quantum controls’, Science Bulletin, vol. 64, no. 19, pp. 1402–1408, 2019.

E. Zahedinejad, J. Ghosh, and B. C. Sanders, ‘High-fidelity single-shot Toffoli gate via quantum control’, Physical review letters, vol. 114, no. 20, p. 200502, 2015.

E. Zahedinejad, J. Ghosh, and B. C. Sanders, ‘Designing high-fidelity single-shot three-qubit gates: a machine-learning approach’, Physical Review Applied, vol. 6, no. 5, p. 054005, 2016.

T. Kondo and K. Ito, ‘A reinforcement learning with evolutionary state recruitment strategy for autonomous mobile robots control’, Robotics and Autonomous Systems, vol. 46, no. 2, pp. 111–124, 2004.

C. Chen, D. Dong, and Z. Chen, ‘Grey reinforcement learning for incomplete information processing’, in International Conference on Theory and Applications of Models of Computation, 2006, pp. 399–407.

P. He and S. Jagannathan, ‘Reinforcement learning-based output feedback control of nonlinear systems with input constraints’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 35, no. 1, pp. 150–154, 2005.

Y. Shahar, ‘A framework for knowledge-based temporal abstraction’, Artificial intelligence, vol. 90, no. 1–2, pp. 79–133, 1997.

B. Atal, ‘Efficient coding of LPC parameters by temporal decomposition’, in ICASSP’83. IEEE International Conference on Acoustics, Speech, and Signal Processing, 1983, vol. 8, pp. 81–84.

J. A. Flanagan, ‘Self-organisation in Kohonen’s SOM’, Neural networks, vol. 9, no. 7, pp. 1185–1197, 1996.

C. Szepesvári, ‘Algorithms for reinforcement learning’, Synthesis lectures on artificial intelligence and machine learning, vol. 4, no. 1, pp. 1–103, 2010.

A. J. Smith, ‘Applications of the self-organising map to reinforcement learning’, Neural networks, vol. 15, no. 8–9, pp. 1107–1124, 2002.

C. J. Watkins and P. Dayan, ‘Q-learning’, Machine learning, vol. 8, no. 3–4, pp. 279–292, 1992.

P. Y. Glorennec and L. Jouffe, ‘Fuzzy Q-learning’, in Proceedings of 6th international fuzzy systems conference, 1997, vol. 2, pp. 659–662.

S. G. Tzafestas and G. G. Rigatos, ‘Fuzzy reinforcement learning control for compliance tasks of robotic manipulators’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 32, no. 1, pp. 107–113, 2002.

M. J. Er and C. Deng, ‘Online tuning of fuzzy inference systems using dynamic fuzzy Q-learning’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 34, no. 3, pp. 1478–1489, 2004.

D. Vengerov, N. Bambos, and H. R. Berenji, ‘A fuzzy reinforcement learning approach to power control in wireless transmitters’, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 35, no. 4, pp. 768–778, 2005.

C. Chen, H.-X. Li, and D. Dong, ‘Hybrid control for robot navigation-a hierarchical q-learning algorithm’, IEEE Robotics & Automation Magazine, vol. 15, no. 2, pp. 37–47, 2008.

S. Whiteson, ‘Evolutionary function approximation for reinforcement learning’, Journal of Machine Learning Research, vol. 7, 2006.

M. A. Nielsen and I. L. Chuang, ‘Quantum computation and quantum information’, Phys. Today, vol. 54, no. 2, p. 60, 2001.

A. Ekert and R. Jozsa, ‘Quantum computation and Shor’s factoring algorithm’, Reviews of Modern Physics, vol. 68, no. 3, p. 733, 1996.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, ‘Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance’, Nature, vol. 414, no. 6866, pp. 883–887, 2001.

L. K. Grover, ‘Quantum mechanics helps in searching for a needle in a haystack’, Physical review letters, vol. 79, no. 2, p. 325, 1997.

M. O. Scully and M. S. Zubairy, ‘Quantum optical implementation of Grover’s algorithm’, Proceedings of the National Academy of Sciences, vol. 98, no. 17, pp. 9490–9493, 2001.

