  Perturbation theory is an important technique for reducing computational cost
and providing physical insights in simulating quantum systems with classical
computers. Here, we provide a quantum algorithm to obtain perturbative energies
on quantum computers. The benefit of using quantum computers is that we can
start the perturbation from a Hamiltonian that is classically hard to solve.
The proposed algorithm uses quantum signal processing (QSP) to achieve this
goal. Along with the perturbation theory, we construct a technique for ground
state preparation with detailed computational cost analysis, which can be of
independent interest. We also estimate a rough computational cost of the
algorithm for simple chemical systems such as water clusters and polyacene
molecules. To the best of our knowledge, this is the first of such estimates
for practical applications of QSP. Unfortunately, we find that the proposed
algorithm, at least in its current form, does not exhibit practical numbers
despite of the efficiency of QSP compared to conventional quantum algorithms.
However, perturbation theory itself is an attractive direction to explore
because of its physical interpretability; it provides us insights about what
interaction gives an important contribution to the properties of systems. This
is in sharp contrast to the conventional approaches based on the quantum phase
estimation algorithm, where we can only obtain values of energy. From this
aspect, this work is a first step towards ``explainable'' quantum simulation on
fault-tolerant quantum computers.
