Perturbation theory with quantum signal processing

Kosuke Mitarai1,2, Kiichiro Toyoizumi3, and Wataru Mizukami2

1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.
2Center for Quantum Information and Quantum Biology, Osaka University, Japan.
3Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan.

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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.

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