Hitting statistics from quantum jumps

A. Chia1, T. Paterek2,3, and L. C. Kwek1,3,4,5

1Centre for Quantum Technologies, National University of Singapore
2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
3Majulab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore
4Institute of Advanced Studies, Nanyang Technological University, Singapore
5National Institute of Education, Nanyang Technological University, Singapore

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


We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys. Rev. A 73, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.

► BibTeX data

► References

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, United Kingdom, 2000).

[2] S. E. Venegas-Andraca, Quantum Walks for Computer Scientists (Morgan and Claypool, 2008).

[3] R. Portugal, Quantum Walks and Search Algorithms (Springer, New York, Heidelberg, Dordrecht, London, 2013).

[4] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Man$\check{\rm c}$al, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems, Nature 446, 782 (2007).

[5] M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, Environment-assisted quantum walks in photosynthetic energy transfer, J. Chem. Phys. 129, 174106 (2008).

[6] T. Oka, N. Konno, R. Arita, and H. Aoki, Breakdown of an electric-field driven system: A mapping to a quantum walk, Phys. Rev. Lett. 94, 100602 (2005).

[7] H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Moran-Dotti, and Y. Silberberg. Realization of quantum walks with negligible decoherence in waveguide lattices, Phys. Rev. Lett. 100, 170506 (2008).

[8] M. Karski, L. Förster, J. M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Quantum walk in position space with single optically trapped atoms, Science 325, 174 (2009).

[9] H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, Quantum walk of a trapped ion in phase space, Phys. Rev. Lett. 103, 090504 (2009).

[10] F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, Realization of a quantum walk with one and two trapped ions, Phys. Rev. Lett. 104, 100503 (2010).

[11] D. Bouwmeester, A. Ekert, and A. Zeilinger (Eds.), The Physics of Quantum Information (Springer-Verlag, Berlin, Heidelberg, New York, 2001).

[12] P. Kok and B. W. Lovett, Introduction to Optical Quantum Information Processing (Cambridge University Press, New York, 2010).

[13] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, New York, 2010).

[14] J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).

[15] R. Dum, P. Zoller, and H. Ritsch, Monte Carlo simulation of the atomic master equation for spontaneous emission, Phys. Rev. A 45, 4879 (1992).

[16] R. Dum, A. S. Parkins, P. Zoller, and C. W. Gardiner, Monte Carlo simulation of master equations in quantum optics for vacuum, thermal, and squeezed reservoirs, Phys. Rev. A 46, 4382 (1992).

[17] K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, J. Opt. Soc. Am. B 10, 524 (1993).

[18] H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice, Photoelectron waiting times and atomic state reduction in resonance fluorescence, Phys. Rev. A 39, 1200 (1989).

[19] H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, Heidelberg, 1993).

[20] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).

[21] D. B. Horoshko and S. Ya. Kilin, Direct detection feedback for preserving quantum coherence in an open cavity, Phys. Rev. Lett. 78, 840 (1997).

[22] J. Gambetta and H. M. Wiseman, State and dynamical parameter estimation for open quantum systems, Phys. Rev. A 64, 042105 (2001).

[23] C. Di Fidio, W. Vogel, M. Khanbekyan, and G.-G. Welsch, Photon emission by an atom in a lossy cavity, Phys. Rev. A 77, 043822 (2008).

[24] A. H. Kiilerich and K. Mølmer, Parameter estimation by multichannel photon counting, Phys. Rev. A 91, 012119 (2015).

[25] H. J. Carmichael, Statistical Methods in Quantum Optics 2 (Springer, Berlin, Heidelberg, New York, 2008).

[26] B. Jones, S. Ghose, J. P. Clemens, P. R. Rice, and L. M. Pedrotti, Photon statistics of a single atom laser, Phys. Rev. A 60, 3267 (1999).

[27] A. Kronwald, M. Ludwig, and F. Marquadt, Full photon statistics of a light beam transmitted through an optomechanical system, Phys. Rev. A 87, 013847 (2013).

[28] P. L. Kelley and W. H. Kleiner, Theory of electromagnetic field measurement and photoelectron counting, Phys. Rev. 136, A316 (1964).

[29] H. Kobayashi, B. L. Mark, and W. Turin, Probability, Random Processes, and Statistical Analysis (Cambridge University Press, New York, 2012).

[30] S. Redner, A Guide to First-Passage Processes, (Cambridge University Press, United States of America, 2001).

[31] T. Taillefumier and M. O. Magnasco, A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding, Proc. Natl. Acad. Sci. 110, E1438 (2013).

[32] S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage times in complex scale-invariant media, Nature 450, 77 (2007).

[33] K. Kraus, General state changes in quantum theory, Ann. Phys. 64, 311 (1971).

[34] Y. Aharanov and L. Davidovich and N. Zagury, Quantum random walks, Phys. Rev. A 48, 1687 (1993).

[35] S. E. Venegas-Andraca, Quantum walks: a comprehensive review, Quantum Inf. Process. 12, 1015 (2012).

