Splitting between k-points for Periodic Systems Quantum Simulation

Quantum computation of periodic systems suffers the problem of the linear scaling of qubit requirement with respect to the number of sampled k-points. As a result, it may not be practical to include N_k N_orb qubits for a VQE calculation, where N_k is the number of k-points and N_orb is the number of spin-orbitals per k-point. Here, we propose an approximate ansatz for VQE that splits the single N_k N_orb circuit to multiple circuits with 4N_orb qubits each. To allow for such splitting, the ansatz should be chosen such that the parity of the occupied states of each k-points are fixed to either even or odd. This fixes the parity such that only one k-point needs to be included to calculate the expectation value of a one-body term and at most four k-points needs to be included to calculate the expectation value of a two-body term. Next, the Hamiltonian is partitioned into multiple circuits with each circuit containing 4 k-points. Finally, the cost function is calculated as the sum of the expectation value of each circuit. VQE calculations will be carried out using an exact simulator and the accuracy and efficiency of this ansatz will be compared against classical quantum chemical calculations.