A Shadow Enhanced Greedy Quantum Eigensolver

TL;DR

The paper introduces SEGQE, a measurement-efficient quantum eigensolver that uses classical shadows and a greedy approach to prepare ground states with fewer measurements, suitable for early fault-tolerant quantum computers.

Contribution

It proposes a novel shadow-assisted greedy framework for ground-state preparation that offers rigorous complexity bounds and practical scalability.

Findings

Converges in a number of iterations approximately linear with system size.

Maintains high-fidelity ground-state approximations.

Offers logarithmic per-iteration sample complexity bounds.

Abstract

While ground-state preparation is expected to be a primary application of quantum computers, it is also an essential subroutine for many fault-tolerant algorithms. In early fault-tolerant regimes, logical measurements remain costly, motivating adaptive, shot-frugal state-preparation strategies that efficiently utilize each measurement. We introduce the Shadow Enhanced Greedy Quantum Eigensolver (SEGQE) as a greedy, shadow-assisted framework for measurement-efficient ground-state preparation. SEGQE uses classical shadows to evaluate, in parallel and entirely in classical post-processing, the energy reduction induced by large collections of local candidate gates, greedily selecting at each step the gate with the largest estimated energy decrease. We derive rigorous worst-case per-iteration sample-complexity bounds for SEGQE, exhibiting logarithmic dependence on the number of candidate gates. Numerical benchmarks on finite transverse-field Ising models and ensembles of random local Hamiltonians demonstrate convergence in a number of iterations that scales approximately linearly with system size, while maintaining high-fidelity ground-state approximations and competitive energy estimates. Together, our empirical scaling laws and rigorous per-iteration guarantees establish SEGQE as a measurement-efficient state-preparation primitive well suited to early fault-tolerant quantum computing architectures.