Refining Quantum Phase Estimation Precision Conditions on Unitaries for Many-Electron Systems

Abstract

Beyond ground state energy estimation, quantum phase estimation (QPE) applied to many-electron systems has the potential to output a projection on the ground state, that would enable the evaluation of observables other than the energy. In this article, after recalling the role of QPE free parameters, we detail the derivation of first-order and unified conditions on unitaries that allow us to control the energy estimation precision and lead to tighter bounds than in previous works. We then introduce a novel condition that allows us to also control the state projection precision. We apply these conditions to a Trotterization case, leading to tighter bounds than the previous ones. The main results in this article are formal, with a first numerical illustration on the H2 molecule that allows us to derive useful insights.

Figures (2)

• Figure 1: QPE results for H_2 with $t=1/(2\sum_\beta \lvert\gamma_\beta\rvert)$. Plot of the various quantities in the caption related to \ref{['eq:unit']}, \ref{['eq:cond_unit']}, \ref{['eq:cond_unit3']}, \ref{['eq:dHproj']}, \ref{['trotter_1st']} and \ref{['trotter_1st-bis']}, as a function of the number of Trotter steps $n(0,t)$. Trace distance and unitaries are rescaled on an energy using the good factors ($\Delta E_0/A_0$ and $1/(2\pi t)$ respectively). The chemical precision area appears in grey and corresponds to a trace distance precision $\alpha_{\rm ch}=2.3\times10^{-3}$ (unitless), see \ref{['eq:alpha.ch']}.
• Figure 2: QPE results for H_2 with $t=1/(2\sum_\beta \lvert\gamma_\beta\rvert)$. The plots are function of the number of phase qubits $N$ and correspond to various Trotterization steps $n_{\min}^{(1)}(q,t)$. Evolution of energy differences (red) and trace distance (blue, rescaled on an energy using $\Delta E_0/A_0$) w.r.t. initial (HF) and exact ground state quantities are shown. The chemical precision area appears in grey and corresponds to a trace distance precision $\alpha_{\rm ch}=2.3\times10^{-3}$ (unitless), see \ref{['eq:alpha.ch']}.