Diabatic Quantum Annealing

• Diabatic quantum annealing is a non-adiabatic approach that exploits controlled transitions between low-energy states to overcome bottlenecks from minimal spectral gaps.
• It employs engineered mechanisms such as catalysts, schedule shaping, and counterdiabatic controls to optimize population transfer and enhance ground-state fidelity.
• DQA techniques extend to Boltzmann sampling and machine learning applications, demonstrating versatility in both optimization and low-energy state enumeration.

Diabatic quantum annealing (DQA) is a non-adiabatic variant of quantum annealing in which the evolution is not required to remain in the instantaneous ground state throughout the anneal. Instead, the protocol deliberately or constructively uses diabatic transitions to low-lying excited states, with the aim that population transferred out of the ground state can later be returned before the end of the evolution. In the standard adiabatic picture, the minimum spectral gap controls the runtime needed for high final ground-state fidelity; exponentially small gaps therefore imply exponentially slow schedules. DQA is motivated by the possibility of bypassing this bottleneck through controlled out-of-equilibrium dynamics, including avoided-crossing engineering, non-monotonic schedules, reverse-annealing variants, and shortcut-to-adiabaticity constructions (Feinstein et al., 2024, Crosson et al., 2020).

1. Definition and conceptual scope

In conventional quantum annealing, the time-dependent Hamiltonian is typically written as

$H_{\rm TFHI}(t) = A(t) H_X + B(t) H_Z,$

with a transverse-field driver $H_X$ and a problem Hamiltonian $H_Z$ diagonal in the computational basis (Crosson et al., 2020). The adiabatic formulation seeks to keep the system in the instantaneous ground state, whereas DQA allows coherent population transfer within a low-energy subspace rather than restricting the dynamics to a single eigenstate (Crosson et al., 2020).

A central rationale for DQA is that many hard annealing instances exhibit exponentially small minimum gaps, often associated with perturbative crossings or first-order transitions, making adiabatic runtimes impractical (Feinstein et al., 2024, Côté et al., 2022, Rajak et al., 2022). DQA instead uses finite-rate evolution so that the system can leave the ground state at one avoided crossing and, under suitable spectral structure or schedule design, return to the ground state at a later crossing (Feinstein et al., 2024, Fry-Bouriaux et al., 2021). This is the sense in which DQA is often described as exploiting “diabatic paths” through the spectrum.

The term also encompasses several related continuous-time Hamiltonian protocols that go beyond traditional ground-state annealing while remaining simpler than fully gate-based implementations. The broader framework includes reverse annealing, continuous-time quantum walks, QAOA-like schedule optimization in the continuous limit, and analog parameterized protocols implemented natively by transverse-field Ising hardware (Crosson et al., 2020).

2. Spectral mechanisms and population-transfer dynamics

The basic dynamical picture in DQA is controlled excitation and recombination. In some problems, the native annealing spectrum already contains a structure that supports beneficial diabatic motion. In other cases, the spectrum is deliberately modified to introduce an additional small gap so that the system first transfers amplitude from the ground state to an excited state and later transfers it back (Feinstein et al., 2024).

A representative engineered form is

$H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$

where $H_d$ is the driver, $H_p$ is the problem Hamiltonian, and $H_c$ is a catalyst, often an $XX$-coupling term such as $J_{\text{xx}} \sigma^x_i \sigma^x_j$ (Feinstein et al., 2024). In the Maximum Weighted Independent Set (MWIS) instances studied there, the catalyst strength $J_{\text{xx}}$ controls both the size and the location of the additional gap. At an optimal value, the engineered gap can become extremely small and enable nearly unit ground-state fidelity at anneal times orders of magnitude shorter than the adiabatic requirement (Feinstein et al., 2024).

