Enhanced Shadow Tomography of Molecular Excited States from Enforcing $N$-representability Conditions by Semidefinite Programming

TL;DR

This paper introduces an improved quantum algorithm combining shadow tomography with $N$-representability constraints via semidefinite programming to efficiently compute excited states of strongly correlated molecules, reducing measurement costs.

Contribution

It extends $N$-representability-enhanced shadow tomography to excited states, enabling more accurate and measurement-efficient quantum simulations of complex molecular systems.

Findings

Successfully computed excited-state energies of H4 chain.

Retained multireference correlation effects in simulations.

Significantly reduced measurement requirements.

Abstract

Excited-state properties of highly correlated systems are key to understanding photosynthesis, luminescence, and the development of novel optical materials, but accurately capturing their interactions is computationally costly. We present an algorithm that combines classical shadow tomography with physical constraints on the two-electron reduced density matrix (2-RDM) to treat excited states. The method reduces the number of measurements of the many-electron 2-RDM on quantum computers by (i) approximating the quantum state through a random sampling technique called shadow tomography and (ii) ensuring that the 2-RDM represents an $N$-electron system through imposing $N$-representability constraints by semidefinite programming. This generalizes recent work on the $N$-representability-enhanced shadow tomography of ground-state 2-RDMs. We compute excited-state energies and 2-RDMs of the H$_4$ chain and analyze the critical points along the photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection. The results show that the generalized shadow tomography retains critical multireference correlation effects while significantly reducing the number of required measurements, offering a promising avenue for the efficient treatment of electronically excited states on quantum devices.