Abstract
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.
Highlights
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]
The quantum-jump approach to dissipative quantum dynamics has traditionally been used for efficiently solving master equations [14, 15, 16, 17], or calculating photon statistics in photon counting [18, 19]
We emphasise that diverging hitting times are a purely quantum phenomenon as a classical walker always reaches the final state in a finite time
Summary
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]. 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]. 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 conclude with a summary of our results and analyses in Sec. 7 and comment on the relationship of our work with other studies not mentioned in the literature review of Secs. 1.1 and 1.2
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