Perturbation theory with quantum signal processing

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 using quantum signal processing (QSP). The benefit of using quantum computers is that we can start the perturbation from a Hamiltonian that is classically hard to solve. Along with the perturbation theory, we construct a technique for ground state preparation with detailed computational cost analysis. We also estimate a rough computational cost of the algorithm for simple chemical systems such as water clusters and polyacene molecules. 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.