Requirements for building effective Hamiltonians using quantum-enhanced density matrix downfolding

TL;DR

This paper introduces a hybrid quantum-classical approach to improve the accuracy of density matrix downfolding by leveraging quantum computers to efficiently sample low-energy states, with applications to complex many-body systems.

Contribution

It proposes a quantum-enhanced protocol for density matrix downfolding, including new requirements and resource estimates for practical implementation on challenging models.

Findings

Quantum DMD can potentially reduce systematic errors in effective Hamiltonian construction.

Resource estimates suggest feasibility for models like the doped 2-D Fermi-Hubbard and cuprate superconductors.

The protocol relies on specific compressibility conditions for Hamiltonians and low-energy subspaces.

Abstract

Density matrix downfolding (DMD) is a technique for regressing low-energy effective Hamiltonians from quantum many-body Hamiltonians. One limiting factor in the accuracy of classical implementations of DMD is the presence of difficult-to-quantify systematic errors attendant to sampling the observables of quantum many-body systems on an approximate low-energy subspace. We propose a hybrid quantum-classical protocol for circumventing this limitation, relying on the prospective ability of quantum computers to efficiently prepare and sample from states in well-defined low-energy subspaces with systematically improvable accuracy. We introduce three requirements for when this is possible, including a notion of compressibility that quantifies features of Hamiltonians and low-energy subspaces thereof for which quantum DMD might be efficient. Assuming that these requirements are met, we analyze design choices for our protocol and provide resource estimates for implementing quantum-enhanced DMD on both the doped 2-D Fermi-Hubbard model and an ab initio model of a cuprate superconductor.