[36] J. Kempe, Quantum walks: an introductory overview, Contemp. Phys. 44, 307 (2003).

[37] J. Kempe, Discrete quantum walks hit exponentially faster, Proceedings of the 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM03), 354 (2003).

[38] H. Krovi and T. A. Brun, Hitting time for quantum walks on the hypercube, Phys. Rev. A 73, 032341 (2006).

[39] N. Shenvi, J. Kempe, and K. B. Whaley, Quantum random-walk search algorithm, Phys. Rev. A 67, 052307 (2003).

[40] A. Ambainis, J. Kempe, and A. Rivosh, Coins make quantum walks faster, Proc. 16th ACM-SIAM Symposium on Discrete Alogrithms, 1099 (2005) (http:/​/​dl.acm.org/​citation.cfm?id=1070432.1070590).

[41] N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev. A 81, 042330 (2010).

[42] E. Farhi and S. Gutmann, Quantum computation and decision trees, Phys. Rev. A 58, 915 (1998).

[43] A. M. Childs, E. Farhi, and S. Gutmann, An example of the difference between quantum and classical random walks, Quant. Inf. Process. 1, 35 (2002).

[44] M. Varbanov, H. Krovi, and T. A. Brun, Hitting time for the continuous quantum walk, Phys. Rev. A 78, 022324 (2008).

[45] T. A. Brun, A simple model of quantum trajectories, Am. J. Phys. 70, 719 (2002).

[46] A. M. Childs and J. Goldstone, Spatial search by quantum walk, Phys. Rev. A 70, 022314 (2004).

[47] A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, and D. A. Spielman, Exponential algorithmic speedup by a quantum walk, Proc. of the 35th Annual ACM Symposium on Theory of Computing, 59 (2003).

[48] A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett. 102, 180501 (2009).

[49] S. Attal, F. Petruccione, and I. Sinayskiy, Open quantum walks on graphs, Phys. Lett. A 376, 1545 (2012).

[50] S. Attal, F. Petruccione, C. Sabot, and I. Sinayskiy, Open quantum random walks, J. Stat. Phys. 147, 832 (2012).

[51] C. Pellegrini, Continuous time open quantum random walks and non-Markovian Lindblad master equations, J. Stat. Phys. 154, 838 (2014).

[52] C. Liu and R. Balu, Steady states of continuous-time open quantum walks, C. Liu and R. Balu, Steady states of continuous-time open quantum walks, Quantum Inf. Process. 16, 173 (2017).

[53] C. F. Lardizabal and R. R. Souza, Open quantum random walks: Ergodicity, hitting times, gambler's ruin and potential theory, J. Stat. Phys. 164, 1122 (2016).

[54] R.-T. Qiu, W.-S. Dai, and M. Xie, Mean-first passage time of quantum transition processes, Physica A, 4748 (2012).

[55] S. Daryanoosh and H. M. Wiseman, Quantum jumps are more quantum than quantum diffusion, New. J. Phys. 16, 063028 (2014).

[56] A. Chia, A. Górecka, P. Kurzyński, T. Paterek, D. Kaszlikowski, Coherent chemical kinetics as quantum walks. II. Radical-pair reactions in Arabidopsis thaliana, Phys. Rev. E 93, 032408 (2016).

[57] H. M. Wiseman and L. Diósi, Complete parameterization, and invariance, of diffusive quantum trajectories for Markovian open systems, Chem. Phys. 268, 91 (2001).

[58] A. Chia and H. M. Wiseman, Complete parameterizations of diffusive quantum monitorings, Phys. Rev. A 84, 012119 (2011).

[59] K. Jacobs, How to project qubits faster using quantum feedback, Phys. Rev. A 67, 030301(R) (2003).

[60] K. Jacobs, Optimal feedback control for rapid preparation of a qubit, Proc. SPIE 5468, 355 (2004).

[61] J. Combes and K. Jacobs, Rapid state reduction of quantum systems using feedback control, Phys. Rev. Lett. 96, 010504 (2006).

[62] H. M. Wiseman and J. F. Ralph, Reconsidering rapid qubit purification by feedback, New J. Phys. 8, 90 (2006).

[63] C. W. Gardiner, Handbook of Stochastic Methods Third Edition (Springer-Verlag, Berlin, Heidelberg, New York, 2004).

[64] A. H. Kiilerich and K. Mølmer, Estimation of atomic interaction parameters by photon counting, Phys. Rev. A. 89, 052110 (2014).

Cited by

[1] Sergey Filippov, "Multipartite Correlations in Quantum Collision Models", Entropy 24 4, 508 (2022).

[2] Newton Loebens, "Open quantum jump chain for a class of continuous-time open quantum walks", Quantum Studies: Mathematics and Foundations (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 14:47:16) and SAO/NASA ADS (last updated successfully 2024-05-26 14:47:17). The list may be incomplete as not all publishers provide suitable and complete citation data.