The local structure of such avoided crossings is commonly interpreted through Landau-Zener theory. The diabatic transition probability at an avoided crossing is given by

$H_X$0

where $H_X$1 is the off-diagonal coupling and $H_X$2 is the instantaneous level separation (Feinstein et al., 2024). In the simplest two-level description, this explains why the transfer probabilities depend exponentially on anneal speed and gap size, and why incomplete transfer at an engineered gap can degrade the intended return to the ground state.

A distinct but related mechanism is Landau-Zener-Stückelberg interference. A 2026 study identifies this interference as the underlying mechanism behind speedups previously observed numerically in the frustrated Ising ring, and uses repeated sweeps and waits through a small-gap region to generate coherent interference between alternative population-transfer paths (Werner et al., 8 Jun 2026). The proposed ansatz uses three linear ramps with two intermediate plateaus and only seven parameters, while retaining competitive performance on the frustrated ring and on a benchmark MAXCUT instance with a nearly vanishing gap (Werner et al., 8 Jun 2026). This suggests that some DQA speedups arise not merely from crossing gaps quickly, but from phase-sensitive interference that requires repeated coherent traversals of the same low-energy subspace.

3. Protocol families and control constructions

Several distinct control paradigms fall under DQA. One class uses catalysts to manipulate the annealing spectrum. Nonstoquastic diabatic quantum annealing introduces a catalyst Hamiltonian $H_X$3 with time profile $H_X$4, producing an extra avoided crossing later in the schedule. In unitary dynamics on MWIS instances, this allows the system to be intentionally excited at one small gap and return to the ground state at the engineered one (Salatino et al., 11 Feb 2025).

A second class uses schedule shaping rather than new interaction terms. The Sweep-Quench-Sweep protocol consists of two quasi-adiabatic sweeps separated by a rapid quench,

$H_X$5

so that the quench induces strong nonadiabatic mixing and later evolution partially recombines amplitude into the ground state (Salatino et al., 11 Feb 2025). For the frustrated ring model, optimized schedules need not be monotonic at all: numerically optimized piecewise-linear schedules show “up-down-up” structure and may cross the perturbative gap multiple times, yielding polynomial scaling $H_X$6 up to $H_X$7, in contrast to the exponential scaling of linear schedules (Côté et al., 2022).

A third class uses local control on stoquastic hardware. The locally suppressed transverse-field protocol locally reduces the transverse field on a target qubit in a magnetically frustrated problem so as to create a “double minimum” in the gap between the ground and first excited states (Fry-Bouriaux et al., 2021). The system can then undergo a diabatic transition to the first excited state at the engineered minimum and return to the ground state at the frustration-induced one (Fry-Bouriaux et al., 2021). This protocol is explicitly designed to be compatible with stoquastic optimization problems, provided independent controls exist for local transverse fields.

A fourth class is counterdiabatic control. In the transverse-field Ising setting, the two-parameter counterdiabatic construction generalizes earlier single-parameter approaches by allowing both $H_X$8 and $H_X$9 to vary independently,

$H_Z$0

with local approximate adiabatic gauge potentials

$H_Z$1

After a single-site rotation, the resulting protocol becomes equivalent to unconventional diabatic control of longitudinal and transverse fields and therefore avoids introducing exotic nonstoquastic terms for implementation (Prielinger et al., 2020). Closely related ideas appear in greedy parameter optimization with site-dependent $H_Z$2-fields, where the non-stoquastic term can likewise be eliminated by a rotation, yielding nontrivial diabatic control of a stoquastic transverse-field Ising model (Kadowaki et al., 2021).

A fifth class modifies only local diagonal terms. A 2025 study introduces a controlled diagonal catalyst

$H_Z$3

with $H_Z$4 (Hattori et al., 19 Mar 2025). On MWIS instances with small gaps, the variationally optimized schedule for this local $H_Z$5-field catalyst yields time-to-solution scaling $H_Z$6 versus $H_Z$7, characterized in the paper as an approximate square speed up in time-to-solution compared to conventional quantum annealing (Hattori et al., 19 Mar 2025).

