Fighting Exponentially Small Gaps by Counterdiabatic Driving

TL;DR

This work examines the efficiency of approximate counterdiabatic driving (CD) for speeding up adiabatic transitions through exponentially small gaps in spin-glass–like systems. It shows that local CD can suppress excitations away from the adiabatic limit but cannot fully overcome the bottleneck, whose gap scales as Δ_min ∼ e^{-α L}, and introduces quantum brachistochrone CD (QBCD) that leverages approximate information about the ground and first excited states near the critical point to achieve exponential speedups, even when nonlocal terms are sparsified. The authors demonstrate the approach on NP-hard problems such as Max Cut and 3-XORSAT, where sparsified QBCD maintains finite ground-state fidelity with reduced nonlocality and lower resource costs than local CD methods. These results indicate that targeted, minimal nonlocal CD terms can dramatically improve adiabatic performance and offer a practical route to implementing fast quantum control in complex many-body systems.

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 \textsc{Max Cut} and $3$-\textsc{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 \textsc{NP}-hard problems, we exponentially reduce the non-locality 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 (COLD).

Figures (19)

• Figure 1: (a) Ising chain of Eqs. \ref{['eq:H']}--\ref{['eq:Jj']}. The chain is $L = 2\ell+1$ sites long, with uniform ferromagnetic bonds (green lines), except for two neighboring, weakly ferromagnetic bonds (light green lines) and a single antiferromagnetic bond (red line). The dashed gray line indicates the presence of the $\mathbb{Z}_2$ reflection symmetries, Eqs. \ref{['eq:reflection_parity']} and \ref{['eq:chiral_reflection_parity']}. (b) The ground state at $\lambda=1$ (fixing $L=9$) has only the antiferromagnetic bond frustrated (dotted line). The other degenerate ground state with opposite fermion parity is not shown. (c)--(d) Two degenerate first excited states at $\lambda=1$ (with $L=9$), characterized by only one weak ferromagnetic bond frustrated (dotted line). Two other degenerate first excited states are obtained by flipping all the spins. (e) The restriction to the even fermion parity sector makes the ground state nondegenerate and the first excited manifold two-dimensional. The positive fermion parity states are obtained by means of symmetric combinations, as shown here for the ground state.
• Figure 2: (a) Sketch of the boundary-localized eigenmodes $|\psi_\mathrm{L})$,$|\psi_\mathrm{R})$ of the effective hopping Hamiltonian \ref{['eq:H_hopping']}. (b) Spectrum of the hopping Hamiltonian \ref{['eq:H_hopping']} for $\ell=8$, $J=0.5$, $J'=0.27$. The many-body ground state of the $\Gamma$ Hamiltonian in Eq. \ref{['eq:H_Gamma']} is formed by populating all the negative-energy modes at $\lambda=0$ (solid lines) and leaving the positive-energy modes empty (dashed lines). The two localized modes $\epsilon_\mathrm{L}$ (blue) and $\epsilon_\mathrm{R}$ (orange) undergo an avoided crossing at $\lambda=\lambda_\mathrm{c}$: while $\epsilon_\mathrm{R}$ contributes to the ground state for $\lambda<\lambda_\mathrm{c}$ (continuous portion of the orange line), it is replaced by $\epsilon_\mathrm{L}$ for $\lambda>\lambda_\mathrm{c}$ (the continuous portion of the blue line). Since the spatial overlap between $|\psi_\mathrm{L})$ and $|\psi_\mathrm{R})$ is exponentially small in $L$, also the gap $\Delta_\mathrm{min}$ remains exponentially small according to Eq. \ref{['eq:Delta_min']}.
• Figure 3: The local approximate counterdiabatic Hamiltonians $H_1^{(1)}$, Eq. \ref{['eq:H1_1st_order']}, and $H_1^{(2)}$, Eq. \ref{['eq:H1_2nd_order']}, take a simple form in the $\Gamma$ free fermionic representation, Eq. \ref{['eq:H_Gamma']}: they are a longer-range hopping terms, with fine-tuned coefficients $i\alpha_j,i\beta_j$. Higher-order variational ansatzes yield a similar structure with longer-range couplings.
• Figure 4: Gap amplification from counterdiabatic driving (CD). (a) For any driving rate, the minimal gap $\Delta_\mathrm{min,CD}$ along the CD-assisted adiabatic path remains exponentially small in the system size, both for the 1st order (Eq. \ref{['eq:H1_1st_order']}, filled markers) and 2nd order (Eq. \ref{['eq:H1_2nd_order']}, empty markers) variational ansatzes. The bare minimal gap $\Delta_\mathrm{min}$ is shown for comparison with black crosses. (b) The CD-assisted gap $\Delta_\mathrm{min,CD}$, despite being exponentially small in the system size, is still exponentially larger than the bare gap $\Delta_\mathrm{min}$: all the log-log fits (lines) are compatible with a growth $\Delta_\mathrm{min,CD} \sim \Delta_\mathrm{min} e^{c(T)L}$, with $c(T) = \alpha-\alpha_\mathrm{CD}(T)$.
• Figure 5: The approximate CD path displays a minimal gap $\Delta_\mathrm{min,CD} \propto e^{-\alpha_\mathrm{CD}(T)L}$. (a) Rate $\alpha_\mathrm{CD}$ as a function of the driving time $T$: the value of $\alpha_\mathrm{CD}$ (dots) saturates at the analytical value $\alpha$ in the absence of CD in Eq. \ref{['eq:Delta_min']} (solid lines), already for moderately short times. (b) The approach to the asymptotic value is well fitted by a power law: $\alpha-\alpha_\mathrm{CD}(T) \sim T^{-2}$.