Quantum Random Evolutions

Ming‐Xing Luo, Yun Deng, Song-Ya Ma +2 more

Matheus Guedes de Andrade, Franklin de Lima Marquezino, Daniel R. Figueiredo

Quantum and random walks were proven to be equivalent on finite graphs by demonstrating how to construct a time-dependent random walk sharing the exact same evolution of vertex probability of any given discrete-time coined quantum walk. Such equivalence stipulated a deep connection between the processes that is far stronger than simply considering quantum walks as quantum analogues of random walks. This article expands on the connection between quantum and random walks by demonstrating a procedure that constructs a time-dependent quantum walk matching the evolution of vertex probability of any given random walk in a unitary way. It is a trivial fact that a quantum walk measured at all time steps of its evolution degrades to a random walk. More interestingly, the method presented describes a quantum walk that matches a random walk without measurement operations, such that the unitary evolution of the quantum walk captures the probability evolution of the random walk. The construction procedure is general, covering both homogeneous and non-homogeneous random walks. For the homogeneous random walk case, the properties of unitary evolution imply that the quantum walk described is time-dependent since homogeneous quantum walks do not converge for arbitrary initial conditions

David Cienfuegos Salgado, J L S aacute nchez-G oacute mez

We prove how the form of the most general invariant stochastic unravelling for Markovian (recently given in the literature by Wiseman and Diósi) and non-Markovian but Lindblad-type open quantum systems can be attained by imposing a single mathematical condition upon the random evolution operator of the system, namely a.s. trace preservation (a.s. stands for almost surely ). The use of random operators ensures the complete positivity of the density operator evolution and characterizes the linear/non-linear character of the evolution in a straightforward way. It is also shown how three quantum stochastic evolution models—continuous spontaneous localization, quantum state diffusion and quantum mechanics with universal position localization—appear as concrete choices for the noise term of the evolution random operators are assumed. We finally conjecture how these operators may in the future be used in two different directions: both to connect quantum stochastic evolution models with random properties of space-time and to handle noisy quantum logical gates.

G. O. S. Ekhaguere

Ph. Blanchard, M.‐O. Hongler

In the continuous space and time limit, we show that the probability density to find the quantum random walk (QRW) driven by the Hadamard "coin" solves a hyperbolic evolution equation similar to the one obtained for a random two-velocity evolution with spatially inhomogeneous transition rates between the velocity states. In spite of the presence of a nonlinear drift term, it is remarkable that the QRW position can easily be described in simple analytical terms. This allows us to derive the quadratic time dependence of the variance typical for the QRW.

Winton Brown