J. A. Jones, ‘Fast searches with nuclear magnetic resonance computers’, Science, vol. 280, no. 5361, pp. 229–229, 1998.

J. Zhao and R.-H. Xie, ‘Genetic algorithms for the geometry optimization of atomic and molecular clusters’, Journal of Computational and Theoretical Nanoscience, vol. 1, no. 2, pp. 117–131, 2004.

G. G. Rigatos and S. G. Tzafestas, ‘Parallelization of a fuzzy control algorithm using quantum computation’, IEEE Transactions on Fuzzy Systems, vol. 10, no. 4, pp. 451–460, 2002.

T. Hogg and D. Portnov, ‘Quantum optimization’, Information Sciences, vol. 128, no. 3–4, pp. 181–197, 2000.

S. Naguleswaran and L. B. White, ‘Quantum search in stochastic planning’, in Noise and Information in Nanoelectronics, Sensors, and Standards III, 2005, vol. 5846, pp. 34–45.

D. Dong, C. Chen, J. Chu, and T.-J. Tarn, ‘Robust quantum-inspired reinforcement learning for robot navigation’, IEEE/ASME transactions on mechatronics, vol. 17, no. 1, pp. 86–97, 2010.

M. Bukov, A. G. R. Day, D. Sels, P. Weinberg, A. Polkovnikov, and P. Mehta, ‘Reinforcement learning in different phases of quantum control’, Physical Review X, vol. 8, no. 3, p. 031086, 2018.

M. Y. Niu, S. Boixo, V. N. Smelyanskiy, and H. Neven, ‘Universal quantum control through deep reinforcement learning’, npj Quantum Information, vol. 5, no. 1, pp. 1–8, 2019.

T. Fösel, P. Tighineanu, T. Weiss, and F. Marquardt, ‘Reinforcement learning with neural networks for quantum feedback’, Physical Review X, vol. 8, no. 3, p. 031084, 2018.

C.-L. Chen and D.-Y. Dong, ‘Superposition-inspired reinforcement learning and quantum reinforcement learning’, in Reinforcement Learning, IntechOpen, 2008.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, ‘Grover’s search algorithm: an optical approach’, Journal of Modern Optics, vol. 47, no. 2–3, pp. 257–266, 2000.

C. Chen, D. Dong, H.-X. Li, J. Chu, and T.-J. Tarn, ‘Fidelity-based probabilistic Q-learning for control of quantum systems’, IEEE transactions on neural networks and learning systems, vol. 25, no. 5, pp. 920–933, 2013.

P. Vettori, ‘On the convergence of a feedback control strategy for multilevel quantum systems’, in Proceedings of the MTNS Conference, 2002, pp. 1–6.

K. Zyczkowski and W. Slomczynski, ‘The Monge metric on the sphere and geometry of quantum states’, Journal of Physics A: Mathematical and General, vol. 34, no. 34, p. 6689, 2001.

J. Wen, S. Cong, and X. Zou, ‘Realization of quantum hadamard gate based on Lyapunov method’, in Proceedings of the 10th World Congress on Intelligent Control and Automation, 2012, pp. 5096–5101.

S. M. Nejad and M. Mehmandoost, ‘Realization of quantum Hadamard gate by applying optimal control fields to a spin qubit’, in 2010 2nd International Conference on Mechanical and Electronics Engineering, 2010, vol. 2, pp. V2-292.

M. Sugawara, ‘General formulation of locally designed coherent control theory for quantum system’, The Journal of chemical physics, vol. 118, no. 15, pp. 6784–6800, 2003.

W. Liang, N. H. Amini, and P. Mason, ‘Feedback exponential stabilization of GHZ states of multi-qubit systems’, arXiv preprint arXiv:2011. 00097, 2020.

M. Mirrahimi, G. Turinici, and P. Rouchon, ‘Reference trajectory tracking for locally designed coherent quantum controls’, The Journal of Physical Chemistry A, vol. 109, no. 11, pp. 2631–2637, 2005.

S. Grivopoulos and B. Bamieh, ‘Lyapunov-based control of quantum systems’, in 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475), 2003, vol. 1, pp. 434–438.

M. Mirrahimi, P. Rouchon, and G. Turinici, ‘Lyapunov control of bilinear Schrödinger equations’, Automatica, vol. 41, no. 11, pp. 1987–1994, 2005.

