  Quantum phase estimation (QPE) is the key subroutine of several quantum
computing algorithms as well as a central ingredient in quantum computational
chemistry and quantum simulation. While QPE strategies have focused on the
estimation of a single phase, applications to the simultaneous estimation of
several phases may bring substantial advantages; for instance, in the presence
of spatial or temporal constraints. In this work, we study a Bayesian algorithm
for the parallel (simultaneous) estimation of multiple arbitrary phases. The
protocol gives access to correlations in the Bayesian multi-phase distribution
resulting in covariance matrix elements scaling inversely proportional to the
square of the total number of quantum resources. The parallel estimation allows
to surpass the sensitivity of sequential single-phase estimation strategies for
optimal linear combinations of phases. Furthermore, the algorithm proves a
certain noise resilience and can be implemented using single photons and
standard optical elements in currently accessible experiments.
