On the robustness of Quantum Phase Estimation to compute ground properties of many-electron systems

TL;DR

This work analyzes the application of Quantum Phase Estimation to electronic-structure problems, focusing on how to constructively choose free parameters to accurately estimate the ground energy $E_0$ and potentially project the ground state $|\psi_0\rangle$. It derives explicit conditions for the time step $t$, the number of phase qubits $N$, initial-state quality, the number of measurement shots, and the accuracy of unitary implementations (e.g., via Trotterization), ensuring chemical accuracy $\varepsilon_{\rm ch.acc}$. A key finding is that, to first order, the Trotterized QPE cost scales with physical system features and not with $N$, making the approach more favorable as system size grows; the overall effort is dominated by the unitary decomposition and its approximation. Numerical experiments on H$_2$ illustrate the practical impact of parameter choices on energy accuracy and ground-state projection, and provide actionable guidance for automating QPE workflows in predictive chemistry and materials science.

Abstract

We propose an analysis of the Quantum Phase Estimation (QPE) algorithm applied to electronic systems by investigating its free parameters such as the time step, number of phase qubits, initial state preparation, number of measurement shots, and parameters related to the unitary operators implementation. A deep understanding of these parameters is crucial to pave the way towards more automation of QPE applied to predictive computational chemistry and material science. To our knowledge, various aspects remain unexplored and a holistic parameter selection method remains to be developed. After reviewing key QPE features, we propose a constructive method to set the QPE free parameters. We derive, among other things, explicit conditions for achieving chemical accuracy in ground energy estimation. We also demonstrate that, using our conditions, the complexity of the Trotterized version of QPE tends to depend only on physical system properties and not on the number of phase qubits. Numerical simulations on the H2 molecule provide a first validation of our approach.

Figures (7)

• Figure 1: QPE algorithm, with outputs $\left(\textcolor{brown}{l^*}, \textcolor{brown}{\ket{\psi_{\rm out}^{(l^*)}}}\right)$ and free parameters $\left(\pmb{t},\pmb{N},\pmb{\ket{\psi_{\rm init}}},\pmb{m_\epsilon},\pmb{n}\right)$ studied in this article. Our objective is to provide a constructive method to set these parameters.
• Figure 2: $|f(\theta_j^{(t)}-\frac{l}{2^N})|^2$ as a function of the discrete variable $\frac{l}{2^N}$ with a fixed value of $\theta_j^{(t)}$, for several values of $N$. The right figure is a zoom of the left figure around $\theta_j^{(t)}$. Vertical lines denotes the $\frac{l_j^{(N)}}{2^N}$ value (with highest probability $\lvert f(\theta_j^{(t)}-\frac{l_j^{(N)}}{2^N}) \rvert^2$).
• Figure 3: System with $4$ eigenstates and randomly generated $\theta_j^{(t)}$. Top: $\lvert f(\theta_j^{(t)}-\frac{l}{2^N}) \rvert^2$ as a function of $\frac{l}{2^N}$ for several $N$ values. Middle: $P(l)$ as a function of $\frac{l}{2^N}$ for several $N$ values. Bottom : Overlap between each exact eigenstate of $H$ with the initial state ($|c_j|^2$, grey) and the final state after phase measurement ($|c_j^{(l^*)}|^2$, color), for several $N$ values.
• Figure 4: Upper bound of the probability $\epsilon=e^{-m_\epsilon\frac{\Delta_{(l^*)}^2}{2}}$ such that $l^*$ is not the most read phase as a function of the number of shots (taking a sufficiently small value such as $\epsilon=10^{-2}$ is pertinent for our applications). The values of $\theta_j^{(t)}$ and $c_j$ are the same than in Fig. \ref{['fig_f_discrete_1']}.
• Figure 5: Implementation of $U^{2^q}$ using first-order Trotterization ($p=1$).