4. Robustness, noise, and dynamical diagnostics

The chief practical issue in DQA is not merely whether a beneficial diabatic path exists, but whether it is robust. In catalyst-assisted DQA, there is a trade-off between the precision needed in catalyst strength and the anneal time. For the MWIS example with an $H_Z$8-catalyst, high ground-state fidelity is robust to anneal time when the catalyst is tuned exactly to its optimal value, but if the catalyst is detuned then increasing the anneal time can reduce the final ground-state fidelity (Feinstein et al., 2024). The paper visualizes this through two-dimensional sweeps showing ridges of high fidelity in the $H_Z$9 plane and notes that the full-width-at-half-maximum in fidelity versus $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$0 narrows as $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$1 increases (Feinstein et al., 2024).

Open-system effects further constrain DQA. In dissipative simulations of MWIS, both Nonstoquastic DQA and Sweep-Quench-Sweep outperform standard quantum annealing in unitary dynamics, but their advantages are strongly reduced in the presence of dephasing and thermal gain-and-loss baths (Salatino et al., 11 Feb 2025). At short times, coherent Landau-Zener transitions still dominate; at intermediate times there is an optimal regime beyond which longer runtimes are detrimental because dissipation dominates; and at long times all protocols approach limiting infidelity set by the Hilbert-space dimension (Salatino et al., 11 Feb 2025). Dephasing is particularly damaging because these protocols rely on phase coherence to de-excite back into the ground state, while high-temperature thermal baths drive the system toward high-entropy states with near-zero ground-state probability (Salatino et al., 11 Feb 2025).

The need for coherence is reinforced by Landau-Zener-Stückelberg-based analyses. The interference mechanism requires coherent phase accumulation between repeated sweeps, and a 2026 study provides both analytical and numerical evidence that coherence is an essential resource for the resulting speedup (Werner et al., 8 Jun 2026). This directly counters the common simplification that DQA is merely “faster annealing”; many successful DQA protocols are in fact coherent control protocols.

Several studies emphasize that dynamical fingerprints matter. In the Sherrington-Kirkpatrick model, a “diabatic bump” appears in the fidelity at intermediate annealing times for hard instances under unitary QA, whereas simulated quantum annealing and simulated annealing show monotonic behavior and no bump (Rakcheev et al., 2022). Correlation-function dynamics also differ qualitatively at intermediate times, though all methods coincide more closely at very short times because the short-time expansion of QA resembles a high-temperature expansion (Rakcheev et al., 2022). Likewise, simulated quantum annealing reproduces the Kibble-Zurek scaling of the average number of defects in the 1D transverse-field Ising model, but deviates in higher defect cumulants and therefore cannot be assumed to reproduce detailed quantum diabatic dynamics faithfully (Bando et al., 2021).

5. Performance regimes, scaling claims, and common misconceptions

The literature does not support a single universal performance statement for DQA. Rather, performance depends strongly on spectral structure, schedule flexibility, control precision, and environmental conditions. For the frustrated ring model, sufficiently optimized non-adiabatic schedules avoid the exponential slowdown seen for linear schedules, with numerical evidence of polynomial scaling up to $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$2 qubits (Côté et al., 2022). For the MWIS examples with a local diagonal catalyst, the reported improvement is an approximate square speed up in time-to-solution, not a change to polynomial time (Hattori et al., 19 Mar 2025). For the $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$3-spin model with $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$4, two-parameter counterdiabatic driving improves ground-state fidelity, residual energy, and time-to-solution over both traditional QA and single-parameter CD control, but the scaling advantage vanishes in the $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$5 limit of that specific model because the CD coefficient scales as $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$6 (Prielinger et al., 2020).

