Counterdiabatic Driving in Quantum Systems

• Counterdiabatic driving is a quantum control technique that supplements a Hamiltonian with auxiliary terms to enforce adiabatic evolution during rapid parameter changes.
• It enables arbitrarily fast state preparation, quantum annealing, and simulation by suppressing nonadiabatic transitions, even in critical many-body systems.
• Exact formulations in models like the quantum Ising chain benchmark the method, highlighting trade-offs in experimental implementation due to nonlocal multispin interactions.

Counterdiabatic driving, also referred to as transitionless quantum driving or shortcuts to adiabaticity, is a quantum control technique that supplements a time-dependent Hamiltonian with specifically engineered auxiliary terms. These terms are constructed to enforce adiabatic evolution along instantaneous eigenstates, even for rapid parameter sweeps. In contrast to traditional adiabatic protocols, which require slow time evolution to prevent nonadiabatic transitions between eigenstates—especially problematic near critical points and in large many-body systems—counterdiabatic driving provides a route to arbitrarily fast and high-fidelity state preparation, quantum annealing, and quantum simulation.

1. Formalism and Principles of Counterdiabatic Driving

The central object in counterdiabatic driving is the adiabatic gauge potential (AGP), which encodes the generator of parameter translations in the instantaneous eigenbasis of the Hamiltonian. For a bare time-dependent Hamiltonian

$$\hat{H}_0(\lambda(t))|n(\lambda)\rangle = \varepsilon_n(\lambda)|n(\lambda)\rangle,$$

the counterdiabatic (CD) Hamiltonian is

$$\hat{H}_{CD}(\lambda(t)) = \hat{H}_0(\lambda(t)) + \dot{\lambda}(t) \, \hat{A}_\lambda,$$

where the AGP is given by

$$\hat{A}_\lambda = i\sum_n \left( |\partial_\lambda n(\lambda)\rangle \langle n(\lambda)| - \langle n(\lambda)|\partial_\lambda n(\lambda)\rangle |n(\lambda)\rangle \langle n(\lambda)| \right).$$

This form guarantees that, regardless of how quickly the parameter $\lambda$ is varied, the state remains in the instantaneous eigenstate $|n(\lambda)\rangle$ at all times (Damski, 2014).

Operationally, the action of $\hat{A}_\lambda$ can be equivalently written in the instantaneous eigenbasis as

$$\langle m | \hat{A}_\lambda | n \rangle = i \langle m | \partial_\lambda n \rangle = -i \frac{\langle m | \partial_\lambda \hat{H}_0 | n \rangle}{\varepsilon_m - \varepsilon_n}, \quad m \neq n.$$

Therefore, the CD correction contains all off-diagonal terms in the energy basis weighted by the inverse energy gap.

The generalization of this protocol to many-body, periodically driven (Floquet), non-Hermitian, and open quantum systems has led to a hierarchy of implementation and optimization strategies (Schindler et al., 2023, Takahashi, 2022, Song et al., 2022).

2. Exact Construction and Symmetries in the Quantum Ising Model

The quantum Ising chain in a transverse field serves as a paradigmatic platform for counterdiabatic driving. Its Hamiltonian,

$$\hat{H}_0 = -\sum_{i=1}^N \left[ \sigma^x_i \sigma^x_{i+1} + g \, \sigma^z_i \right],$$

permits an exact closed-form counterdiabatic term of the form

$$\hat{H}_1 = -g'(t) \left[ \sum_{m=1}^{N/2-1} h_m(g) \hat{H}_1^{[m]} + \frac{1}{2} h_{N/2}(g) \hat{H}_1^{[N/2]} \right],$$

with multispin operators

$$\hat{H}_1^{[m]} = \sum_{n=1}^{N} \left[ \sigma^x_n \left( \prod_{j=n+1}^{n+m-1} \sigma^z_j \right) \sigma^y_{n+m} + \sigma^y_n \left( \prod_{j=n+1}^{n+m-1} \sigma^z_j \right) \sigma^x_{n+m} \right].$$

The central result is the derivation of the closed-form coefficients

$$h_m(g) = \frac{g^{2m} + g^N}{8 g^{m+1}(1 + g^N)}, \qquad m = 0, 1, \dots, N-1,$$

which demonstrate nontrivial behavior under duality $g \leftrightarrow 1/g$ and encode the critical scaling near the quantum phase transition through their dependence on $g^m$ and $g^N$, reflecting the correlation length $\xi(g) \sim |g-1|^{-1}$ (Damski, 2014).

This exact solution allows, in principle, arbitrarily fast adiabatic dynamics. In practice, the implementation of the nonlocal, high-range multispin terms required is experimentally formidable; consequently, effective truncations or thermodynamic approximations are pursued.

3. Approximations and Finite-Size Effects

Two main approximation strategies are analyzed:

• Truncated Counterdiabatic Driving: Only multispin interactions up to a finite range $M < N/2$ are retained:

$$\bar{h}_m(g) = \begin{cases} h_m(g), & 1 \leq m \leq M, \ 0, & m > M. \end{cases}$$

This approach provides substantial improvement over no CD driving, but to achieve greater than 95% fidelity nearly all terms must be present due to the multiscale interaction profile.

