  Two main obstacles for observing quantum advantage in noisy
intermediate-scale quantum computers (NISQ) are the finite precision effects
due to control errors, or disorders, and decoherence effects due to thermal
fluctuations. It has been shown that dissipative quantum computation is
possible in presence of an idealized fully-engineered bath. However, it is not
clear, in general, what performance can be achieved by NISQ when internal bath
degrees of freedom are not controllable. In this work, we consider the task of
quantum search of a marked node on a complete graph of $n$ nodes in the
presence of both static disorder and non-zero coupling to an environment. We
show that, given fixed and finite levels of disorder and thermal fluctuations,
there is an optimal range of bath temperatures that can significantly improve
the success probability of the algorithm. Remarkably for a fixed disorder
strength $\sigma$, the system relaxation time decreases for higher temperatures
within a robust range of parameters. In particular, we demonstrate that for
strong disorder, the presence of a thermal bath increases the success
probability from $1/(n \sigma^2)$ to at least $1/2$. While the asymptotic
running time is approximately maintained, the need to repeat the algorithm many
times and issues associated with unitary over-rotations can be avoided as the
system relaxes to an absorbing steady state. Furthermore, we discuss for what
regimes of disorder and bath parameters quantum speedup is possible and mention
conditions for which similar phenomena can be observed in more general families
of graphs. Our work highlights that in the presence of static disorder, even
non-engineered environmental interactions can be beneficial for a quantum
algorithm.