C. Altafini, ‘Feedback stabilization of isospectral control systems on complex flag manifolds: application to quantum ensembles’, IEEE Transactions on Automatic Control, vol. 52, no. 11, pp. 2019–2028, 2007.

C. Altafini, ‘Feedback control of spin systems’, Quantum Information Processing, vol. 6, no. 1, pp. 9–36, 2007.

S. H. Tersigni, P. Gaspard, and S. A. Rice, ‘On using shaped light pulses to control the selectivity of product formation in a chemical reaction: An application to a multiple level system’, The Journal of chemical physics, vol. 93, no. 3, pp. 1670–1680, 1990.

J. Cohen and M. Mirrahimi, ‘Dissipation-induced continuous quantum error correction for superconducting circuits’, Physical Review A, vol. 90, no. 6, p. 062344, 2014.

K. Maruyama, F. Nori, and V. Vedral, ‘Colloquium: The physics of Maxwell’s demon and information’, Reviews of Modern Physics, vol. 81, no. 1, p. 1, 2009.

J. Zhang, Y.-X. Liu, R.-B. Wu, K. Jacobs, and F. Nori, ‘Quantum feedback: theory, experiments, and applications’, Physics Reports, vol. 679, pp. 1–60, 2017.

K. Jacobs and D. A. Steck, ‘A straightforward introduction to continuous quantum measurement’, Contemporary Physics, vol. 47, no. 5, pp. 279–303, 2006.

M. B. Mensky, Continuous quantum measurements and path integrals. CRC Press, 1993.

H.-S. Goan, G. J. Milburn, H. M. Wiseman, and H. B. Sun, ‘Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach’, Physical Review B, vol. 63, no. 12, p. 125326, 2001.

J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, and S. M. Girvin, ‘Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect’, Physical Review A, vol. 77, no. 1, p. 012112, 2008.

J. Jing and T. Yu, ‘Non-Markovian relaxation of a three-level system: quantum trajectory approach’, Physical review letters, vol. 105, no. 24, p. 240403, 2010.

Y. Fuji and Y. Ashida, ‘Measurement-induced quantum criticality under continuous monitoring’, Physical Review B, vol. 102, no. 5, p. 054302, 2020.

L. Bouten, R. Van Handel, and M. R. James, ‘An introduction to quantum filtering’, SIAM Journal on Control and Optimization, vol. 46, no. 6, pp. 2199–2241, 2007.

V. P. Belavkin, ‘Measurement, filtering and control in quantum open dynamical systems’, Reports on Mathematical Physics, vol. 43, no. 3, pp. A405–A425, 1999.

M. R. James, ‘Risk-sensitive optimal control of quantum systems’, Physical Review A, vol. 69, no. 3, p. 032108, 2004.

P. Rouchon and J. F. Ralph, ‘Efficient quantum filtering for quantum feedback control’, Physical Review A, vol. 91, no. 1, p. 012118, 2015.

J. Dalibard, Y. Castin, and K. Mølmer, ‘Wave-function approach to dissipative processes in quantum optics’, Physical review letters, vol. 68, no. 5, p. 580, 1992.

W. K. Wootters, ‘Quantum measurements and finite geometry’, Foundations of Physics, vol. 36, no. 1, pp. 112–126, 2006.

G. G. Gillett et al., ‘Experimental feedback control of quantum systems using weak measurements’, Physical review letters, vol. 104, no. 8, p. 080503, 2010.

J. Zhang, Y.-X. Liu, R.-B. Wu, C.-W. Li, and T.-J. Tarn, ‘Transition from weak to strong measurements by nonlinear quantum feedback control’, Physical Review A, vol. 82, no. 2, p. 022101, 2010.

G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, ‘Quantum tomography’, Advances in Imaging and Electron Physics, vol. 128, pp. 206–309, 2003.

L. M. Artiles, R. D. Gill, and M. I. Gutc ă, ‘An invitation to quantum tomography’, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 67, no. 1, pp. 109–134, 2005.

R. O’Donnell and J. Wright, ‘Efficient quantum tomography’, in Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, 2016, pp. 899–912.

