On Hitting Times for General Quantum Markov Processes

Lorenzo Laneve1,2, Francesco Tacchino2, and Ivano Tavernelli2

1Faculty of Informatics — Università della Svizzera Italiana, 6900 Lugano, Switzerland
2IBM Quantum, IBM Research — Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland

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Random walks (or Markov chains) are models extensively used in theoretical computer science. Several tools, including analysis of quantities such as hitting and mixing times, are helpful for devising randomized algorithms. A notable example is Schöning's algorithm for the satisfiability (SAT) problem. In this work, we use the density-matrix formalism to define a $\textit{quantum Markov chain}$ model which directly generalizes classical walks, and we show that a common tools such as hitting times can be computed with a similar formula as the one found in the classical theory, which we then apply to known quantum settings such as Grover's algorithm.

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