Certification failure in variational-principle observables
Abstract
Algorithms which prepare states via the variational principle assume that closeness in energy implies closeness in state; we formalise how this assumption can fail by showing that a ground-state energy estimate converged to within ε of the true value is compatible with observable expectation values in error by up to √(2εχO) at leading order, where χO is the static susceptibility of the observable. As a result, the observable error can be orders of magnitude larger than the energy error, a mechanism we call non-linear error amplification, and because a variational algorithm cannot guarantee the correctness of these properties from energy alone, energy does not necessarily certify an observable. In light of our findings, we examine three algorithms: the variational quantum eigensolver on the transverse-field Ising chain, the density-matrix renormalisation group on a pair of weakly coupled hydrogen chains, and Krylov quantum diagonalisation on lithium fluoride. In each case we demonstrate that a well-converged energy can leave observables uncertified.
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BibTeX
@misc{manchanda2026certification,
title = {{Certification failure in variational-principle observables}},
author = {Ganga Singh Manchanda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03989},
eprint = {2610.03989},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03989},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Certification failure in variational-principle observables
AU - Ganga Singh Manchanda
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03989
UR - https://arxiv.org/abs/2610.03989
AB - Algorithms which prepare states via the variational principle assume that closeness in energy implies closeness in state; we formalise how this assumption can fail by showing that a ground-state energy estimate converged to within ε of the true value is compatible with observable expectation values in error by up to √(2εχ_O) at leading order, where χ_O is the static susceptibility of the observable. As a result, the observable error can be orders of magnitude larger than the energy error, a mechanism we call non-linear error amplification, and because a variational algorithm cannot guarantee the correctness of these properties from energy alone, energy does not necessarily certify an observable. In light of our findings, we examine three algorithms: the variational quantum eigensolver on the transverse-field Ising chain, the density-matrix renormalisation group on a pair of weakly coupled hydrogen chains, and Krylov quantum diagonalisation on lithium fluoride. In each case we demonstrate that a well-converged energy can leave observables uncertified.
ER -