D. Yang, A. Grankin, L. M. Sieberer, D. V. Vasilyev, and P. Zoller, ‘Quantum non-demolition measurement of a many-body hamiltonian’, Nature communications, vol. 11, no. 1, pp. 1–8, 2020.

T. Nakajima et al., ‘Quantum non-demolition measurement of an electron spin qubit’, Nature nanotechnology, vol. 14, no. 6, pp. 555–560, 2019.

M. J. Woolley, A. C. Doherty, and G. J. Milburn, ‘Continuous quantum nondemolition measurement of Fock states of a nanoresonator using feedback-controlled circuit QED’, Physical Review B, vol. 82, no. 9, p. 094511, 2010.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, ‘Real-time quantum feedback control of atomic spin-squeezing’, Science, vol. 304, no. 5668, pp. 270–273, 2004.

R. Inoue, R. Namiki, T. Sagawa, Y. Takahashi, and Others, ‘Unconditional quantum-noise suppression via measurement-based quantum feedback’, Physical review letters, vol. 110, no. 16, p. 163602, 2013.

L. Gregory, ‘Quantum Filtering Theory and the Filtering Interpretation’, in AIP Conference Proceedings, 2005, vol. 750, pp. 95–103.

C. Gardiner, P. Zoller, and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics. Springer Science and Business Media, 2004.

C. W. Gardiner and M. J. Collett, ‘Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation’, Physical Review A, vol. 31, no. 6, p. 3761, 1985.

J. E. Gough, M. R. James, and H. I. Nurdin, ‘Single photon quantum filtering using non-Markovian embeddings’, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 370, no. 1979, pp. 5408–5421, 2012.

Z. Dong, G. Zhang, and N. H. Amini, ‘Quantum filtering for a two-level atom driven by two counter-propagating photons’, Quantum Information Processing, vol. 18, no. 5, pp. 1–27, 2019.

V. N. Kolokoltsov, ‘The law of large numbers for quantum stochastic filtering and control of many particle systems’, arXiv preprint arXiv:2008. 07375, 2020.

K. P. Knutsen, J. C. Johnson, A. E. Miller, P. B. Petersen, and R. J. Saykally, ‘High spectral resolution multiplex CARS spectroscopy using chirped pulses’, Chemical Physics Letters, vol. 387, no. 4–6, pp. 436–441, 2004.

H. M. Wiseman and G. J. Milburn, Quantum measurement and control. Cambridge university press, 2009.

H. M. Wiseman, S. Mancini, and J. Wang, ‘Bayesian feedback versus Markovian feedback in a two-level atom’, Physical Review A, vol. 66, no. 1, p. 013807, 2002.

J. Wang and H. M. Wiseman, ‘Feedback-stabilization of an arbitrary pure state of a two-level atom’, Physical Review A, vol. 64, no. 6, p. 063810, 2001.

Y. Li, B. Luo, and H. Guo, ‘Entanglement and quantum discord dynamics of two atoms under practical feedback control’, Physical Review A, vol. 84, no. 1, p. 012316, 2011.

A. R. R. Carvalho, A. J. S. Reid, and J. J. Hope, ‘Controlling entanglement by direct quantum feedback’, Physical Review A, vol. 78, no. 1, p. 012334, 2008.

L. K. Thomsen, S. Mancini, and H. M. Wiseman, ‘Spin squeezing via quantum feedback’, Physical Review A, vol. 65, no. 6, p. 061801, 2002.

F. Ticozzi and L. Viola, ‘Quantum Markovian subsystems: invariance, attractivity, and control’, IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2048–2063, 2008.

C. Ahn, H. M. Wiseman, and G. J. Milburn, ‘Quantum error correction for continuously detected errors’, Physical Review A, vol. 67, no. 5, p. 052310, 2003.

S. S. Ge, T. L. Vu, and T. H. Lee, ‘Quantum measurement-based feedback control: a nonsmooth time delay control approach’, SIAM Journal on Control and Optimization, vol. 50, no. 2, pp. 845–863, 2012.

K. Nishio, K. Kashima, and J.-I. Imura, ‘Effects of time delay in feedback control of linear quantum systems’, Physical Review A, vol. 79, no. 6, p. 062105, 2009.

