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Hitting statistics from quantum jumps
A. Chia
1
, T. Paterek
2 3
, and L. C. Kwek
1 3 4 5
1
Centre for Quantum Technologies, National University of Singapore
2
Division of Physics and Applied Physics, School of Physical and Mathematical Sc iences, Nanyang Technological University, Singapore
3
Majulab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore
4
Institute of Advanced Studies, Nanyang Technological University, Singapore
5
National Institute of Education, Nanyang Technological University, Singapore
July 17, 2017
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.
1 Introduction
With the advent of quantum information science and the desire to build a quantum computer, the study of quantum
algorithms have become an integral part of quantum information theory [1]. Quantum walks have played a special
role in quantum computing by providing a platform on which quantum algorithms may be analysed [2, 3]. Moreover,
they have served as a useful mechanism to describe and explain coherent transport processes in photosynthesis [4, 5]
and the breakdown of a driven system in an electric field [6]. This has in turn stimulated much experimental effort
to realise quantum walks, see e.g. Refs. [7, 8, 9, 10]. Theoretical quantum optics on the other hand has had fruitful
applications in the analyses of quantum technologies [11, 12, 13]. In this paper we will apply the well-known theory
of quantum jumps developed in quantum optics to define and calculate the distribution of hitting times in open
(i.e. non-unitary) quantum walks.
The paper is organised as follows. We first review quantum jumps and motivate its application to hitting
problems in quantum walks in Sec. 1.1. The theory of quantum walks and some existing works related to quantum
jumps are then reviewed in Sec. 1.2. We then introduce the necessary background in Sec. 2 with the presentation
of the quantum-jump formalism in Sec. 2.1, followed by our approach to continuous-time open quantum walks in
Sec. 2.2. In Sec. 3 we explain how the quantum-jump method is to be applied to quantum walks and obtain our first
result—the hitting-time distribution. Then in Sec. 4 we derive explicit expressions for the hitting-time statistics.
We then relate our result to a previous definition of the hitting time devised for a discrete-time measured walk in
Sec. 5. Here the hitting-time statistics of the discrete-time measured walk is shown to converge to the quantum-
jump approach in the continuous-time limit. A problem with the quantum-jump definition of hitting times is that
it becomes inaccurate when there are coherent transitions to the final state. We overcome this problem in Sec. 6,
1
arXiv:1608.05510v4 [quant-ph] 14 Jul 2017