- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantum Control without Quantum States

Modesto Orozco-Ruiz*,†, Nguyen H. Le†, and Florian Mintert

  • Blackett Laboratory, Physics Department, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom

  • *Contact author: m.orozco-ruiz21@imperial.ac.uk
  • †These authors contributed equally to this work.
PDF

PRX Quantum 5, 040346 – Published 20 December, 2024

DOI: https://doi.org/10.1103/PRXQuantum.5.040346

Abstract

We show that combining ideas from the fields of quantum invariants and optimal control can be used to design optimal quantum control solutions without explicit reference to quantum states. The states are specified only implicitly in terms of operators to which they are eigenstates. The scaling in numerical effort of the resultant approach is not given by the typically exponentially growing effort required for the specification of a time-evolved quantum state, but it is given by the effort required for the specification of a time-evolved operator. For certain Hamiltonians, this effort can be polynomial in the system size. We describe how control problems for state preparation and the realization of propagators can be formulated in this approach, and we provide explicit control solutions for a spin chain with an extended Ising Hamiltonian. The states considered for state-preparation protocols include eigenstates of Hamiltonians with more than pairwise interactions and these Hamiltonians are also used for the definition of target propagators. The cost of describing suitable time-evolving operators grows only quadratically with the system size, allowing us to construct explicit control solutions for up to 50 spins. While subexponential scaling is obtained only in special cases, we provide several examples that demonstrate favorable scaling beyond the extended Ising model.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

The expectation that quantum computers will be able to outperform even our advanced classical computers stems from the fact that the amount of information that can be stored grows exponentially in the number of qubits. This, however, also implies an exponentially growing effort for any task that involves simulating the dynamics of a qubit register. Here, we show that this scaling can sometimes be overcome in quantum control applications.

The underlying idea is that quantum states can be determined in terms of eigenvalue relations that can be specified with polynomial effort, even if the information required to explicitly write down the state vector grows exponentially. Formulating a control problem exclusively in terms of operators for eigenvalue relations then helps us to avoid the computationally prohibitive exponential scaling.

The quadratic scaling obtained for several geometries of interacting qubits allows us to devise explicit control protocols for diabatic state preparation and quantum simulation for up to 50 qubits.

Article Text

References (83)

