Tensor Factorized Recursive Hamiltonian Downfolding To Optimize The Scaling Complexity Of The Electronic Correlations Problem on Classical and Quantum Computers

Abstract: This paper presents a new variant of post-Hartree-Fock Hamiltonian downfolding-based quantum chemistry methods with optimized scaling for high-cost simulations like coupled cluster (CC), full configuration interaction (FCI), and multi-reference CI (MRCI) on classical and quantum hardware. This improves the applicability of these calculations to practical use cases. High-accuracy quantum chemistry calculations, such as CC, involve memory and time-intensive tensor operations, which are the primary bottlenecks in determining the properties of many-electron systems. The complexity of those operations scales exponentially with system size. We aim to find properties of chemical systems by optimizing this scaling through mathematical transformations on the Hamiltonian and the state space. By defining a bi-partition of the many-body Hilbert space into electron-occupied and unoccupied blocks for a given orbital, we perform a downfolding transformation that decouples the electron-occupied block from its complement. We represent high-rank electronic integrals and cluster amplitude tensors as low-rank tensor factors of a downfolding transformation, mapping the full many-body Hamiltonian into a smaller dimensional block Hamiltonian recursively. This reduces the computational complexity of solving the residual equations for Hamiltonian downfolding on CPUs from $\mathcal{O}(N^7)$ for CCSD(T) and $\mathcal{O}(N^9)$ - $\mathcal{O}(N^{10})$ for CI and MRCI to $\mathcal{O}(N^3)$. Additionally, we create a quantum circuit encoding of the tensor factors, generating circuits of $\mathcal{O}(N^2)$ depth with $\mathcal{O}(\log N)$ qubits. We demonstrate super-quadratic speedups of expensive quantum chemistry algorithms on both classical and quantum computers.