Certification failure in variational-principle observables
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 $\sqrt{2εχ_O}$ 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.