J. K. Stockton, R. Van Handel, and H. Mabuchi, ‘Deterministic Dicke-state preparation with continuous measurement and control’, Physical Review A, vol. 70, no. 2, p. 022106, 2004.

M. Sarovar, H.-S. Goan, T. P. Spiller, and G. J. Milburn, ‘High-fidelity measurement and quantum feedback control in circuit QED’, Physical Review A, vol. 72, no. 6, p. 062327, 2005.

Z. Liu et al., ‘Deterministic creation and stabilization of entanglement in circuit QED by homodyne-mediated feedback control’, Physical Review A, vol. 82, no. 3, p. 032335, 2010.

W. Feng, P. Wang, X. Ding, L. Xu, and X.-Q. Li, ‘Generating and stabilizing the Greenberger-Horne-Zeilinger state in circuit QED: Joint measurement, Zeno effect, and feedback’, Physical Review A, vol. 83, no. 4, p. 042313, 2011.

A. C. Doherty and K. Jacobs, ‘Feedback control of quantum systems using continuous state estimation’, Physical Review A, vol. 60, no. 4, p. 2700, 1999.

A. C. Doherty, S. Habib, K. Jacobs, H. Mabuchi, and S. M. Tan, ‘Quantum feedback control and classical control theory’, Physical Review A, vol. 62, no. 1, p. 012105, 2000.

D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya, and S. Habib, ‘Quantum feedback control of atomic motion in an optical cavity’, Physical review letters, vol. 92, no. 22, p. 223004, 2004.

K. Jacobs, J. Finn, and S. Vinjanampathy, ‘Real-time feedback control of a mesoscopic superposition’, Physical Review A, vol. 83, no. 4, p. 041801, 2011.

N. Galioto and A. A. Gorodetsky, ‘Bayesian Identification of Hamiltonian Dynamics from Symplectic Data’, in 2020 59th IEEE Conference on Decision and Control (CDC), 2020, pp. 1190–1195.

A. S. Holevo, Probabilistic and statistical aspects of quantum theory, vol. 1. Springer Science & Business Media, 2011.

M. F. Emzir, M. J. Woolley, and I. R. Petersen, ‘A quantum extended Kalman filter’, Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 22, p. 225301, 2017.

S. M. Ross, Introduction to stochastic dynamic programming. Academic press, 2014.

L. Bouten and R. Van Handel, ‘On the separation principle in quantum control’, in Quantum stochastics and information: statistics, filtering and control, World Scientific, 2008, pp. 206–238.

S. Lloyd, ‘Coherent quantum feedback’, Physical Review A, vol. 62, no. 2, p. 022108, 2000.

R. J. Nelson, Y. Weinstein, D. Cory, and S. Lloyd, ‘Experimental demonstration of fully coherent quantum feedback’, Physical Review Letters, vol. 85, no. 14, p. 3045, 2000.

H. M. Wiseman and G. J. Milburn, ‘All-optical versus electro-optical quantum-limited feedback’, Physical Review A, vol. 49, no. 5, p. 4110, 1994.

S. Lloyd, ‘Quantum controllers for quantum systems’, arXiv preprint quant-ph/9703042, 1997.

N. B. Dehaghani and A. H. Alizadeh, ‘Design, simulation and optimization of a nonreciprocal guided-wave optical isolator based on Mach-Zehnder interferometer’, Journal of Physics Communications, vol. 3, no. 12, p. 125006, 2019.

M. J. Collett and C. W. Gardiner, ‘Squeezing of intracavity and traveling-wave light fields produced in parametric amplification’, Physical Review A, vol. 30, no. 3, p. 1386, 1984.

J. Gough and M. R. James, ‘The series product and its application to quantum feedforward and feedback networks’, IEEE Transactions on Automatic Control, vol. 54, no. 11, pp. 2530–2544, 2009.

R. L. Hudson and K. R. Parthasarathy, ‘Quantum Ito’s formula and stochastic evolutions’, Communications in mathematical physics, vol. 93, no. 3, pp. 301–323, 1984.

J. Gough and M. R. James, ‘Quantum feedback networks: Hamiltonian formulation’, Communications in Mathematical Physics, vol. 287, no. 3, pp. 1109–1132, 2009.

M. Yanagisawa and H. Kimura, ‘Transfer function approach to quantum control-part II: Control concepts and applications’, IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2121–2132, 2003.

