Gram-Certified Resource Continuation for Structured Quantum Representation Audits

• URL: http://arxiv.org/abs/2607.10360v1
• Date: Sat, 11 Jul 2026 15:38:24 GMT
• Title: Gram-Certified Resource Continuation for Structured Quantum Representation Audits
• Abstract summary: We develop Gram-certified resource continuation for structured quantum-representation workloads.<n>We show that continuation cannot overcome a final Schmidt-rank ceiling.<n>These noise-free classical results neither establish generic 40-qubit simulability nor claim hardware performance or quantum advantage.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Dense representation of an $n$-qubit pure state requires $2^n$ complex amplitudes, precluding dense classical materialization at large $n$. We develop Gram-certified resource continuation for structured quantum-representation workloads and ask when a solution obtained under a lower-cost resource model remains a justified initialization for a richer one. For coarse and fine state ensembles connected by a declared isometry, fine, coarse, and cross complex amplitude overlaps form a positive-semidefinite block Gram matrix. A signed operator of dimension at most twice the sample count has the nonzero signed spectrum of the fine density minus the lifted coarse density, yielding trace- and operator-norm diagnostics without constructing either density operator. We prove that a coarse weighted spectral flag with objective suboptimality $δ_c$ has fine-level suboptimality at most $δ_c+2\varepsilon$, where $\varepsilon$ is the empirical trace distance; the factor two is attainable. We distinguish encoder change from exact feasible-family prolongation, give a gap-dependent subspace-stability test, and show that continuation cannot overcome a final Schmidt-rank ceiling. In deterministic synthetic controls over an 8-to-40-qubit ladder, exact ancilla lifts agree to numerical precision. Transferred initialization reduces final-rung block updates from 30 to 20, but the complete cascade costs $4.80$--$5.43$ times a direct final-rung solve, without material objective improvement. Reordering eight Bell pairs reduces the maximum matrix-product-state bond from 256 to 2. Thus continuation is justified only when cross-rung mismatch, feasible-family inclusion, topology, and total work jointly satisfy prespecified audits. These noise-free classical results neither establish generic 40-qubit simulability nor claim hardware performance or quantum advantage.

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