Optimizing and Comparing Quantum Resources of Statistical Phase Estimation and Krylov Subspace Diagonalization

Abstract

We develop a framework that enables direct and meaningful comparison of two early fault-tolerant methods for the computation of eigenenergies, namely \gls{qksd} and \gls{spe}, within which both methods use expectation values of Chebyshev polynomials of the Hamiltonian as input. For \gls{qksd} we propose methods for optimally distributing shots and ensuring sufficient non-linearity of states spanning the Krylov space. For \gls{spe} we improve rigorous error-bounds, achieving roughly a factor $2/3$ reduction of circuit depth. We provide insights into the scalability of and the practical realization of these methods by computing the maximum Chebyshev degree, linearly related to circuit depth, and the respective number of repetitions required for the simulation of molecules with active spaces up to 54 electrons in 36 orbitals by leveraging \gls{mps}/\gls{dmrg}.

Figures (12)

• Figure 1: Maximum Chebyshev polynomial degree $K$ (proportional to circuit depth) and total shot count $M$ required to achieve a shot-noise error below $10^{-3}$ Hartree for various molecules. For (symbols connected by lines), the generalized eigenvalue problem is solved while retaining the dominant two (crosses) or three (circles) eigenvalues of the overlap matrix $\widetilde{S}$ (see Table \ref{['tab:data']} for the achieved rmse and additional details). The scattered points indicate the circuit depth and shot count for which, with 99% success probability, the is guaranteed to achieve an error of $10^{-3}$ Hartree or be on par with the of at the largest $K$ shown if that value is higher. Small markers correspond to $M$ and $K$ estimates with the error bounds from Ref. Wan_2022, while large markers are based on improved bounds derived in Section \ref{['sec:tight_bounds']} (see Figure \ref{['fig:delta_E_vs_K']} for an analysis of their tightness).
• Figure 2: Bounds on the truncation error of the scaled error function as a function of the truncation order $K$. Orange is the upper bound from Ref. Wan_2022. Blue is the bound from Eq. \ref{['eq:tighter_bound']} and . green is the actual infinity norm difference between $Q_{\beta,K}$ and the scaled error function $\erf(\sqrt{2 \beta}x)$ estimated numerically by evaluation of both functions on a fine grid.
• Figure 3: Illustration of $C_{+}+C_{-}$ and its noisy approximation respectively for the DMRG-approximated spectrum of $\mathrm{Fe_4S_4}$ with the jumps, due to lowest five $\arccos(\lambda_k)$ visible. The values $\tilde{C}_{+}(x) + \tilde{C}{-}(x)$ for all $x$ were evaluated from a single set of Chebychev expectation values according to Eq. \ref{['eq:sum_approx']} with $K$ and $M$ chosen such that $\delta \leq 5 \times 10^{-5}$. The initial state and $p_0$ are identical to those used in Figure \ref{['fig:fig1']} and elsewhere in this work.
• Figure 4: Comparison of various bounds on the error $\Delta E$ to the actual performance as a function of the maximal Chebyshev polynomial degree $K$ for the different molecules considered in this work. The dashed line is the upper bound from Ref. Wan_2022. The solid line is the bound implied by Eq. \ref{['eq:tighter_bound']} and the dash-dotted line is additionally taking into account the error propagation due to spectral amplification according to Eq. \ref{['eq:error_prop']}. The energy precision actually achieved by in Hartree is plotted as dots for the $M$ and $K$ inferred from the tightest bound, as was done in Figure \ref{['fig:fig1']}
• Figure 5: Absolute energy error as a function of $K$, the maximum Chebyshev polynomial order used in the algorithm, for different molecules, Hamiltonian representations, and initial-state overlaps.