Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian

Achieving an accurate description of fermionic systems typically requires considerably many more orbitals than fermions. Previous resource analyses of quantum chemistry simulation often failed to exploit this low fermionic number information in the implementation of Trotter-based approaches and overestimated the quantum-computer runtime as a result. They also depended on numerical procedures that are computationally too expensive to scale up to large systems of practical interest. Here we propose techniques that solve both problems by using various factorized decompositions of the electronic structure Hamiltonian. We showcase our techniques for the uniform electron gas, finding substantial (over $100\times$) improvements in Trotter error for low-filling fraction and pushing to much higher numbers of orbitals than is possible with existing methods. Finally, we calculate the $T$-count to perform phase estimation on Jellium. In the low-filling regime, we observe improvements in gate complexity of over $10\times$ compared to the best Trotter-based approach reported to date. We also report gate counts competitive with qubitization-based approaches for Wigner-Seitz values of physical interest.

Figure 1

First (left) and second (right) order commutator bounds for a 2D uniform electron gas system with $r_s=5$, resolved with 200 spin orbitals. The electronic density is kept fixed, such that the volume of the simulation cell increases with the number of electrons considered, which alters the Hamiltonian coefficients. This effect competes with the electron number dependence of the fermionic seminorm to determine the resulting error bounds. For the Cholesky decomposition the chemical potential was shifted by the minimum value that ensured $V$ was positive definite.

Figure 2

First (left) and second (right) order commutator bounds for a 2D uniform electron gas system with $r_s=5$, and 49 electrons, as a function of the number of spin orbitals used to resolve the system. Fermionic and Pauli commutator bounds could not be calculated for all data points, due to the large memory requirements of those approaches. The “projected Pauli bounds” were obtained as described in Appendix pp6. For the Cholesky decomposition the chemical potential was shifted by the minimum value that ensured $V$ was positive definite.

Figure 3

The ratio between the second-order fermionic commutator bound $W_2^F$ and the second-order Pauli commutator bound $W_2^P$, as a function of the number of spin orbitals used, for a homogeneous electron gas system with $r_s=5$. The number of electrons in each calculation is varied, in order to match the specified filling fraction as closely as possible.