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arXiv:2410.02520 · PRX Quantum

Fighting Exponentially Small Gaps by Counterdiabatic Driving

András Grabarits, Federico Balducci, Adolfo del Campo

Published Nov 19, 2025• 4 citations
arXivHTMLPDF

Abstract

We investigate the efficiency of approximate counterdiabatic driving (CD) in accelerating adiabatic passage through exponentially small gaps. First, we analyze a minimal spin-glass bottleneck model that is analytically tractable and exhibits both an exponentially small gap at the transition point and a change in the ground state that involves a macroscopic rearrangement of spins. Using the variational Floquet-Krylov expansion to construct CD terms, we find that while the formation of excitations is significantly suppressed, achieving a fully adiabatic evolution remains challenging. Extending our investigation to realistic NP-hard spin-glass problems—specifically, the 3-regular and 3-—we find again that local CD expansions lead to negligible improvements in the final ground-state fidelity. These results highlight the limited impact of local CD methods in overcoming the bottlenecks associated with first-order quantum phase transitions. To address this limitation, we propose an alternative method, termed quantum brachistochrone counterdiabatic driving (QBCD), which employs the approximate full CD connecting the ground state and the first-excited state at a single-parameter value close to the critical point. In the minimal spin-glass model, QBCD enables exponentially faster adiabatic evolution than the local strategies. To alleviate the challenges of its experimental and classical implementation for realistic -hard problems, we exponentially reduce the nonlocality of the QBCD Hamiltonian by sparsifying its matrix elements to the density of the local expansions. Despite this drastic simplification, sparsified QBCD maintains finite ground-state fidelity at driving times exponentially shorter than in local strategies and counterdiabatic optimized local driving, demonstrating that even an exponentially reduced fraction of nonlocality near the critical point is sufficient to achieve significant speedup across spin-glass bottlenecks. At the same time, reduced nonlocality facilitates classical implementations via advanced numerical methods, while on quantum hardware its truncation serves as a resource-efficient approximation scheme for higher-order CD realizations.