• Thermodynamic Approximation: The momentum sum is replaced by an integral, leading to

$$\tilde{h}_m(g) = \frac{1}{8} \begin{cases} g^{m-1}, & 0 \leq g < 1, \ g^{-m-1}, & g \ge 1. \end{cases}$$

While this approximation offers fidelities up to 96% in large systems, it fails to capture critical finite-size corrections, especially near $g \to 1$ where the energy gap vanishes as $1/N$. The exact structure, carrying $g^N$, encodes the dramatic finite-size and criticality-induced corrections; lacking these, the thermodynamic approximation triggers spurious excitations close to criticality.

Approximation	Key Formula	Critical Limitations
Truncated CD	$\bar{h}_m(g)$ up to $m = M < N/2$	High fidelity only with $M \simeq N/2$
Thermodynamic CD	$\tilde{h}_m(g)$ as above	Misses finite-size, critical corrections

Consequently, the exact closed-form result provides an essential benchmark and guidance for designing practical, high-fidelity protocols.

4. Extensions, Work Fluctuations, and Thermodynamic Implications

In fast quantum protocols—particularly in thermal machines, quantum annealers, or NMR experiments—counterdiabatic driving not only eliminates average dissipated work but also fundamentally alters work fluctuations. Universal results show that while the mean work under CD protocols matches the adiabatic limit, work fluctuations are increased, quantified as

$$\delta (\Delta W)^2 = \mathrm{Var}[W(t)] - \mathrm{Var}[W(t)]_{\text{ad}} = \hbar^2 \sum_n p_n^0 g^{(n)}_{\mu \nu} \dot{\lambda}^\mu \dot{\lambda}^\nu,$$

where $g^{(n)}_{\mu\nu}$ is the quantum geometric tensor associated with the eigenstate $|n(t)\rangle$ (Funo et al., 2016). This leads to a quantum speed limit for frictionless processes: $$\tau \geq \frac{ \hbar \mathcal{L}(\rho(0),\rho(\tau)) }{ \overline{\delta \Delta W_\tau} },$$ with $\mathcal{L}$ the Bures length between initial and final mixed states, and $\overline{\delta \Delta W_\tau}$ the time-averaged excess fluctuation.

These theoretical advances establish a bound between operational time and the quantum geometric (thermodynamic) cost, further strengthened near criticality as work fluctuations diverge with system size and inverse gap.

5. Impact of Quantum Criticality and Scaling Behavior

Quantum criticality is fundamental in counterdiabatic driving, as the vanishing gap and diverging correlation length ($\xi(g) \to \infty$ at $g \approx 1$) cause extraordinary finite-size corrections. In the quantum Ising chain, explicit dependence of $h_m(g)$ on $g^N \sim e^{\pm N/\xi(g)}$ ensures that counterdiabatic corrections scale correctly with proximity to the phase transition. The scaling of the energy gap as $1/N$ and the associated need for high-range multispin terms mean that simplified approximations (thermodynamic/CD truncations) become inadequate precisely where state preparation is most difficult, highlighting a limitation of global shortcut-to-adiabaticity protocols (Damski, 2014).

Additionally, in the work fluctuation framework, the excess fluctuation scales as

$$\delta (\Delta W)^2 \sim N / |\lambda - \lambda_c|^{2 - \nu d}, \qquad \mathrm{at}\ \lambda_c: \sim N^{2/(d\nu)},$$

distinguishing critical from noncritical behavior (Funo et al., 2016).

6. Practical Implementation and Outlook

The exact counterdiabatic driving protocol formulated for the quantum Ising chain, with a compact closed-form for all coefficients, allows for both theoretical exploration and practical assessment of experimental viability. Key practical insights include:

• Full CD driving demands nonlocal, high-order multispin interactions, raising economic and technical barriers for laboratory realization.
• Truncated or thermodynamic protocols offer a trade-off: partial counterdiabaticity provides significant suppression of diabatic excitations, but high-fidelity preparation near criticality remains demanding.
• The explicit formula for $h_m(g)$ is essential for quantifying the importance of each multispin term, and it clarifies the symmetry (duality) and scaling structure of the counterdiabatic Hamiltonian.
• These results benchmark all future efforts using local or variational approximations to counterdiabatic protocols, as well as extensions to more complex models and settings (periodically driven, open, or non-Hermitian systems).

Overall, the formalism of counterdiabatic driving—especially its exact construction for the quantum Ising model—provides a rigorous foundation and quantitative guide for the design and understanding of shortcut-to-adiabaticity protocols in strongly correlated and critical systems. The explicit connections to geometric quantum mechanics, thermodynamic cost, and finite-size scaling further extend its scope to the analysis of speed limits, quantum phase transitions, and practical quantum information processing.