J. E. Gough, M. R. James, and H. I. Nurdin, ‘Squeezing components in linear quantum feedback networks’, Physical Review A, vol. 81, no. 2, p. 023804, 2010.

H. Zhang and H. Rabitz, ‘Robust optimal control of quantum molecular systems in the presence of disturbances and uncertainties’, Physical Review A, vol. 49, no. 4, p. 2241, 1994.

C. D’Helon and M. R. James, ‘Stability, gain, and robustness in quantum feedback networks’, Physical Review A, vol. 73, no. 5, p. 053803, 2006.

A. I. Maalouf and I. R. Petersen, ‘Finite Horizon H infinity Control for a Class of Linear Quantum Measurement Delayed Systems: A Dynamic Game Approach’, SIAM Journal on Control and Optimization, vol. 52, no. 3, pp. 1787–1808, 2014.

A. I. Maalouf and I. R. Petersen, ‘Coherent $H^infty $ Control for a Class of Annihilation Operator Linear Quantum Systems’, IEEE Transactions on Automatic Control, vol. 56, no. 2, pp. 309–319, 2010.

D. Dong and I. R. Petersen, ‘Notes on sliding mode control of two-level quantum systems’, Automatica, vol. 48, no. 12, pp. 3089–3097, 2012.

C. D’Helon, A. C. Doherty, M. R. James, and S. D. Wilson, ‘Quantum risk-sensitive control’, in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 3132–3137.

M. R. James, ‘A quantum Langevin formulation of risk-sensitive optimal control’, Journal of Optics B: Quantum and Semiclassical Optics, vol. 7, no. 10, p. S198, 2005.

P. Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations, vol. 902. John Wiley & Sons, 2011.

N. Yamamoto, ‘Robust observer for uncertain linear quantum systems’, Physical Review A, vol. 74, no. 3, p. 032107, 2006.

Z. Leghtas, A. Sarlette, and P. Rouchon, ‘Adiabatic passage and ensemble control of quantum systems’, Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 44, no. 15, p. 154017, 2011.

J.-S. Li and N. Khaneja, ‘Control of inhomogeneous quantum ensembles’, Physical review A, vol. 73, no. 3, p. 030302, 2006.

M. H. Levitt, ‘Composite pulses’, Progress in Nuclear Magnetic Resonance Spectroscopy, vol. 18, no. 2, pp. 61–122, 1986.

Z.-C. Shi et al., ‘Robust single-qubit gates by composite pulses in three-level systems’, Physical Review A, vol. 103, no. 5, p. 052612, 2021.

N. Augier, U. Boscain, and M. Sigalotti, ‘Adiabatic ensemble control of a continuum of quantum systems’, SIAM Journal on Control and Optimization, vol. 56, no. 6, pp. 4045–4068, 2018.

C. Chen, D. Dong, B. Qi, I. R. Petersen, and H. Rabitz, ‘Quantum ensemble classification: A sampling-based learning control approach’, IEEE transactions on neural networks and learning systems, vol. 28, no. 6, pp. 1345–1359, 2016.

Y. Sun, H. Ma, C. Wu, C. Chen, and D. Dong, ‘Ensemble control of open quantum systems using differential evolution’, in 2015 10th Asian Control Conference (ASCC), 2015, pp. 1–6.

D. Dong, X. Xing, H. Ma, C. Chen, Z. Liu, and H. Rabitz, ‘Learning-based quantum robust control: algorithm, applications, and experiments’, IEEE transactions on cybernetics, vol. 50, no. 8, pp. 3581–3593, 2019.

Y. Zhang, ‘General robust-optimization formulation for nonlinear programming’, Journal of optimization theory and applications, vol. 132, no. 1, pp. 111–124, 2007.

D. Bertsimas, O. Nohadani, and K. M. Teo, ‘Nonconvex robust optimization for problems with constraints’, INFORMS journal on computing, vol. 22, no. 1, pp. 44–58, 2010.

D. Bertsimas, O. Nohadani, and K. M. Teo, ‘Robust optimization for unconstrained simulation-based problems’, Operations research, vol. 58, no. 1, pp. 161–178, 2010.

