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Fighting Exponentially Small Gaps by Counterdiabatic Driving

András Grabarits1,*, Federico Balducci1,2,†, and Adolfo del Campo1,3

  • *Contact author: andras.grabarits@uni.lu
  • †Contact author: fbalducci@pks.mpg.de
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PRX Quantum 7, 010322 – Published 2 February, 2026

DOI: https://doi.org/10.1103/tgzt-dy3h

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 Max Cut and 3-xorsat—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 np-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.

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Popular Summary

The control of quantum systems via adiabatic processes lies at the heart of current quantum technological developments. Of particular interest are control schemes speeding up adiabatic dynamics across first-order quantum phase transitions. In these scenarios, the adiabatic dynamics can only be maintained by slowing down exponentially with the system size due to the presence of avoided level crossings with exponentially small gaps between macroscopically distinct states. The effect is typically present in models with disorder, such as spin-glass systems encoding combinatorial optimization problems. A successful framework for engineering additional control terms, and accelerating the adiabatic dynamics, is counterdiabatic driving (CD). However, to apply this powerful analytical method to many-body systems such that it can be subsequently realized on real quantum devices, it is necessary to perform some approximations and truncate the nonlocal operators required by the exact theory.

Our work examines how efficiently approximate CD can accelerate the adiabatic passage through the difficult first-order quantum phase transitions in a generic spin-glass bottleneck. We find that while approximate CD can reduce excitation formation during time evolution, it struggles to achieve full adiabatic evolution due to the presence of a small gap. To overcome the problem, we suggest an alternative approach, termed quantum brachistochrone counterdiabatic driving (QBCD). The main idea is to tackle directly the critical point relying solely on the approximate knowledge of the ground state and the first-excited state at a single value of the control parameter. We find that QBCD significantly improves accuracy in reaching the target quantum state, offering a better solution for optimization problems like NP-hard Ising models where local CD strategies fall short.

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PRX Quantum 7, 010322– Published 2 February, 2026
  • Received 17 October 2024
  • Revised 10 October 2025
  • Accepted 19 November 2025
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