This diversity of outcomes underlies a recurrent misconception: DQA is not synonymous with generic non-adiabaticity. Merely shortening the anneal can populate unwanted excited states or local minima. The box model study of continuous-space annealing finds that diabatic transitions are ubiquitous for practical runtimes and that residual energy typically shows a fast regime, a diabatic regime with exponential decay, and an adiabatic regime with $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$7 decay (Koh et al., 8 May 2026). It also identifies “flat gaps,” extended spectral plateaus rather than sharp avoided crossings, and argues that these are associated with wavefunction localization in local minima and diabatic trapping (Koh et al., 8 May 2026). The paper explicitly notes that the standard Landau-Zener formula captures the exponential form of decay only qualitatively in that setting because the assumptions of isolated avoided crossings are violated (Koh et al., 8 May 2026).

Another misconception is that DQA always targets optimization only. Some DQA work uses excited-state population not as a nuisance but as the resource of interest. In Bayesian multiple-target data association, DQA is used to enumerate low-energy feasible assignments rather than to isolate a single optimum, because the posterior estimator requires summing over many high-probability assignments (McCormick et al., 2022). In that context, the non-adiabatic regime is intentionally chosen to populate a useful ensemble of low-energy states.

6. Sampling, machine learning, and broader applications

DQA has also been developed as a controllable Boltzmann sampler. In a purely unitary formulation, the annealing protocol

$H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$8

can be tuned so that the output distribution approximates the Boltzmann distribution of an Ising model in the high-temperature regime (Gyhm et al., 2024). The effective inverse temperature is determined analytically by the schedule: $H(s) = (1-s) H_d + s(1-s) H_c + s H_p,$9 For a linear schedule $H_d$0, $H_d$1, the paper derives

$H_d$2

and shows that the two-body term must be rescaled by $H_d$3 for correct first-order matching to the Boltzmann distribution (Gyhm et al., 2024). Using infinite-range and two-dimensional ferromagnetic Ising models, the study reports bounded sampling errors in the paramagnetic phase regardless of system size, while accuracy deteriorates in the ordered phase and near the transition (Gyhm et al., 2024).

This sampling perspective was extended to energy-based generative models. A 2025 study applies the analytic relation between annealing schedule and effective inverse temperature to training restricted Boltzmann machines on a quantum annealer, replacing classical MCMC sampling with DQA-based Boltzmann samples (Kim et al., 11 Sep 2025). The workflow embeds the model weights into the hardware graph, programs a schedule to realize the target $H_d$4, and uses the resulting samples in the RBM gradient update

$H_d$5

The study reports faster convergence, lower validation error, and per-sample time up to $H_d$6 faster than classical persistent contrastive divergence with $H_d$7 Gibbs steps, while also identifying a systematic temperature misalignment on analog hardware and introducing a coupling rescaling factor

$H_d$8

to correct it (Kim et al., 11 Sep 2025).

Beyond machine learning, DQA has been proposed as a hybrid subroutine for Bayesian tracking and data association. There the time-dependent Hamiltonian

$H_d$9

is run intentionally outside the adiabatic regime so that low-energy feasible assignments are sampled rather than only the ground state, after which a classical Bayesian update sums over the sampled assignments (McCormick et al., 2022). This application illustrates a broader point: DQA can function either as an optimization heuristic or as a low-energy sampler, depending on whether excited states are an intermediate transport channel or part of the desired output distribution.

The current state of the field suggests that DQA is best understood not as a single algorithm but as a family of coherent nonequilibrium annealing protocols. Across catalysts, local suppression, reverse annealing, schedule optimization, and counterdiabatic control, the recurring structure is the same: small gaps need not imply adiabatic slowdown if the protocol can exploit controlled excursions in a dynamically isolated low-energy sector. A plausible implication is that future progress will depend less on a universal schedule prescription than on hardware that supports precise timing, local control, tunable pauses, and sufficiently long coherence to preserve the interference and recombination processes on which successful DQA depends (Feinstein et al., 2024, Salatino et al., 11 Feb 2025, Werner et al., 8 Jun 2026).