Quantum Inf Process (2014) 13:559–572
Decoherence in quantum Markov chains
Raqueline Azevedo Medeiros Santos ·
Renato Portugal · Marcelo Dutra Fragoso
Received: 18 February 2013 / Accepted: 28 October 2013 / Published online: 12 November 2013
© Springer Science+Business Media New York 2013
Abstract It is known that under some assumptions, the hitting time in quantum
Markov chains is quadratically smaller than the hitting time in classical Markov chains.
This work extends this result for decoherent quantum Markov chains. The decoher-
ence is introduced using a percolation-like graph model, which allows us to deﬁne
a decoherent quantum hitting time and to establish a decoherent-intensity range for
which the decoherent quantum hitting time is quadratically smaller than the classical
hitting time. The detection problem under decoherence is also solved with quadratic
speedup in this range.
Keywords Quantum Markov chains · Percolation · Decoherence ·
Quantum hitting time
In Computer Science, Markov chains are employed in randomized algorithms such as
searching algorithms on graphs. The expected time to reach a vertex for the ﬁrst time,
known as hitting time, plays an important role in those algorithms as the running time
to ﬁnd a solution. For instance, randomized algorithms are used to address the k-SAT
and the graph connectivity problem .
In the classical case, Markov chains and random walks are equivalent formalisms. In
the quantum case, there are three versions of quantum walks: (1) discrete-time quantum
walks , (2) continuous-time quantum walks , and (3) Szegedy’s formalism .
All of them have been used for developing quantum algorithms that outperform their
classical versions [3,4,7,11,19,23]. These models use Hilbert spaces of different size,
R. A. M. Santos (
) · R. Portugal · M. D. Fragoso
Laboratório Nacional de Computação Cientíﬁca (LNCC), Av. Getúlio Vargas 333,
Petrópolis, RJ 25651-075, Brazil