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

**Authors:** Ritam Banerjee, Ananthakrishna Gopal, Soham Bhandary, Janani Seshadri, Anirban Mukherjee  
**arXiv:** [2303.07051](https://arxiv.org/abs/2303.07051)  
**Date:** 2023-03-13  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2303.07051v3/

## Key points

- Novel approach to post-Hartree-Fock Hamiltonian downfolding-based quantum chemistry methods
- Focus on enhancing practical applicability of calculations for high-cost simulations like CC, FCI, and MRCI
- Strategy involving mathematical transformations on the Hamiltonian and state space to address memory and time-intensive tensor operations
- Downfolding transformation to decouple electron-occupied block from its complement and represent high-rank electronic integrals as low-rank tensor factors
- Significant reduction in computational complexity for solving residual equations from O(N^7) to O(N^3) for CCSD(T)
- Quantum circuit encoding of tensor factors with O(N^2) depth using O(log N) qubits
- Experimental results showcasing super-quadratic speedups in implementing expensive quantum chemistry algorithms on classical and quantum computers

## Summary

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This paper presents a novel approach to post-Hartree-Fock Hamiltonian downfolding-based quantum chemistry methods, specifically tailored for high-cost simulations such as coupled cluster (CC), full configuration interaction (FCI), and multi-reference CI (MRCI) on both classical and quantum hardware. The primary focus of this research is to enhance the practical applicability of these calculations by addressing the memory and time-intensive tensor operations that are inherent bottlenecks in determining the properties of complex many-electron systems. The complexity of these operations escalates exponentially with system size, posing significant challenges for accurate quantum chemistry calculations like CC. To overcome this hurdle, the authors propose a strategy that involves mathematical transformations on the Hamiltonian and state space. By partitioning the many-body Hilbert space into electron-occupied and unoccupied blocks for a given orbital, a downfolding transformation is performed to effectively decouple the electron-occupied block from its complement. Furthermore, high-rank electronic integrals and cluster amplitude tensors are represented as low-rank tensor factors through this downfolding transformation, thereby mapping the full many-body Hamiltonian into a smaller dimensional block Hamiltonian recursively. This innovative approach significantly reduces the computational complexity involved in solving residual equations for Hamiltonian downfolding on CPUs – from O(N^7) for CCSD(T) to O(N^3). Additionally, a quantum circuit encoding of these tensor factors is created, generating circuits with O(N^2) depth using O(log N) qubits. The experimental results presented in this study showcase super-quadratic speedups achieved by implementing expensive quantum chemistry algorithms on both classical and quantum computers. Overall, this research contributes valuable insights into optimizing the scaling complexity of electronic correlations problems in quantum chemistry through tensor factorized recursive Hamiltonian downfolding techniques.