Counterdiabatic Driving with Performance Guarantees

Abstract

Counterdiabatic (CD) driving has the potential to speed up adiabatic quantum state preparation by suppressing unwanted excitations. However, existing approaches either require intractable classical computations or are based on approximations which do not have performance guarantees. We propose and analyze a non-variational, system-agnostic CD expansion method and analytically show that it converges exponentially quickly in the expansion order. In finite systems, the required resources scale inversely with the spectral gap, which we argue is asymptotically optimal. To extend our method to the thermodynamic limit and suppress errors stemming from high-frequency transitions, we leverage finite-time adiabatic protocols. In particular, we show that a time determined by the quantum speed limit is sufficient to prepare the desired ground state, without the need to optimize the adiabatic trajectory. Numerical tests of our method on the quantum Ising chain show that our method can outperform state-of-the-art variational CD approaches.

Figures (8)

• Figure 1: Universal counterdiabatic driving. (a) Our CD scheme seeks to suppress transitions from the ground state to the rest of the eigenspectrum during adiabatic quantum many-body state preparation. The figure shows the transition frequencies as a function of time with $\Delta$ the minimum spectral gap and $\Omega$ the largest transition frequency. In our scheme, we target the transitions with frequencies in the interval $[\Delta, \omega_{\max}]$ (grey shaded area), where $\omega_{\max}$ is a cutoff frequency chosen to enhance the protocol's efficiency. (b) Universal CD driving suppresses these transitions by approximating the adiabatic gauge potential (AGP) matrix elements over these transition frequencies by a polynomial of degree $2d-1$, whose coefficients depend only on the interval bounds $[\Delta,\omega_{\max}]$. For frequencies $\omega>\omega_{\max}$, the polynomial grows as $\omega^{2d-1}$.
• Figure 2: Universal CD driving at finite sizes. (a) Ground-state infidelity for an integrable and nonintegrable point of the Ising model with $L=6$ in the limit of $\tau\to 0$, as a function of the expansion order $d$. The universal scheme is displayed for the choice $\omega_{\max}=\Omega$ (teal squares) and for the optimal choice of $\omega_{\max}$ at each expansion order $d$ (see inset in (a) for $d=5$ and purple triangles). Our scheme has comparable performance to variational AGP (red circles). The analytical scaling from Eq. \ref{['eq:infidelity-only-poly']} is shown in dashed black. (b) System size scaling of the infidelity in the nonintegrable case for $d=14$.
• Figure 3: Universal CD driving at finite timescales. (a) Combining the universal polynomial approximation of order $d\sim 1/\Delta$ with adiabatic protocols of duration $\tau\sim 1/\Delta$ allows us to construct a scheme that converges in the TD limit. The cut-off frequency $\omega_{\max}$ can be chosen as some system size independent local energy scale $J$. Above $\omega_{\max}$ the transitions are compensated by the frequency space decay of $\dot{\lambda}$, here shown in purple. (b) We show the time fidelities for the Universal and Variational approach for the nonintegrable Ising model with $h_z=J/\varphi$ and $L=6$. Fidelities of a bare adiabatic evolution plotted (dashed black lines) are plotted for comparison. For large $\tau$ the universal scheme improves upon the variational approach due to the additional flexibility of tuning $\omega_{\max}$ as $\tau$ is increased.
• Figure S1: (a) Error of the polynomial approximation for $\Delta=0.1$ and $\omega_{\max}=1$, compared to the analytically predicted scaling. (b) The expansion order $d$ required to construct a polynomial that deviates at most $\varepsilon=10^{-2}$ from $1/\omega$ on the interval $[\Delta, \omega_{\max}]$. For comparison, we plot the asymptotically predicted scaling of the degree.
• Figure S2: The derivatives (a) and the protocols defined in Eq. \ref{['seq:protocols']} themselves (b), for different convolution orders $k$. In (c) these protocols are used to adiabatically prepare the ground state for the system defined in the main text, for $h_z=J/\varphi$.