A. Mutapcic and S. Boyd, ‘Cutting-set methods for robust convex optimization with pessimizing oracles’, Optimization Methods & Software, vol. 24, no. 3, pp. 381–406, 2009.

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, ‘Robust design of slow-light tapers in periodic waveguides’, Engineering Optimization, vol. 41, no. 4, pp. 365–384, 2009.

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, ‘Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides’, Optics express, vol. 20, no. 19, pp. 21558–21575, 2012.

J. R. West, B. H. Fong, and D. A. Lidar, ‘Near-optimal dynamical decoupling of a qubit’, Physical review letters, vol. 104, no. 13, p. 130501, 2010.

H. Y. Carr and E. M. Purcell, ‘Effects of diffusion on free precession in nuclear magnetic resonance experiments’, Physical review, vol. 94, no. 3, p. 630, 1954.

G. S. Uhrig, ‘Keeping a quantum bit alive by optimized π-pulse sequences’, Physical Review Letters, vol. 98, no. 10, p. 100504, 2007.

K. Khodjasteh and D. A. Lidar, ‘Fault-tolerant quantum dynamical decoupling’, Physical review letters, vol. 95, no. 18, p. 180501, 2005.

A. M. Souza, G. A. Álvarez, and D. Suter, ‘Robust dynamical decoupling’, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 370, no. 1976, pp. 4748–4769, 2012.

K. Khodjasteh, D. A. Lidar, and L. Viola, ‘Arbitrarily accurate dynamical control in open quantum systems’, Physical review letters, vol. 104, no. 9, p. 090501, 2010.

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, ‘Optimized dynamical decoupling in a model quantum memory’, Nature, vol. 458, no. 7241, pp. 996–1000, 2009.

B. Pokharel, N. Anand, B. Fortman, and D. A. Lidar, ‘Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits’, Physical review letters, vol. 121, no. 22, p. 220502, 2018.

P. Z. Zhao, X. Wu, and D. M. Tong, ‘Dynamical-decoupling-protected nonadiabatic holonomic quantum computation’, Physical Review A, vol. 103, no. 1, p. 012205, 2021.

G. A. Alvarez, A. Ajoy, X. Peng, and D. Suter, ‘Performance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin bath’, Physical Review A, vol. 82, no. 4, p. 042306, 2010.

G. A. Paz-Silva, S.-W. Lee, T. J. Green, and L. Viola, ‘Dynamical decoupling sequences for multi-qubit dephasing suppression and long-time quantum memory’, New Journal of Physics, vol. 18, no. 7, p. 073020, 2016.

C. A. Ryan, J. S. Hodges, and D. G. Cory, ‘Robust decoupling techniques to extend quantum coherence in diamond’, Physical Review Letters, vol. 105, no. 20, p. 200402, 2010.

T. E. Hodgson, L. Viola, and I. D’Amico, ‘Towards optimized suppression of dephasing in systems subject to pulse timing constraints’, Physical Review A, vol. 81, no. 6, p. 062321, 2010.

A. M. Souza, G. A. Alvarez, and D. Suter, ‘Robust dynamical decoupling for quantum computing and quantum memory’, Physical review letters, vol. 106, no. 24, p. 240501, 2011.

T. Gullion, D. B. Baker, and M. S. Conradi, ‘New, compensated carr-purcell sequences’, Journal of Magnetic Resonance (1969), vol. 89, no. 3, pp. 479–484, 1990.

H. K. Ng, D. A. Lidar, and J. Preskill, ‘Combining dynamical decoupling with fault-tolerant quantum computation’, Physical Review A, vol. 84, no. 1, p. 012305, 2011.

G. A. Álvarez and D. Suter, ‘Measuring the spectrum of colored noise by dynamical decoupling’, Physical review letters, vol. 107, no. 23, p. 230501, 2011.

I. Almog, Y. Sagi, G. Gordon, G. Bensky, G. Kurizki, and N. Davidson, ‘Direct measurement of the system--environment coupling as a tool for understanding decoherence and dynamical decoupling’, Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 44, no. 15, p. 154006, 2011.

J. M. Taylor et al., ‘High-sensitivity diamond magnetometer with nanoscale resolution’, Nature Physics, vol. 4, no. 10, pp. 810–816, 2008.