# Reduced Density Matrix and Cumulant Approximations of Quantum Linear Response

## TL;DR

This paper explores approximations to reduce the quantum workload in quantum linear response calculations, but finds that these methods often fail for strongly correlated systems.

## Contribution

The paper introduces and evaluates approximations to quantum linear response using reduced density matrices and cumulants for NISQ-era quantum computing.

## Key findings

- Approximations to 4-body RDMs and RDCs work well for equilibrium geometries and core excitations.
- Approximations to 3-body RDMs and RDCs severely affect results and are not viable.
- Strongly correlated systems cause failures in both 3-body and 4-body approximations.

## Abstract

Linear response (LR) is an important tool in the computational
chemist’s toolbox. It is therefore unsurprising that the emergence
of quantum computers has led to a quantum counterpart known as quantum
LR (qLR). However, the current quantum era of near-term intermediate-scale
quantum (NISQ) computers is dominated by noise, short decoherence
times, and slow measurement speeds. It is therefore of interest to
find approximations that can greatly reduce the quantum workload while
only slightly impacting the quality of a method. In an effort to achieve
this, we approximate the naive qLR with the singles and doubles (qLRSD)
method, by either directly approximating the reduced density matrices
(RDMs) or indirectly through their respective reduced density cumulants
(RDCs). We present an analysis of the measurement costs associated
with qLR using RDMs and report qLR results for model hydrogen ladder
systems; for varying active space sizes in OCS, SeH2, and
H2S; and for symmetrically stretched H2O and
BeH2. Discouragingly, while approximations to the 4-body
RDMs and RDCs seem to produce good results for systems at the equilibrium
geometry and for some types of core excitations, they both tend to
fail when the system exhibits strong correlation. All approximations
to the 3-body RDMs and/or RDCs severely affect the results and cannot
be applied.