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

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 \(γ\); 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.