Exponential Scaling Barriers for Variational Quantum Eigensolvers

Abstract

The Variational Quantum Eigensolver (VQE) is widely regarded as a promising algorithm for calculating ground states of quantum systems that are intractable for classical computers. This promise is typically motivated by the hope of mitigating the exponential growth of Hilbert space with system size. Here we scrutinize how the computational cost of adaptive VQE scales with the size of the target system. We demonstrate that the Rényi entropy derived from classical simulations predicts the required number of adaptive iterations of VQE with high accuracy ($R^2 \approx 0.99$). We validate this on a benchmarking set of more than 20 different molecules with active spaces ranging from four to ten orbitals. For these molecules, we find an exponential scaling of the number of adaptive iterations, and in turn, of the circuit depth with the system size. We therefore conclude that it is unlikely that VQE in its current form is able to simulate large molecular systems with high fidelity without exponential resource requirements.

Figures (16)

• Figure 1: (a) Schematic overview of the study. For each molecule in the benchmark set (Table \ref{['tab:molecule_set']}), ADAPT-VQE simulations determine the number of iterations $n_\mathrm{ADAPT}$ required to reach a target energy error $\varepsilon$. The Rényi entropy $h_\alpha$, computed from the CI coefficient distribution of classical CASSCF calculations, serves as a complexity metric to predict $n_\mathrm{ADAPT}$. (b) Correlation between the Rényi entropy $h^\ast$ and $n_\mathrm{ADAPT}$ at chemical accuracy ($\varepsilon_\mathrm{chem}$) across the full benchmark set. Here, $h^\ast \equiv h_{\alpha^\ast}$ denotes the Rényi entropy at the order $\alpha^\ast$ that maximizes the coefficient of determination (R$^2$) at chemical accuracy. We find that $\alpha^\ast \approx 0.25$, yielding R$^2 = 0.948$ (see Section \ref{['sec:quantifying_problem_complexity']}). The linear relationship in this semi-logarithmic plot implies that $n_\mathrm{ADAPT}$ scales exponentially with the Rényi entropy.
• Figure 2: CEO-ADAPT-VQE with TETRIS extension applied to all molecules in Table \ref{['tab:molecule_set']} except hydrogen chains. (a) Energy error $\varepsilon$ versus number of ADAPT iterations $n_{\mathrm{ADAPT}}$. (b) Energy error versus largest gradient magnitude $g^\ast$ among all operators in the pool at each iteration. (c) Energy error versus deviation $|\langle\hat{S}^2\rangle - S(S+1)|$ from the expected spin eigenvalue. The shaded region indicates $\varepsilon < \varepsilon_{\mathrm{chem}}$ (chemical accuracy).
• Figure 3: Relationship between the Rényi entropy $h_\alpha$ and the required number of ADAPT iterations $n_{\mathrm{ADAPT}}$ to reach an energy error relative to the target state $\varepsilon$. The coefficient of determination (R$^2$) score of a linear fit between log($n_{\mathrm{ADAPT}}(\varepsilon)$) and $h_\alpha$ is shown for the seven six-orbital molecules described in Table \ref{['tab:molecule_set']} in the given basis and spin states using QEB-ADAPT-VQE without TETRIS. The $\alpha$ resulting in the highest R$^2$ scores is indicated in red for each $\varepsilon$ and the best $\alpha$ at chemical accuracy $\varepsilon_{\mathrm{chem}}$ is marked (black).
• Figure 4: Comparison of (a) the number of ADAPT iterations $n_{\mathrm{ADAPT}}$ and (b) the scaling of $n_{\mathrm{ADAPT}}$ required to reach chemical accuracy $\varepsilon_{\mathrm{chem}}$ as a function of system size $N$, for hydrogen chains consisting of four to ten atoms in equilibrium geometry (1.0 Å interatomic distance). In panel (a), the vertical axis shows the energy error relative to the target state $\varepsilon$, with the chemical accuracy $\varepsilon_{\mathrm{chem}}$ highlighted. Panel (b) shows an exponential scaling behavior (black line) with the fit equation and coefficient of determination $R^2$ indicated. The data point for H$_{10}$ is marked with an "X" as it was obtained by extrapolation.
• Figure 5: Extrapolated number of ADAPT iterations $n_\mathrm{ADAPT}$ required to reach chemical accuracy ($\varepsilon_\mathrm{chem}$), estimated from the Rényi entropy $h^\ast$ using the linear fit in Figure \ref{['fig:paper_overview_sketch']}(b). Here, $h^\ast \equiv h_{\alpha^\ast}$ denotes the Rényi entropy at the order $\alpha^\ast \approx 0.25$ that maximizes the coefficient of determination (R$^2$) at chemical accuracy. The "pred. band" indicates the 95% confidence interval. For H$_{15}$, the square marker shows the independent estimate obtained from the system-size scaling fit in Figure \ref{['fig:Hn_chain_equi']}(b).