Improving variational counterdiabatic driving with weighted actions and computer algebra

TL;DR

This work tackles the challenge of implementing high-fidelity, finite-time adiabatic processes in quantum many-body systems by refining variational counterdiabatic driving. By exploiting the non-uniqueness of the adiabatic gauge potential, the authors introduce weighted actions, where a fictitious polynomial of the Hamiltonian concentrates the variational effort on chosen energy sectors, and incorporate nonlocal information through higher-degree polynomials. They develop a computer-algebra-based algorithm to compute refined driving protocols in polynomial time and demonstrate substantial fidelity gains in quantum Ising models, including ferromagnetic, antiferromagnetic, and spin-glass cases, with gains growing with system size. The framework is broadly applicable and can, in principle, replace conventional variational CD driving, potentially enabling more robust state preparation and quantum information tasks in large-scale quantum systems.

Abstract

Variational counterdiabatic (CD) driving is a disciplined and widely used method to robustly control quantum many-body systems by mimicking adiabatic processes with high fidelity and reduced duration. Central to this technique is a universal structure of the adiabatic gauge potential (AGP) over a parameterized Hamiltonian. Here, we reveal that introducing a new degree of freedom into the theory of the AGP can significantly improve variational CD driving. Specifically, we find that the algebraic characterization of the AGP is not unique, and we exploit this non-uniqueness to develop the weighted variational method for deriving a refined driving protocol. This approach extends the conventional method in two aspects: it assigns customized weights to matrix elements relevant to specific problems, and it effectively incorporates nonlocal information. We also develop an efficient numerical algorithm to compute the refined driving protocol using computer algebra. Our framework is broadly applicable and, in principle, it can replace any previous use of variational CD driving. We demonstrate its practicality by applying it to adiabatic evolution along the ground state of a parameterized Hamiltonian. This proposal outperforms the conventional method in terms of fidelity, as confirmed by extensive numerical simulations on quantum Ising models.

Figures (11)

• Figure 1: Schematics of the methods of CD driving. (a) Concept of exact CD driving, depicted in the energy eigenbasis of the original Hamiltonian $H(\lambda)$. Gray curves represent the energy of the eigenstates of $H(\lambda)$. Without CD driving, the population escapes from the target eigenstates (e.g., the ground state in this figure) during the time evolution, as depicted by the red curve (left). CD driving uses a driving force $V(\lambda) =\Phi(\lambda)$, depicted as a "hammer" hitting the system, to ensure that the system correctly tracks the target eigenstate during time evolution (right). Exact CD driving keeps the system perfectly on the target, but it is impractical for many-body systems. (b) Comparison between the conventional and weighted variational methods of CD driving. The conventional variational method tries to design a hammer $V(\bm\alpha(\lambda))$ that hits all eigenstates equally well (left). However, this goal may not be fully achieved with a limited number of driving fields. In contrast, our weighted variational method designs a specialized hammer for important eigenstates (e.g., low-energy states in this figure), providing a tailored approach to a specific problem (right). As a result, our method can more effectively suppress the escape of the population from the target eigenstates, as shown by the red curves.
• Figure 2: Determination of the driving coefficients using the weighted variational method with different values of $K$, exemplified with a ferromagnetic Ising model with $N = 12$. (a) The weight on the eigenstates $w_n^{(K)}(\lambda)$ plotted against the energy eigenvalues $\epsilon_n(\lambda)$ for $K = 1,\dots,5$ (purple--yellow). We use $\lambda = 0.25$ as an example. Each weight is normalized so that the area under the curve is constant for better visualization, as the overall normalization is unimportant. The arrows show the energy shift $E_{\lambda}^{(K)}$ for $K = 2,\dots,5$, which corresponds to the center of the weight function. The energies $\epsilon_{1}$ and $\epsilon_{D}$ are the lowest and the highest energy eigenvalues, and the gray histogram shows the density of states. (b) The driving coefficients $\alpha^{(K)}_{\mu}(\lambda)$ at two representative sites of the one-body driving obtained from the weighted variational method with $K = 1,\dots,5$. Solid curves represent $\alpha^{(K)}_{\mu}(\lambda)$ of one of the sites, and dashed curves are for another site. We plot the coefficients for all sites in Fig. S1 in Supplemental Material Suppli.
• Figure 3: Performance of the weighted variational method in a ferromagnetic Ising model of $N=12$ with the one-body driving. We simulate the time evolution with the driving protocol obtained by the weighted variational method with $K = 1,\dots,5$, where $K = 1$ is the conventional variational method, and $K \geq 2$ are our proposal. We also simulate the time evolution without CD driving, represented by $K = \varnothing$. (a) The time evolution of the fidelity to the ground state $\mathcal{F}^{(K)}(t)$. (b) The final fidelity $\mathcal{F}_{\mathrm{f}}^{(K)}$ plotted over different protocol durations $t_{\mathrm{d}}$.
• Figure 4: Numerical test of the weighted variational method in ferromagnetic systems. We generate 100 instances for every system size $N = 9,12,15$ and perform the weighted variational method for $K = 1,\dots,5$ for each instance. We use the one-body driving and a short duration $t_{\mathrm{d}} = 0.01$. (a) Final fidelity $\mathcal{F}_{\mathrm{f}}^{(K)}$ for different values of degree $K$ and system size $N$, where $K = \varnothing$ is the absence of CD driving, $K = 1$ is the conventional variational method, and $K\geq2$ are our proposal. Circle represents the final fidelity for each instance. Black $\times$ mark shows the median, and the error bar indicates the quartile range calculated for each $(N,K)$. (b) Gain $\mathcal{G}^{(K)}_{\mathrm{f}}$, i.e., the relative increase of the final fidelity compared to the conventional method. The final fidelity $\mathcal{F}_\mathrm{f}^{(K)}$ is divided by $\mathcal{F}_\mathrm{f}^{(1)}$ of the same instance. Black $\times$ mark and error bar are the median and quartile range of the gain. Some outliers are above the upper bound of the axis.
• Figure 5: Numerical test of the weighted variational method in antiferromagnetic systems. We generate $100$ instances for every system size $N = 9,12,15$ and perform simulations similarly to Fig. \ref{['fig:ferromagnetic']}. (a) Final fidelity $\mathcal{F}_{\mathrm{f}}^{(K)}$ with the one-body driving, varying the system size $N$ and the degree $K$. (b) Final fidelity $\mathcal{F}_{\mathrm{f}}^{(K)}$ with the two-body driving. (c) Gain $\mathcal{G}_{\mathrm{f}}^{(K)}$ with the two-body driving. In all panels, some outliers are above the upper bound of the axes.