Gap Sensitivity and Bottleneck Migration in Sparse Constraint-Preserving Quantum Annealing

The gist The exchange-resolved sensitivity reliably predicts beneficial infinitesimal changes, but its reliability decreases when the same direction is extended to a finite edge replacement

Local Response of the Minimum Gap

The central question is now local: given a driver w, which exchange modification most strongly raises its current minimum gap The quantity Ge measures how strongly that exchange channel distinguishes the two states forming the current low-energy bottleneck Proposition 1 states that if Λ(w) is simple and the minimum gap has a unique minimizer s⋆, then ∂J/∂we=1 − s⋆Λ[Ge(s⋆) − le δD(s⋆)]

Bottleneck Migration and Relinearization

The central narrative of this work is the distinction between local repair and bottleneck migration The loss of finite-step reliability is associated with migration of the active bottleneck and stronger low-energy reorganization When two minima become co-active, the appropriate local response is to choose dact ∈ arg max d∈Dc∪0 min b∈Ac gb(d) This max–min rule protects all currently active minima to first order by choosing the one with the largest worst response across the tied bottlenecks

Spectral and Problem-Informed Approximations

The exact sensitivity requires current quantum low-energy states, so classical information is used to approximate favorable exchange directions At the beginning of an anneal, a symmetric exclusion process gap identity gives an exact initial-gap descriptor For interior avoided crossings, the diagonal exchange matrix elements give Ge ≃ cos 2θ (d(e)b − d(e)a) The first approximation for the dense quadratic family is a relax–swap construction which uses no exact combinatorial optimum but recovers a favorable classical swap

Numerical Protocol and Results

The numerical experiments are arranged to probe progressively higher levels of the framework, including bottleneck migration and relinearization at a resolved exchange The exhaustive study uses 30 matched N = 6, K = 3 objectives for budgets B = 5..., 9 Bottleneck migration occurs on 421/798 paths, with the order being that the original bottleneck is first raised before a different competing minimum becomes globally limiting Relinearization at a resolved exchange shows that the active-minima rule improves 278 of 279 exchanges and abstains at the remaining event, with no harmful outcome observed

Conclusion

The relevant object for driver design is not a universally preferred graph topology, but the evolving low-energy bottleneck of the annealing path The present results establish this local sensitivity–migration–relinearization mechanism; they do not establish global convergence, scalable autonomous driver design, finite-time speedup, or quantum advantage

How it works

The exchange-resolved sensitivity is derived by decomposing the normalized minimum-gap response into different physical exchange channels through the lowenergy Dirichlet-energy contrast This local response has a direct physical meaning where every physical exchange induces a family of swaps between feasible configurations, and its Dirichlet form quantifies how the wavefunction varies over those swaps

Bottleneck Migration

The mechanism of bottleneck migration is observed when several smooth local minima of ∆(s, w) coexist, and raising the currently limiting minimum can expose a previously higher minimum which then becomes the new global bottleneck This reorganization is quantified through the rank-two projector Π = ψ0⟩⟨ψ0 + ψ1⟩⟨ψ1, and a smaller value signals stronger reorganization

Relinearization at a Bottleneck Exchange

When two minima become co-active, the derivative should no longer be optimized for only the bottleneck from which the update began, and the relevant local object is instead the lower envelope of the active gap branches The active-minima rule improves 278 of 279 exchanges and abstains at the remaining event, with no harmful outcome observed

Classical Approximations

Two problem-informed approximations are used for the dense quadratic family: a relax–swap construction which uses no exact combinatorial optimum but recovers a favorable classical swap and a second approximation replacing individual configurations by classical basins, where the amplitude weighted inter-basin kinetic coupling Fe(A, B) is defined These approximations leave several potential sources of error, including identifying the pair of relevant basins and modeling the internal amplitudes

Classical Selector Analysis

The classical-selector analysis should be read as a diagnostic of which information is missing rather than as a scalable replacement for the quantum-state response The basin-pair experiment identifies one source of error, showing that a useful coupling score depends on identifying the competitors that actually define the bottleneck The results do not support a universal classical replacement for the quantum sensitivity

How it works

The numerical protocol involves four stages: first verifying local sensitivity directly, then following the selected direction to finite strength to identify loss of predictive power, testing whether bottleneck-migration persists in different objective families, and finally testing relinearization at a resolved exchange The guided best-of-six median is R = 1 at every budget, but the lower tail shows that the discrete landscape remains nontrivial

Conclusion

The paper establishes a finite-size mechanism rather than a scalable prescription or finite-time quantum advantage A scalable method would additionally need to detect competing bottlenecks, estimate their active-set responses from accessible information, and choose steps that retain local validity The present analysis specifies the response and lower-envelope structure such an extension must respect

