Trotter Error and Orbital Transformations in Quantum Phase Estimation

TL;DR

It is found that localised orbital bases do not produce large Trotter errors in molecular calculations, which is an important result for efficient QPE set-ups.

Abstract

Quantum computation with Trotter product formulae is straightforward and requires little overhead in terms of logical qubits. The choice of the orbital basis significantly affects circuit depth, with localised orbitals yielding lowest circuit depths. However, literature results point to large Trotter errors incurred by localised orbitals. Here, we therefore investigate the effect of orbital transformations on Trotter error. We consider three strategies to reduce Trotter error by orbital transformation: (i) The a priori selection of an orbital basis that produces low Trotter error. (ii) The derivation of an orbital basis that produces a ground state energy free of Trotter error (as we observed that the Trotter error is a continuous function in the Givens-rotation parameter, from which continuity of this error upon orbital transformation can be deduced). (iii) Application of propagators that change the computational basis between Trotter steps. Our numerical results show that reliably reducing Trotter error by orbital transformations is challenging. General recipes to produce low Trotter errors cannot be easily derived, despite analytical expressions which suggest ways to decrease Trotter error. Importantly, we found that localised orbital bases do not produce large Trotter errors in molecular calculations, which is an important result for efficient QPE set-ups.

Figures (8)

• Figure 1: Top: Trotter energy errors $\lvert \Delta E_0 \rvert$ obtained to precision $\epsilon=10^{-5}$$E_{\mathrm{h}}$ (see Eqs. \ref{['dE_to_precision_epsilon']}-\ref{['E_0_Trotter_via_IFT']}) in various common orbital bases. Bottom: perturbation theory energy error estimate $\epsilon_2$ defined in Eq. \ref{['epsilon_2_def']}. Results were obtained for small atomic systems in a sto-3g basis. For evolution times $t$ per Trotter step see Table \ref{['tab:system_oview']}. The fermionic Hamiltonian representation and a magnitude ordering of the Trotter series were employed.
• Figure 2: Trotter energy error $\lvert \Delta E_0 \rvert$ obtained to precision $\epsilon=10^{-5}$$E_{\mathrm{h}}$ (see Eqs. \ref{['dE_to_precision_epsilon']}-\ref{['E_0_Trotter_via_IFT']}) for $\pi$-systems with active spaces of up to 6 spatial orbitals in various common orbital bases. The fermionic Hamiltonian representation and a magnitude ordering of the Trotter series were employed. For evolution times $t$ per Trotter step and for the abbreviations of the different molecules, we refer to Table \ref{['tab:system_oview']}.
• Figure 3: Estimates of Trotter error for the investigated $\pi$-systems. The fermionic Hamiltonian representation and magnitude ordering of the Trotter series were employed. Top: perturbation theory energy error estimate $\epsilon_2 t^2$. Middle: ACF offset $g(t)$. Bottom: infidelity $|1-f(t)|$.
• Figure 4: Trotter errors $\lvert \Delta E_0 \rvert$ obtained to precision $\epsilon=10^{-5}$$E_{\mathrm{h}}$ (see Eqs. \ref{['dE_to_precision_epsilon']}-\ref{['E_0_Trotter_via_IFT']}) for active spaces of various $\pi$-systems in different orbital bases. Thin, dashed bars: a (basis-dependent) magnitude ordering of the Trotter series was employed (data previously shown in Fig. \ref{['fig:med_orb_vs_dE']}). Thick bars: the terms in the Trotter series were reordered to match the magnitude ordering of the canonical orbital basis.
• Figure 5: Trotter energy error estimates $\epsilon_2 t^2$ (top) and time-independent Trotter energy error estimates $\epsilon_2$ (bottom) for active spaces of various $\pi$-systems in different orbital bases. Thin, dashed bars: a magnitude ordering of the Trotter series was employed (data previously shown in Fig. \ref{['fig:correlating_quantites_larger_pi_sys']}). Thick bars: the terms in the Trotter series were reordered to match the magnitude ordering of the canonical orbital basis.