  1. C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe, EPJ Quantum Technol. 9, 1 (2022).
  2. T. H. Johnson, S. R. Clark, and D. Jaksch, What is a quantum simulator? EPJ Quantum Technol. 1, 1 (2014).
  3. D. Marcos, P. Rabl, E. Rico, and P. Zoller, Superconducting circuits for quantum simulation of dynamical gauge fields, Phys. Rev. Lett. 111, 110504 (2013).
  4. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature 464, 45 (2010).
  5. M. Horowitz and E. Grumbling, Quantum Computing: Progress and Prospects (National Academies Press, Washington, DC, 2019).
  6. L. Gyongyosi and S. Imre, A survey on quantum computing technology, Comput. Sci. Rev. 31, 51 (2019).
  7. I. Buluta and F. Nori, Quantum simulators, Science 326, 108 (2009).
  8. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  9. Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, et al., Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  10. H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  11. L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, et al., Quantum computational advantage with a programmable photonic processor, Nature 606, 75 (2022).
  12. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature 607, 667 (2022).
  13. 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, J. Magn. Reson. 172, 296 (2005).
  14. T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity, Phys. Rev. A 72, 042331 (2005).
  15. W. J. Chetcuti, C. Sanavio, S. Lorenzo, and T. J. G. Apollaro, Perturbative many-body transfer, New J. Phys. 22, 033030 (2020).
  16. P. Doria, T. Calarco, and S. Montangero, Optimal control technique for many-body quantum dynamics, Phys. Rev. Lett. 106, 190501 (2011).
  17. D. Burgarth, K. Maruyama, M. Murphy, S. Montangero, T. Calarco, F. Nori, and M. B. Plenio, Scalable quantum computation via local control of only two qubits, Phys. Rev. A 81, 040303 (2010).
  18. N. H. Le, M. Cykiert, and E. Ginossar, Scalable and robust quantum computing on qubit arrays with fixed coupling, npj Quantum Inf. 9, 1 (2023).
  19. A. Castro, J. Werschnik, and E. K. Gross, Controlling the dynamics of many-electron systems from first principles: A combination of optimal control and time-dependent density-functional theory, Phys. Rev. Lett. 109, 153603 (2012).
  20. F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  21. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  22. D. Gottesman, Ph.D. thesis, Stabilizer Codes and Quantum Error Correction, California Institute of Technology (Caltech), Pasadena, California, 1997.
  23. R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68, 022312 (2003).
  24. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  25. A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  26. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  27. T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  28. H. R. Lewis Jr and W. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 (1969).
  29. R. Kaushal and S. Mishra, Dynamical algebraic approach and invariants for time-dependent Hamiltonian systems in two dimensions, J. Math. Phys. 34, 5843 (1993).
  30. H. Korsch, Dynamical invariants and time-dependent harmonic systems, Phys. Lett. A 74, 294 (1979).
  31. D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  32. X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga, Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity, Phys. Rev. Lett. 104, 063002 (2010).
  33. E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Hamiltonian engineering via invariants and dynamical algebra, Phys. Rev. A 89, 043408 (2014).
  34. S. Simsek and F. Mintert, Quantum control with a multi-dimensional Gaussian quantum invariant, Quantum 5, 409 (2021).
  35. A. P. Peirce, M. A. Dahleh, and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A 37, 4950 (1988).
  36. D. d’Alessandro, Introduction to Quantum Control and Dynamics (Chapman and Hall–CRC Press, New York, New York, 2021).
  37. F. Iachello, Lie Algebras and Applications (Springer, Heidelberg, Germany, 2006), Vol. 708.
  38. S. G. Schirmer, A. I. Solomon, and J. V. Leahyv, Criteria for reachability of quantum states, J. Phys. A: Math. Gen. 35, 8551 (2002).
  39. M. Plesch and Č. Brukner, Quantum-state preparation with universal gate decompositions, Phys. Rev. A 83, 032302 (2011).
  40. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  41. I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014).
  42. A. Gilchrist, N. K. Langford, and M. A. Nielsen, Distance measures to compare real and ideal quantum processes, Phys. Rev. A 71, 062310 (2005).
  43. S. Greenaway, F. Sauvage, K. E. Khosla, and F. Mintert, Efficient assessment of process fidelity, Phys. Rev. Res. 3, 033031 (2021).
  44. D. M. Reich, G. Gualdi, and C. P. Koch, Minimum number of input states required for quantum gate characterization, Phys. Rev. A 88, 042309 (2013).
  45. M. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, and C. Koch, krotov: A python implementation of Krotov’s method for quantum optimal control, SciPost Phys. 7, 080 (2019).
  46. G. B. Mbeng, A. Russomanno, and G. E. Santoro, The quantum Ising chain for beginners, SciPost Phys. Lect. Notes, 082 (2024).