How it works

For all 420 of 421 dense switching paths, the first branch exchange happens above the initial gap, which indicates the operation of the “repair-and-expose” mechanism, in which repairing the active bottleneck exposes a new competing minimum The sparse family demonstrates this same “repair-and-expose” mechanism but with reduced switching frequency This is the level at which numerical evidence supports a mechanism that is not tied to one objective generator

How it works

The local sensitivity correctly predicts small driver changes, finite continuation can change the active bottleneck, and the first resolved exchange generally occurs after the original gap has been raised Switching is accompanied by stronger low-energy reorganization The comparison between guided and random search shows that the discrete landscape remains nontrivial The basin-pair experiment reveals where an important part of the practical error enters, as a useful coupling score depends on identifying the competitors that actually define the bottleneck

How it works

The results use small feasible sectors, exact objective normalization, exhaustive graph landscapes, and oracle low-energy states as benchmark resources They establish a finite-size mechanism rather than a scalable prescription or finite-time quantum advantage The sparse family tests sensitivity and migration but not the full search or selector pipeline Relinearization uses resolved interior switching points and their quantum states, which are repeated observations across 27 objectives

Conclusion

The results use small feasible sectors, exact objective normalization, exhaustive graph landscapes, and oracle low-energy states as benchmark resources They establish a finite-size mechanism rather than a scalable prescription or finite-time quantum advantage The sparse family tests sensitivity and migration but not the full search or selector pipeline Relinearization uses resolved interior switching points and their quantum states, which are repeated observations across 27 objectives The present analysis specifies the response and lower-envelope structure such an extension must respect

How it works

The exchange-resolved sensitivity is derived by decomposing the normalized minimum-gap response into different physical exchange channels through the lowenergy Dirichlet-energy contrast, including the correction arising from driver normalization This local response has a direct physical meaning where every physical exchange induces a family of swaps between feasible configurations, and its Dirichlet form quantifies how the wavefunction varies over those swaps

How it works

The central finite-step effect is bottleneck migration where a direction that repairs the current minimum gap can expose a different competing minimum At resolved exchanges of the active bottleneck, relinearizing the response over all co-active minima is substantially more reliable than continuing the original direction or optimizing only the incoming minimum

Conclusion

The paper establishes a finite-size mechanism rather than a scalable prescription or finite-time quantum advantage The sparse family tests

1. The AI system can perform exchange-resolved sensitivity to identify locally favourable modifications of the driver, which means it can predict which physical exchange modification will most strongly raise the current minimum gap based on a local, infinitesimal analysis.

2. The system can implement a repair-and-expose mechanism by tracking how the active bottleneck changes, allowing it to adapt its strategy when competing minima become co-active, thus avoiding failure when extending a locally favorable direction to a finite edge replacement.

3. The system can utilize a tied-minimum max–min update rule, which is substantially more reliable than continuing the original direction or optimizing only the incoming minimum, to maintain robustness during driver modification steps.

4. The system can employ a lower envelope prescription when two minima are co-active, selecting the direction that protects all active minima by choosing dact ∈ arg max d∈Dc∪0 min b∈Ac gb(d), effectively treating the problem as a local lower-envelope problem.

5. The AI can perform finite discrete search guided by local sensitivity to select the best complete unit-edge replacement, although it must recognize that this is not guaranteed to find a globally optimal trajectory, as shown by the fact that the all-neighbor search misses the exhaustive optimum on some sparse instances.

6. The system can distinguish between local repair and bottleneck migration, allowing it to understand why a locally correct direction might cease to be appropriate for the global minimum gap when extending the driver change.

Constraint-preserving exchange drivers allow many coupling graphs that preserve the same feasible sector, but exhibit different minimum gaps along an annealing path. We derive an exchange-resolved sensitivity of the normalised minimum gap for fixed-cardinality Laplacian drivers sharing a Dicke ground state, and apply it to identify locally favourable modifications of the driver. Numerical experiments indicate this sensitivity reliably predicts beneficial infinitesimal changes, but its reliability decreases when the same direction is extended to a finite edge replacement. The gap landscape tracking reveals one mechanism of this loss of reliability. Improving the active bottleneck can expose a competing minimum, with a more extensive reorganisation of the low-energy subspace. We call this the "repair-and-expose" mechanism and observe it in both dense quadratic and sparse frustrated objective families. This motivates relinearizing the response when competing minima become co-active. In the tested oracle setting, a tied-minimum max-min update is substantially more reliable than continuing the original direction or optimizing only the incoming minimum. Supporting finite-size search and selector tests show that this local information can remain useful beyond the infinitesimal limit, while its practical extraction depends on the driver geometry and identification of the relevant low-energy competitors. These results connect local exchange-resolved gap sensitivity to bottleneck migration and conditional driver redirection.