Resource estimations for the Hamiltonian simulation in correlated electron materials

Correlated electron materials, such as superconductors and magnetic materials, are regarded as fascinating targets in quantum computing. However, the quantitative resources, specifically the number of quantum gates and qubits, required to perform a quantum algorithm to simulate correlated electron materials remain unclear. In this study, we estimate the resources required for the Hamiltonian simulation algorithm for correlated electron materials, specifically for organic superconductors, iron-based superconductors, binary transition metal oxides, and perovskite oxides, using the fermionic swap network. The effective Hamiltonian derived using the $ab~initio$ downfolding method is adopted for the Hamiltonian simulation, and a procedure for the resource estimation by using the fermionic swap network for the effective Hamiltonians including the exchange interactions is proposed. For example, in the system for the $10^2$ unit cells, the estimated number of gates per Trotter step and qubits are approximately $10^7$ and $10^3$, respectively, on average for the correlated electron materials. Furthermore, our results show that the number of interaction terms in the effective Hamiltonian, especially for the Coulomb interaction terms, is dominant in the gate resources when the number of unit cells constituting the whole system is up to $10^2$, whereas the number of fermionic swap operations is dominant when the number of unit cells is more than $10^3$.