Quantum Markov Processes and Applications in Physics

Eugen Grycko, W. Kirsch, Tobias Mühlenbruch

An appropriate generator matrix induces in a natural way a Markov semigroup which can be applied to the construction of a continuous time Markov process whose asymptotic distribution is independent of the initial state.We propose a generator associated with a prescribed (arbitrary) distribution which turns out to be reversible for the corresponding Markov process.As an application, we show that the Markov process offers a physically plausible description of a quantum system exposed to a heat bath.The system approaches its Gibbs state corresponding to the temperature of the heat bath if its initial state commutes with the Hamiltonian.

Luigi Accardi

Crispin W. Gardiner

P. Mazzetti, Claudio Oldano

The application of the theory developed in paper I to problems of physical interest is considered and developed up to the point of finding the correlation matrix of the problem, thus reducing the complete solution to a matter of numerical computation. The considered cases are of rather large generality and refer to (a) fluctuation processes due to carrier trapping, (b) membrane noise, (c) spectral properties of linear atom chains, and (d) fluctuations in quantum systems whose Hamitonian contains an operator which is a random function of time. In the last case, the most general conditions under which the time evolution of a quantum system can be treated as a Markov process will be found and discussed.

Baptiste Claudon, Jean‐Philip Piquemal, Pierre Monmarché

Quantum algorithms can potentially solve a handful of problems more efficiently than their classical counterparts. In that context, it has been discussed that Markov chains problems could be solved significantly faster using quantum computing. Indeed, previous work suggests that quantum computers could accelerate sampling from the stationary distribution of reversible Markov chains. However, in practice, certain physical processes of interest are nonreversible in the probabilistic sense and reversible Markov chains can sometimes be replaced by more efficient nonreversible chains targeting the same stationary distribution. This study constructs Markov chain reversibilizations and develops quantum algorithmic techniques to accelerate nonreversible processes. Such an up-to-exponential quantum speedup goes beyond the predicted quadratic quantum acceleration for reversible chains and is likely to have a decisive impact on many applications ranging from statistics and machine learning to computational modeling in physics, chemistry, biology and finance.

Burkhard Kümmerer