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  • Letter
  • Open Access

Chaotic roots of the modular multiplication dynamical system in Shor's algorithm

Abu Musa Patoary1, Amit Vikram1, Laura Shou2, and Victor Galitski1,3

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Phys. Rev. Research 6, L032046 – Published 28 August, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032046

Abstract

Shor's factoring algorithm, believed to provide an exponential speedup over classical computation, relies on finding the period of an exactly periodic quantum modular multiplication operator. This exact periodicity is the hallmark of an integrable system, which is paradoxical from the viewpoint of quantum chaos, given that the classical limit of the modular multiplication operator is a highly chaotic system that occupies the “maximally random” Bernoulli level of the classical ergodic hierarchy. In this work, we approach this apparent paradox from a quantum dynamical systems viewpoint, and consider whether signatures of ergodicity and chaos may indeed be encoded in such an “integrable” quantization of a chaotic system. We show that Shor's modular multiplication operator, in specific cases, can be written as a superposition of quantized A-baker's maps exhibiting more typical signatures of quantum chaos and ergodicity. This work suggests that the integrability of Shor's modular multiplication operator may stem from the interference of other “chaotic” quantizations of the same family of maps, and paves the way for deeper studies on the interplay of integrability, ergodicity, and chaos in and via quantum algorithms.

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References (43)

  1. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE, 1994), pp. 124–134.
  2. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev. 41, 303 (1999).
  3. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, United Kingdom, 2010).
  4. R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM 21, 120 (1978).
  5. F. Haake, Quantum Signatures of Chaos (Springer, Berlin, 2001).
  6. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, United Kingdom, 2002).
  7. S. W. McDonald and A. N. Kaufman, Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectories, Phys. Rev. Lett. 42, 1189 (1979).
  8. G. Casati, F. Valz-Gris, and I. Guarnieri, On the connection between quantization of nonintegrable systems and statistical theory of spectra, Lett. Nuovo Cimento 28, 279 (1980).
  9. M. V. Berry, Quantizing a classically ergodic system: Sinai's billiard and the KKR method, Ann. Phys. 131, 163 (1981).
  10. O. Bohigas, M.-J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  11. J. H. Hannay and A. M. Ozorio de Almeida, Periodic orbits and a correlation function for the semiclassical density of states, J. Phys. A: Math. Gen. 17, 3429 (1984).
  12. M. V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. London A 400, 229 (1985).
  13. N. Argaman, Y. Imry, and U. Smilansky, Semiclassical analysis of spectral correlations in mesoscopic systems, Phys. Rev. B 47, 4440 (1993).
  14. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Semiclassical foundation of universality in quantum chaos, Phys. Rev. Lett. 93, 014103 (2004).
  15. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Periodic-orbit theory of universality in quantum chaos, Phys. Rev. E 72, 046207 (2005).
  16. A. Vikram and V. Galitski, Dynamical quantum ergodicity from energy level statistics, Phys. Rev. Res. 5, 033126 (2023).
  17. J. Hannay and M. V. Berry, Quantization of linear maps on a torus-fresnel diffraction by a periodic grating, Physica D 1, 267 (1980).
  18. J. P. Keating, The cat maps: Quantum mechanics and classical motion, Nonlinearity 4, 309 (1991).
  19. P. Pakoński, K. Życzkowski, and M. Kuś, Classical 1D maps, quantum graphs and ensembles of unitary matrices, J. Phys. A: Math. Gen. 34, 9303 (2001).
  20. N. L. Balazs and A. Voros, The quantized baker's transformation, Ann. Phys. 190, 1 (1989).
  21. M. Saraceno, Classical structures in the quantized baker transformation, Ann. Phys. 199, 37 (1990).
  22. F. J. Dyson and H. Falk, Period of a discrete cat mapping, Am. Math. Mon. 99, 603 (1992).
  23. A. Lakshminarayan, Shuffling cards, factoring numbers and the quantum baker's map, J. Phys. A: Math. Gen. 38, L597 (2005).
  24. A. Lakshminarayan, Modular multiplication operator and quantized baker's maps, Phys. Rev. A 76, 042330 (2007).
  25. N. Balazs and A. Voros, The quantized baker's transformation, Europhys. Lett. 4, 1089 (1987).
  26. B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, Dynamical stochasticity in classical and quantum mechanics, Sov. Sci. Rev., Sect. C 2, 209 (1981).
  27. D. Shepelyansky, Ehrenfest time and chaos, Scholarpedia 15, 55031 (2020).
  28. N. Anantharaman and S. Nonnenmacher, Entropy of semiclassical measures of the Walsh-quantized baker's map, Ann. Henri Poincaré 8, 37 (2007).
  29. A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8, 477 (1957).
  30. M. Saraceno and A. Voros, Towards a semiclassical theory of the quantum baker's map, Physica D 79, 206 (1994).
  31. R. Rubin and N. Salwen, A canonical quantization of the baker's map, Ann. Phys. 269, 159 (1998).
  32. L. Ermann and M. Saraceno, Generalized quantum baker maps as perturbations of a simple kernel, Phys. Rev. E 74, 046205 (2006).
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L032046 for a detailed discussion of the quantization of A-baker's maps, derivation of the main result, and the semiclassical limit.
  34. S. De Bièvre and M. Degli Esposti, Egorov theorems and equidistribution of eigenfunctions for the quantized sawtooth and baker maps, Ann. Inst. H. Poincaré Phys. Théor. 69, 1 (1998).
  35. M. D. Esposti, S. Nonnenmacher, and B. Winn, Quantum variance and ergodicity for the baker's map, Commun. Math. Phys. 263, 325 (2006).
  36. M. M. Tracy and A. J. Scott, The classical limit for a class of quantum baker's maps, J. Phys. A: Math. Gen. 35, 8341 (2002).
  37. T. A. Brun and R. Schack, Realizing the quantum baker's map on a NMR quantum computer, Phys. Rev. A 59, 2649 (1999).
  38. Y. S. Weinstein, S. Lloyd, J. Emerson, and D. G. Cory, Experimental implementation of the quantum baker's map, Phys. Rev. Lett. 89, 157902 (2002).
  39. D. V. Vasilyev, A. Grankin, M. A. Baranov, L. M. Sieberer, and P. Zoller, Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits, PRX Quantum 1, 020302 (2020).
  40. L. K. Joshi, A. Elben, A. Vikram, B. Vermersch, V. Galitski, and P. Zoller, Probing many-body quantum chaos with quantum simulators, Phys. Rev. X 12, 011018 (2022).
  41. Y. G. Sinai, A weak isomorphism of transformations with invariant measure, Dokl. Akad. Nauk SSSR 147, 797 (1962).
  42. E. Glasner, Ergodic Theory via Joinings (American Mathematical Society, 2003).
  43. I. P. Cornfield, S. V. Fomin, and Y. G. Sinai, Ergodic Theory (Springer, New York, 1982).

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Phys. Rev. Research 6, L032046– Published 28 August, 2024
  • Received 15 September 2023
  • Accepted 22 March 2024
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