  47. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  48. X.-R. Jin, X. Ji, Y.-Q. Zhang, S. Zhang, S.-K. Hong, K.-H. Yeon, and C.-I. Um, Three-party quantum secure direct communication based on GHZ states, Phys. Lett. A 354, 67 (2006).
  49. M. Hillery, V. Bužek, and A. Berthiaume, Quantum secret sharing, Phys. Rev. A 59, 1829 (1999).
  50. F. Fröwis and W. Dür, Stable macroscopic quantum superpositions, Phys. Rev. Lett. 106, 110402 (2011).
  51. A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, et al., Generation and manipulation of Schrödinger cat states in Rydberg atom arrays, Science 365, 570 (2019).
  52. C. Song, K. Xu, H. Li, Y.-R. Zhang, X. Zhang, W. Liu, Q. Guo, Z. Wang, W. Ren, J. Hao, et al., Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits, Science 365, 574 (2019).
  53. M. A. Nielsen, Cluster-state quantum computation, Rep. Math. Phys. 57, 147 (2006).
  54. 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 algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  55. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor Network python (TeNPy), SciPost Phys. Lect. Notes 5 (2018).
  56. M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, and F. Pollmann, Time-evolving a matrix product state with long-ranged interactions, Phys. Rev. B 91, 165112 (2015).
  57. S. Deffner and S. Campbell, Quantum speed limits: From Heisenberg’s uncertainty principle to optimal quantum control, J. Phys. A: Math. Gen. 50, 453001 (2017).
  58. Data for Orozco-Ruiz et al., Quantum control without quantum states, 2024.
  59. D. Joseph, A. Callison, C. Ling, and F. Mintert, Two quantum Ising algorithms for the shortest-vector problem, Phys. Rev. A 103, 032433 (2021).
  60. T. Hogg, Adiabatic quantum computing for random satisfiability problems, Phys. Rev. A 67, 022314 (2003).
  61. N. Goldman and J. Dalibard, Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields, Phys. Rev. X 4, 031027 (2014).
  62. A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective, New J. Phys. 17, 093039 (2015).
  63. A. Stern and N. H. Lindner, Topological quantum computation—from basic concepts to first experiments, Science 339, 1179 (2013).
  64. A. Paetznick, C. Knapp, N. Delfosse, B. Bauer, J. Haah, M. B. Hastings, and M. P. da Silva, Performance of planar Floquet codes with Majorana-based qubits, PRX Quantum 4, 010310 (2023).
  65. C. Gidney, M. Newman, A. Fowler, and M. Broughton, A fault-tolerant honeycomb memory, Quantum 5, 605 (2021).
  66. B. Srivastava, A. F. Kockum, and M. Granath, The XYZ2 hexagonal stabilizer code, Quantum 6, 698 (2022).
  67. P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quantum many-body systems, Phys. Rev. Lett. 123, 090602 (2019).
  68. P. M. Schindler and M. Bukov, Counterdiabatic driving for periodically driven systems, Phys. Rev. Lett. 133, 123402 (2024).
  69. J. Yao, L. Lin, and M. Bukov, Reinforcement learning for many-body ground-state preparation inspired by counterdiabatic driving, Phys. Rev. X 11, 031070 (2021).
  70. I. Čepaitė, A. Polkovnikov, A. J. Daley, and C. W. Duncan, Counterdiabatic optimized local driving, PRX Quantum 4, 010312 (2023).
  71. A. Verdeny, Ł. Rudnicki, C. A. Müller, and F. Mintert, Optimal control of effective Hamiltonians, Phys. Rev. Lett. 113, 010501 (2014).
  72. A. Verdeny, J. Puig, and F. Mintert, Quasi-periodically driven quantum systems, Z. Naturforsch. A 71, 897 (2016).
  73. P. de Fouquieres, S. Schirmer, S. Glaser, and I. Kuprov, Second order gradient ascent pulse engineering, J. Magn. Reson. 212, 412 (2011).
  74. M. Dalgaard and F. Motzoi, Fast, high precision dynamics in quantum optimal control theory, J. Phys. B: At. Mol. Opt. Phys. 55, 085501 (2022).
  75. S. Machnes, E. Assémat, D. Tannor, and F. K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits, Phys. Rev. Lett. 120, 150401 (2018).
  76. G. Wendin, Quantum information processing with superconducting circuits: A review, Rep. Prog. Phys. 80, 106001 (2017).
  77. S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, et al., Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021).
  78. J. Carrasco, A. Elben, C. Kokail, B. Kraus, and P. Zoller, Theoretical and experimental perspectives of quantum verification, PRX Quantum 2, 010102 (2021).
  79. D. G. Cory, M. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental quantum error correction, Phys. Rev. Lett. 81, 2152 (1998).
  80. J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, et al., Realization of quantum error correction, Nature 432, 602 (2004).
  81. F. Sauvage and F. Mintert, Optimal quantum control with poor statistics, PRX Quantum 1, 020322 (2020).
  82. Y. Yi-Zhuang, Mathematica packages for physicists, https://github.com/EverettYou/Mathematica-for-physics.
  83. P. de Fouquieres, S. G. Schirmer, S. J. Glaser, and I. Kuprov, Second order gradient ascent pulse engineering, J. Magn. Reson. 212, 412 (2011).

Outline

Information

PRX Quantum 5, 040346– Published 20 December, 2024
  • Received 25 April 2024
  • Revised 17 October 2024
  • Accepted 2 December 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation