Universal Counterdiabatic Driving

TL;DR

The paper tackles the challenge of designing fast, approximate counterdiabatic driving without detailed knowledge of the system Hamiltonian. It develops universal local CD protocols by formulating the AGP in Krylov space and approximating $1/ ext{ω}$ with odd Chebyshev polynomials within a controlled window, governed by two scales, $oldsymbol{ extμ}$ and $oldsymbol{ extΩ}$, whose optimal values depend on the high-frequency tail of the spectral function $oldsymbol{ extΦ}_oldsymbol{ extλ}(oldsymbol{ extω})$. A key finding is that convergence and effectiveness hinge on the tail exponent $oldsymbol{ extα}$, with $oldsymbol{ extα}>1$ allowing $oldsymbol{ extμ(ℓ)} o 0$ as $ ext{ℓ} o ext{∞}$, while $oldsymbol{ extα}\le 1$ can prevent improvement in the thermodynamic limit; this reveals a deep link between short-time operator growth and long-time nonadiabatic behavior. The authors illustrate the approach on three models (disordered TFI, Ising with NNN interactions, and XXZ) and show distinct scaling laws for $oldsymbol{ extΩ(ℓ)}$ and $oldsymbol{ extμ(ℓ)}$ tied to the high-frequency tails, as well as a connection to hydrodynamic tails via Lanczos coefficients. Overall, the work provides a practical framework for universal CD control and suggests fundamental connections between short- and long-time quantum dynamics with potential impact on quantum annealing and fast state preparation.

Abstract

Local counterdiabatic (CD) driving provides a systematic way of constructing a control protocol to approximately suppress the excitations resulting from changing some parameter(s) of a quantum system at a finite rate. However, designing CD protocols typically requires knowledge of the original Hamiltonian a priori. In this work, we design local CD driving protocols in Krylov space using only the characteristic local time scales of the system set by e.g. phonon frequencies in materials or Rabi frequencies in superconducting qubit arrays. Surprisingly, we find that convergence of these universal protocols is controlled by the asymptotic high frequency tails of the response functions. This finding hints at a deep connection between the long-time, low frequency response of the system controlling non-adiabatic effects, and the high-frequency response determined by the short-time operator growth and the Krylov complexity, which has been elusive until now.

Figures (9)

• Figure 1: A cartoon illustration of the problem of fitting $1/\omega$ by odd polynomials. There are three regions: $\omega < \mu$ (red, IR), where the fit is poor and the approximate AGP $A_\lambda^{(\ell)}$ fails to replicate the true AGP $A_\lambda$, and so does not suppress low energy excitations. Next, $\mu \leq \omega \leq \Omega$ (green) where the approximation is good and so excitations are suppressed. Finally, $\omega > \Omega$ (violet, UV) where high energy excitations are excited, not by the finite rate of changing a parameter, but by the large error in the CD drive approximation of Eq. \ref{['eqn:commutator_ansatz']}.
• Figure 2: For the transverse field Ising model, with $N = 500$, we compute the $\mu(\ell)$ which maximizes fidelity \ref{['eqn:final-fid']}. We observe the asymptotic scaling of $\mu(\ell) \sim (\log \ell) / \ell$, which is consistent with known mathematical results for the polynomial fitting of $1/\omega$childsQuantumAlgorithmSystems2017. As $\mu(\ell) \rightarrow 0$, low energy excitations are suppressed and we approach unit fidelity. This is seen in the inset when the scaled fidelity $\log\mathcal{F}/N \rightarrow 0$ as $\ell\to\infty$.
• Figure 3: Approximation of $1/x$ by Chebyshev polynomials, where $x = \omega/\Omega$. The inset shows the same plot on a log scale. In each case, the optimal $\mu(\ell)$ that is obtained by maximizing fidelity for the TFI model annealing problem is used, with fixed $\Omega(\ell) = \Omega_{max}$.
• Figure 4: The fidelity density of the final state in the thermodynamic limit after annealing across the Ising transition in the presence of modulated fields. While the variational approach is slightly better due to precise knowledge of the spectral function, the universal protocol offers competitive performance using only knowledge of the local energy scale. A similar plot for a non-integrable model at finite system size in shown in Fig. \ref{['fig:compare-cost-funcs']} in Appendix \ref{['appendix:omega-cost-func']}.
• Figure 5: For the next-nearest-neighbor Ising model with $N = 14$, annealing from the paramagnetic to ferromagnetic phases, the optimal $\Omega(\ell)$ is obtained numerically by maximizing the fidelity defined in Eq. \ref{['eqn:final-fid']}. The dashed line is a linear fit, which is consistent with a general argument from Sec. \ref{['subsec:asymptotes']}. The inset shows the scaled fidelity of the final state. Although the asymptotic behavior of the fidelity cannot be reliably obtained due to finite-size effects, it is expected to plateau at a small non-zero value in the thermodynamic limit $N\to \infty$.