An iterative Bayesian variational quantum eigensolver with the von Mises-Fisher distribution

We develop an iterative Bayesian variational quantum eigensolver in which the unknown ground state, parametrised as a real unit vector, is inferred through a von Mises–Fisher distribution sequentially updated from Hadamard-test measurements on a single ancilla qubit. Building on a companion paper that treated a single success outcome via exact moment matching, we generalise the scheme to condition on $m$ success outcomes for the likelihood function per iteration. Since exact moment matching becomes impractical for $m>1$, we exploit the large-concentration regime, in which a first-order expansion of the log-likelihood yields a closed-form von Mises–Fisher posterior with $m$ entering only through a multiplicative weight. We prove that the two convergence properties of the single-outcome scheme—non-decreasing overlap and non-decreasing concentration—are preserved provided $m$ lies within an explicitly characterised range. Based on the theory, we propose a measurement-based algorithm and illustrate the convergence on randomly generated three- and four-qubit Hamiltonian matrices, on ${\mathrm{H}\mathrm{e}\mathrm{H}}^{+}$, and on ${\mathrm{H}}_2$.