Variational procedure yielding upper and lower bounds to eigenvalues of hermitian operators

Francisco Marcelo Fernández, John F. Ogilvie, Richard H. Tipping

H. H. Aly, Roger C. Hill

Maurice Cohen, Tova Feldmann

The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.

Thiago O. Maciel, Giancarlo Camilo, Allan Tosta +3 more

Quantum signal processing (QSP) realizes operator functions through polynomial transformations of a finite degree. In addition to function approximation errors, realistic implementations incur algorithmic errors also when encoding the input operator and in synthesizing gates into a discrete fault-tolerant gate set. We provide an end-to-end analysis of these errors and show that the input error propagation is governed by the operator-Lipschitz constant $L_\star$ of the implemented function. We then derive an efficient classical algorithm to compute tight lower and upper bounds on $L_\star$, and we show that these bounds induce in turn lower and upper bounds on the total quantum-resource cost (for instance, $T$-gate count) needed for QSP implementations compatible with a target total error. We further devise a procedure to search for a polynomial with minimum cost based solely on its coefficients, yielding a resource-aware polynomial-design procedure for QSP. We illustrate the resulting method for ground state energy estimation via eigenvalue thresholding for the H$_6$ molecular Hamiltonian. In the tested hardware profiles, the resource-aware family improves the total-error budget and reduces the modeled $T$-gate count by $51.9\%$--$56.7\%$ relative to the standard analytical polynomial obtained from an error-function approximation.

Robert Blau, A. R. P. Rau, Larry Spruch

Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where $W$ is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the $n$th and $m$th states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for $W$ and only for real wave functions) or have been very conservative. We remove most of these restrictions ($W$ need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on ${W}_{\mathrm{nm}}$ in terms of "simple" upper bounds (which may be poor) on ${{W}^{2}}_{\mathrm{nn}}$. Here again, only for a very few particular choices of $W$ have such simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ been given previously (for example, $n$ restricted to be the ground state, and $W$ the operator ${z}_{i}$, the coordinate of the $i$th electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators $W$ in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on ${{W}^{2}}_{\mathrm{nn}}$ do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of ${r}_{1}$ and ${r}_{1}^{2}$---we, therefore, require simple bounds on ${r}_{1}^{2}$ and ${r}_{1}^{4}$---for the states $1{s}^{2} ^{1}S$ and $1s2s ^{3}S$ of the helium atom, with rather good results.

Frederik G. Scholtz, H. B. Geyer, F.J.W. Hahne