When Does a Quantum Speedup Survive End-to-End?

Pablo Herrero Gómez; Antonio Jimeno Morenilla; David Muñoz Hernández; Higinio Mora Mora

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arXiv · 2609.09850

When Does a Quantum Speedup Survive End-to-End?

Pablo Herrero Gómez·Antonio Jimeno Morenilla·David Muñoz Hernández·Higinio Mora Mora

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Abstract

Primitive quantum speedups are interface-relative: they depend on the input access used to run the primitive and on the output contract used to consume its state or samples. This paper introduces a transcript-level admissibility relation $A_M\preceq_{\mathrm{int}}A_Q$, defined relative to the declared implementation package of the quantum interface. It identifies which adaptive classical access transcripts that same package licenses, with all setup, transcript-generation, and precision overheads charged. The main application is an operational audit for normalized-Betti estimation in clique-complex TDA, separating three declared-interface regimes. Reversible indexed simplex interfaces certify matched classical simplex sampling and local Laplacian row access by evaluating their reversible routines on single computational branches. Membership-based preparations induce a rejection route of overhead $\binom{n}{k+1}/|S_k|$. Abstract spectral or block-encoding interfaces require an accompanying implementation package, transcript reduction, or shared representation. Under the indexed certificate and interface closure, the end-to-end cost is fixed by the imported estimator's spectral dependence on the gap $\gamma$; the concretely realized bounded-treewidth family already admits exact $\mathrm{poly}(n)$ classical Betti computation by rank over $\mathbb{Q}$. A low-rank separation supports the role of access and output contracts.

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Pablo Herrero Gómez, Antonio Jimeno Morenilla, David Muñoz Hernández, Higinio Mora Mora. 2026-09-09. When Does a Quantum Speedup Survive End-to-End?. https://arxiv.org/abs/2609.09850

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