Instance-Independent Drivers in Quantum Annealing

Instance-Independent Drivers in Quantum Annealing

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.

Quantum annealing interpolates from a driver Hamiltonian to an instance-dependent problem Hamiltonian. A transverse driver mixes the problem basis, and instance-independent means that this mixing operator is chosen without knowledge of the particular instance. Quantum speedup must use the time or complexity criterion of the cited no-speedup theorem.

Catalogue wording retained for traceability.

Extend the no-quantum-speedup theorem for the transverse-field driver to every instance-independent transverse Hamiltonian.

Scope and source qualification.

The statement concerns the source instance ensemble and theorem assumptions; it is not an assertion against every quantum algorithm on every optimization problem.

Short English statement

Determine whether the no-speedup result for the standard transverse-field driver holds for every instance-independent transverse Hamiltonian.

Research attempt

Model, date and investigated scope.

OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Computed the exact avoided-crossing gap for an instance-independent dense driver as a test of driver-specific proof transfer; no source no-speedup theorem or admissible counterexample.

Sources and literature check.

Read the supplied definition, catalogue wording, source qualifications and reference entries for 30005555. The historical source/context and inherited status are retained; no fresh literature or novelty certification is claimed. No external theorem is used without proof in this attempt; explicit model assumptions are distinguished from facts about the source model.

Testing whether the driver alone controls the gap.

On an $N$-dimensional search space, $N>1$, let $|s\rangle=N^{-1/2}\sum_x|x\rangle$ and a marked basis state be $|w\rangle$. Consider the interpolation \[ H(t)=-(1-t)|s\rangle\langle s|-t|w\rangle\langle w|,\qquad0\le t\le1. \] The driver is independent of $w$, but is dense in the problem basis. On the span of $|s\rangle,|w\rangle$, the trace is $-1$ and determinant is $t(1-t)(1-1/N)$. Hence the two eigenvalues have separation \[ \Delta(t)=\sqrt{1-4t(1-t)(1-1/N)},\qquad \min_t\Delta(t)=N^{-1/2}. \] The remaining eigenvalues are zero, so this two-level calculation also identifies the low spectral avoided crossing for $N>1$.

Scope and failed extrapolation.

This is not the source instance ensemble and is not presented as a speedup theorem; gap size alone does not prove a runtime or include implementation cost. It demonstrates that an instance-independent driver can have connectivity very different from the transverse bit-flip driver, so a proof using that specific connectivity cannot simply be copied without rechecking hypotheses. Unbounded rescaling of a driver would also alter times unless norms or costs are fixed.

First gap.

A source-theorem extension must control every admissible normalized instance-independent transverse operator for the actual ensemble, including dense choices if allowed, and the stated complexity measure. No universal no-speedup result, nor a counterexample satisfying those source assumptions, is obtained here.

Verification and adversarial audit.

Checked the two-dimensional trace/determinant and endpoint gaps; distinguished spectral scaling, implementation cost and proven algorithmic speedup. This is a GPT-6 Astra Ultra self-audit, not independent human verification. Prompt A drafting followed by prompt B scope/dependency/boundary self-audit. Checked the two-dimensional trace/determinant and endpoint gaps; distinguished spectral scaling, implementation cost and proven algorithmic speedup. The remaining gap is: Obtain an ensemble-specific uniform bound over all allowed normalized transverse drivers with the source runtime/cost assumptions. Unresolved status and zero original-target completion are retained.

Coordinator checked the two-level trace, determinant and avoided-crossing gap for N greater than one, including the zero eigenvalue at endpoint parameters. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.

Python 3 and local file-reading/writing tools for manuscript serialization and structural checks; mathematical calculations were symbolic reasoning, with no numerical experiment or theorem prover. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.

Final status and first remaining gap.

UNRESOLVED. Obtain an ensemble-specific uniform bound over all allowed normalized transverse drivers with the source runtime/cost assumptions.

Completion Estimate

COMPLETION: 0% The proved calculation is an elementary auxiliary result or a test of a restricted strategy; no new portion of the full source target is established.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath, record 30005555 (OWR-13750338-003). Catalogue curation under CC BY 4.0; definitions and English exposition added locally. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Machine Learning for Science: Mathematics at the Interface of Data-driven and Mechanistic Modelling $2023$, Oberwolfach Reports, DOI 10.4171/owr/2023/26, pp. 1453--1484. Original report and mathematical context. https://doi.org/10.4171/owr/2023/26.