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).
We derive the continuous-time limit of discrete quantum walks with topological phases. We show the existence of a continuous-time limit that preserves their topological phases. We consider both simple-step and splitstep walks, and derive analytically equations of motion governing their behavior. We obtain simple analytical solutions showing the existence of bound states at the boundary of two phases, and solve the equations of motion numerically in the bulk.
Quantum games with incomplete information can be studied within a Bayesian framework. We analyze games quantized within the EWL framework [Eisert, Wilkens, and Lewenstein, Phys Rev. Lett. 83, 3077 (1999)]. We solve for the Nash equilibria of a variety of two-player quantum games and compare the results to the solutions of the corresponding classical games. We then analyze Bayesian games where there is uncertainty about the player types in two-player conflicting interest games. The solutions to the Bayesian games are found to have a phase diagram-like structure where different equilibria exist in different parameter regions, depending both on the amount of uncertainty and the degree of entanglement. We find that in games where a Pareto-optimal solution is not a Nash equilibrium, it is possible for the quantized game to have an advantage over the classical version. In addition, we analyze the behavior of the solutions as the strategy choices approach an unrestricted operation. We find that some games have a continuum of solutions, bounded by the solutions of a simpler restricted game. A deeper understanding of Bayesian quantum game theory could lead to novel quantum applications in a multi-agent setting.
The stochastic nature of quantum communication protocols naturally lends itself for expression via probabilistic logic languages. In this work we describe quantized computation using Horn clauses and base the semantics on quantum probability. Turing computable Horn clauses are very convenient to work with and the formalism can be extended to general form of first order languages. Towards this end we build a Hilbert space of H-interpretations and a corresponding non commutative von Neumann algebra of bounded linear operators. We demonstrate the expressive power of the language by casting quantum communication protocols as Horn clauses.
We describe a quantum mechanics based logic programming language that supports Horn clauses, random variables, and covariance matrices to express and solve problems in probabilistic logic. The Horn clauses of the language wrap random variables, including infinite valued, to express probability distributions and statistical correlations, a powerful feature to capture relationship between distributions that are not independent. The expressive power of the language is based on a mechanism to implement statistical ensembles and to solve the underlying SAT instances using quantum mechanical machinery. We exploit the fact that classical random variables have quantum decompositions to build the Horn clauses. We establish the semantics of the language in a rigorous fashion by considering an existing probabilistic logic language called PRISM with classical probability measures defined on the Herbrand base and extending it to the quantum context. In the classical case H-interpretations form the sample space and probability measures defined on them lead to consistent definition of probabilities for well formed formulae. In the quantum counterpart, we define probability amplitudes on Hinterpretations facilitating the model generations and verifications via quantum mechanical superpositions and entanglements. We cast the well formed formulae of the language as quantum mechanical observables thus providing an elegant interpretation for their probabilities. We discuss several examples to combine statistical ensembles and predicates of first order logic to reason with situations involving uncertainty.
Conference Committee Involvement (1)
Quantum Information Science, Sensing, and Computation X