Less is more: subspace reduction for counterdiabatic driving of Rydberg atom arrays

TL;DR

This work addresses the scalability challenge of exact counterdiabatic driving in adiabatic quantum computation by introducing a subspace reduction approach for MIS solved on Rydberg atom arrays. It combines a subspace projection onto independent-set configurations with both exact diagonalization and Krylov-based methods to construct the gauge potential $A_{\lambda}(t)$, and it introduces a subspace-based cost function $\langle \cdot,\cdot\rangle_{F,sub}$ to further enhance fidelity. The results show that restricting to nearest-neighbor and next-nearest-neighbor independent-set subspaces maintains or improves final fidelity while dramatically reducing computational cost, with fidelities approaching or exceeding $0.99$ in favorable cases and Krylov convergence accelerated within the subspace. The findings suggest a practical path to applying counterdiabatic driving to larger quantum systems and other constrained optimization problems by exploiting the structure of the relevant subspace. These advances hold potential for more efficient quantum annealing and Floquet-engineered implementations in MIS and related quantum optimization tasks.

Abstract

This study explores the use of subspace methods in combination with counterdiabatic driving in a Rydberg atom system to solve the Maximum Independent Set (MIS) problem. Although exact counterdiabatic driving offers excellent performance, it comes at an unscalable computational cost. In this work, we demonstrate that counterdiabatic driving can be significantly improved by restricting the analysis to a relevant subspace of the system. We first show that both direct diagonalization and the Krylov method for obtaining the counterdiabatic matrix can be accelerated through the use of subspace techniques, while still maintaining strong performance. We then demonstrate that the cost function used in the standard Krylov method can be further optimized by employing a subspace-based cost function. These findings open up new possibilities for applying counterdiabatic driving in a practical and efficient manner to a variety of quantum systems.

Figures (4)

• Figure 1: (a) Representative 11-vertex graphs illustrating independent sets, maximal independent sets, and maximum independent sets, with each example highlighting the distinctions among these categories. (b) Example control waveforms for the Rabi frequency $\Omega(t)$ and detuning $\Delta(t)$ used in maximum independent set searches. Vertical blue and light-coral lines indicate the time slices used for evaluating the corresponding adiabatic gauge-potential matrices discussed later.
• Figure 2: (a) The error measure $1-F_{s}^2$ plotted on a logarithmic scale as a function of the number of atoms. The shaded bands indicate one standard deviation, and the solid line shows the average over all instances. The gray line gives the final fidelity without any counterdiabatic (CD) driving. The red and amber yellow lines show the fidelities obtained using the counterdiabatic terms derived from the subspace method with nearest-neighbor and next-nearest-neighbor subspaces, while the blue line shows the results from full space counterdiabatic driving. (b) Final fidelity $F_{s}$ as a function of the order of the Krylov expansion (equivalently, the dimension of the Krylov space) $l$, evaluated over 150 distinct 11-atom configurations. Red and amber yellow markers indicate results from the Krylov method using submatrices restricted to nearest-neighbor and next-nearest-neighbor interactions. Results from the full space Krylov method and the no driving case are shown in blue and gray for comparison. Error bars denote one standard deviation. The red and amber yellow shaded regions indicate the upper bound achievable by the Krylov method, corresponding to the mean ± one standard deviation obtained from subspace diagonalization.
• Figure 3: Distribution of occurrences as a function of the negative log final fidelity $-\ln(F_{s})$ for 11-atom configurations. The left panel shows statistics over 150 instances, while the right panel highlights the distribution for the hardest graph instances. Improvements are shown for three approaches: the standard full space Krylov method (blue), the Krylov method with a nearest-neighbor (NN) submatrix cost function (red), and the next-nearest-neighbor (NNN) submatrix variant (amber yellow). For each method, results are provided for the Krylov orders $l=3, 6, 9$.
• Figure 4: (a) Illustration of nearest-neighbor (NN) and next-nearest-neighbor (NNN) exclusions. For each atom, for example the selected red one in the graph, all the states involving the simultaneous excitation of two atoms connected by red edges are excluded from the counterdiabatic calculations. For the next-nearest-neighbor calculations, the simultaneous excitation of the atoms connected by red and amber yellow edges is excluded. This procedure is applied to each atom until all edges are considered. The large semi-transparent circles represent the radius to identify the nearest-neighbor (red) and next-nearest-neighbor (amber yellow) for the selected atom (red) in the graph. (b) Magnitude of elements of the Rabi-drive counterdiabatic term $\dot{\Omega}(t)A_{\Omega}(t)$ in energy and configuration bases, shown in log scale. In the energy eigenspace, each row and column represent an energy eigenstate ranked by their corresponding eigenvalues from low to high. In the configuration space, the elements selected by the nearest-neighbor subspace are marked by light cyan squares, and those selected by the next-nearest-neighbor subspace are marked by chocolate squares. (c) Same as panel (b), but for the detuning counterdiabatic term $\dot{\Delta}(t)A_\Delta(t)$. (d) An example of an 11-atom configuration with one atom highlighted to show its nearest-neighbor and next-nearest neighbor edges. As the number of atoms grows, the number of edges increases and further reduces the dimension of states in the subspace.