Open quantum walks on graphs

Monika Rani, Supriyo Dutta, Subhashish Banerjee

Radhakrishnan Balu, Chaobin Liu

Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. The difference of long-time behaviors between CTOQW and other two continuous-time processes on graphs is exemplified. The examples demonstrate that the structure of a graph can affect a quantum coherence effect on CTOQW through a long time run. Precisely, a quantum coherence effect persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, a quantum coherence effect will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).

Renato Portugal

Alain Joye

In this paper, we consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.

Renato Portugal

Stéphane Attal, Francesco Petruccione, Ilya Sinayskiy

Open quantum walks (OQW) are formulated as quantum Markov chains on graphs. It is shown that OQWs are a very useful tool for the formulation of dissipative quantum computing algorithms and for dissipative quantum state preparation. In particular, single qubit gates and the CNOT-gate are implemented as OQWs on fully connected graphs. Also, dissipative quantum state preparation of arbitrary single qubit states and of all two-qubit Bell-states is demonstrated. Finally, the discrete time version of dissipative quantum computing is shown to be more efficient if formulated in the language of OQWs.