Universal Digitized Counterdiabatic Driving

TL;DR

The paper addresses constructing counterdiabatic driving for parametrized quantum Hamiltonians by digitizing the adiabatic gauge potential $\hat{\mathcal{A}}(\lambda)$ without introducing additional many-body or nonlocal terms. It introduces the digital composite unitary $\hat{U}(\lambda)=\prod_{k=-K,k\neq0}^{K} e^{i\theta_k \hat{H}(\lambda)} e^{-i\frac{\phi_k}{2}\partial_\lambda \hat{H}(\lambda)} e^{-i\theta_k \hat{H}(\lambda)}$ with rotation angles $\theta_k=\frac{k\pi}{\Omega}$ and $\phi_k=-\mathrm{sgn}(k)\frac{2\delta\lambda}{\Omega}\mathrm{Si}(\theta_k\Omega)$. The resulting generator $\hat{V}(\lambda)$ comprises infinite nested commutators via $\mathcal{L}^l\partial_\lambda\hat{H}(\lambda)$ and can be analyzed by Fourier expansion to minimize $\|\hat{\mathcal{A}}(\lambda)-\hat{V}(\lambda)\|$. Numerical tests in a two-level system and a many-body model show very low infidelity and robust periodic behavior as a function of the digitization parameter $K$ and cutoff $\Omega$, with the cutoff not necessarily equal to the maximum gap $\Delta_{\max}$.

Abstract

Counterdiabatic driving realizes parameter displacement of an energy eigenstate of a given parametrized Hamiltonian using the adiabatic gauge potential. In this paper, we propose a universal method of digitized counterdiabatic driving, constructing the adiabatic gauge potential in a digital way with the idea of universal counterdiabatic driving. This method has three advantages over existing universal counterdiabatic driving and/or digitized counterdiabatic driving: it does not introduce any many-body and/or nonlocal interactions to an original target Hamiltonian; it can incorporate infinite nested commutators, which constitute the adiabatic gauge potential; and it gives explicit expression of rotation angles for digital implementation. We show the consistency of our method to the exact theory in an analytical way and the effectiveness of our method with the aid of numerical simulations.

Figures (3)

• Figure 1: The infidelity to the ground state of the many-body system with respect to the integer $K$. The parameters are $J(\lambda)=J_0=-1$, $h^X(\lambda)=h_0^X\lambda=1$ ($h_0^X=1$ and $\lambda=1$), $\delta\lambda=10^{-3}$, $N=10$, and $\Omega=\Delta_\mathrm{max}=20.278\cdots$. The black solid line represents the infidelity under the parameter quench and the black dashed lines represent $kK_p^{\Omega=\Delta_\mathrm{max}}/2$ with $k=1,2,\dots$.
• Figure 2: Difference from the exact adiabatic gauge potential with respect to the energy differences. Each curve represents (red solid curve) $K=4$, (green dashed curve) $K=8$, (blue dotted curve) $K=13$, and (yellow dashed-dotted curve) $K=17$. The parameters are the same as Fig. \ref{['Fig.LMG']}. The gray shaded areas are out of $[\Delta_\mathrm{min},\Delta_\mathrm{max}]$. The inset is an enlarged view around the minimum energy gap $\Delta_\mathrm{min}$.
• Figure 3: The infidelity to the ground state of the many-body system with respect to the integer $K$. The parameters are the same as Fig. \ref{['Fig.LMG']} except for the cutoff $\Omega=15$ instead of the maximum energy gap $\Delta_\mathrm{max}=20.278\cdots$.