Accelerating fermionic system simulation on quantum computers

A potential approach for demonstrating quantum advantage is using quantum computers to simulate fermionic systems. Quantum algorithms for fermionic system simulation usually involve the Hamiltonian evolution and measurements. However, in the second quantization representation, the number of terms in many fermion system Hamiltonians, such as molecular Hamiltonians, is substantial, approximately $O({N}^{4})$, where $N$ is the number of molecular orbitals. Due to this, the computational resources required for Hamiltonian evolution and expectation value measurements could be excessively large. To address this, we introduce a grouping strategy that partitions these $O({N}^{4})$ Hamiltonian terms into $O({N}^{2})$ groups, with the terms in each group mutually commuting. Based on this grouping method, we propose a parallel Hamiltonian evolution scheme that reduces the circuit depth of Hamiltonian evolution by a factor of $N$. Moreover, our grouping measurement strategy reduces the number of measurements needed to $O({N}^{2})$, whereas the current best grouping measurement schemes require $O({N}^{3})$ measurements. Additionally, we find that measuring the expectation value of a group of Hamiltonian terms requires fewer repetitions than measuring a single term individually, thereby reducing the number of quantum circuit executions. Our approach saves a factor of ${N}^{3}$ in the overall time for Hamiltonian evolution and measurements, significantly decreasing the time required for quantum computers to simulate fermionic systems.