DOI
10.1103/tgzt-dy3h

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We investigate the efficiency of approximate counterdiabatic driving (CD) in accelerating adiabatic passage through exponentially small gaps. First, we analyze a minimal spin-glass bottleneck model that is analytically tractable and exhibits both an exponentially small gap at the transition point and a change in the ground state that involves a macroscopic rearrangement of spins. Using the variational Floquet-Krylov expansion to construct CD terms, we find that while the formation of excitations is significantly suppressed, achieving a fully adiabatic evolution remains challenging. Extending our investigation to realistic NP-hard spin-glass problems—specifically, the 3-regular and 3-—we find again that local CD expansions lead to negligible improvements in the final ground-state fidelity. These results highlight the limited impact of local CD methods in overcoming the bottlenecks associated with first-order quantum phase transitions. To address this limitation, we propose an alternative method, termed quantum brachistochrone counterdiabatic driving (QBCD), which employs the approximate full CD connecting the ground state and the first-excited state at a single-parameter value close to the critical point. In the minimal spin-glass model, QBCD enables exponentially faster adiabatic evolution than the local strategies. To alleviate the challenges of its experimental and classical implementation for realistic -hard problems, we exponentially reduce the nonlocality of the QBCD Hamiltonian by sparsifying its matrix elements to the density of the local expansions. Despite this drastic simplification, sparsified QBCD maintains finite ground-state fidelity at driving times exponentially shorter than in local strategies and counterdiabatic optimized local driving, demonstrating that even an exponentially reduced fraction of nonlocality near the critical point is sufficient to achieve significant speedup across spin-glass bottlenecks. At the same time, reduced nonlocality facilitates classical implementations via advanced numerical methods, while on quantum hardware its truncation serves as a resource-efficient approximation scheme for higher-order CD realizations.
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 that do not have performance guarantees. We propose and analyze a nonvariational, 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.
Adiabatic control is a fundamental technique for manipulating quantum systems, guided by the quantum adiabatic theorem, which ensures suppressed nonadiabatic transitions under slow parameter variations. Quantum annealing, a heuristic algorithm leveraging adiabatic control, seeks the ground states of Ising spin glass models and has drawn attention for addressing combinatorial optimization problems. However, exponentially small energy gaps in such models often necessitate impractically long runtime to satisfy the adiabatic condition. Despite this limitation, improving the quality of approximate solutions remains crucial for practical applications. The quantum adiabatic brachistochrone provides a method to enhance adiabaticity by minimizing an action representing nonadiabaticity via the variational principle. While effective, its implementation requires detailed energy spectra, complicating its use in quantum annealing. Shortcuts to adiabaticity by counterdiabatic driving offer alternative approaches for accelerating adiabatic processes. However, the theory of shortcuts to adiabaticity often faces challenges such as nonlocal control requirements, high computational cost, and trade-offs between speed and energy efficiency. In this work, we propose a novel quantum adiabatic brachistochrone protocol tailored for quantum annealing that eliminates the need for energy spectrum information. Our approach builds on advancements in counterdiabatic driving to design efficient parameter schedules. We demonstrate the effectiveness of our method through numerical simulations on the transverse-field Ising chain and axial next-nearest neighbor Ising models.
Counterdiabatic driving (CD) offers a fast and robust strategy to manipulate and prepare the quantum state in multibody systems. However, the exact CD term involving the spectral properties of the system is difficult to calculate and generally takes a complicated form for higher-dimensional complex systems. Recently, Sels and Polkovnikov [Proc. Natl. Acad. Sci. U. S. A. 114, E3909 (2017)] proposed a variational method to optimally approximate the CD term with no need for the full spectral information of the original Hamiltonian. Using a nuclear-magnetic-resonance setup, here we report an experimental demonstration of this method to construct available shortcuts to adiabaticity in a nonintegrable spin-1/21/21/2 chain via only local controls. In this process, the ground state of such a system is rapidly prepared and the final state fidelity is significantly increased compared with the traditional adiabatic method. Our experiments prove the feasibility of this variational method to design CD in practice, and thus it is a promising strategy for fast, high-fidelity quantum-state manipulation in complex and noisy quantum systems.
A system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. The realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model (del Campo et al 2012 Phys. Rev. Lett. 109 115703 ). We derive an exact closed-form expression for all the coefficients of the counterdiabatic Ising Hamiltonian. We discuss two approximations to the exact counterdiabatic Ising Hamiltonian quantifying their efficiency of the dynamical preparation of the desired ground state. In particular, these studies show how quantum criticality enhances finite-size effects in counterdiabatic dynamics.
We introduce an inertial counterdiabatic (CD) protocol that reformulates shortcut-to-adiabaticity design as a moving-frame diagonalization problem. The method employs the Magnus expansion to construct a variational propagator generated by an accessible CD ansatz. In the new rotating frame, the controls are optimized to suppress the off-diagonal Hamiltonian terms, and the CD terms are defined to counteract the corresponding inertial contribution. The propagator generated with the Magnus expansion naturally exploits additional control directions generated through its nested commutator, while the sequential design of the propagator introduces temporal nonlocality into the optimization. We benchmark the method on a two-qubit system and on the one-dimensional quantum Ising chain, where it consistently outperforms variational CD protocols based on action minimization. The largest improvements are observed near gap-closing regions, demonstrating the advantage of combining temporal and algebraic nonlocality for suppressing diabatic excitations.
Local counterdiabatic driving (CD) provides a feasible approach for realizing approximate adiabatic or reversible processes like quantum state preparation using only local controls and without demanding excessively long protocol times. However, in many instances getting high accuracy of such CD protocols requires engineering very complicated new controls or pulse sequences. In this work, we describe a systematic method for altering the adiabatic path by adding extra local controls along which performance of local CD protocols is enhanced, both close to and far from the adiabatic limit. We also identify an iterative procedure to improve the performance of local counterdiabatic driving further without any knowledge of the quantum wave function. We then show that these methods provides dramatic improvement in the preparation of nontrivial Greenberger-Horne-Zeilinger ground states of several different spin systems with both short-range and long-range interactions.