The Quantum Realization Functor and the κ-Obstruction Theorem: Why Quantum Computation is Bounded by the Algebraic Geometry of Motives
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2026
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| _version_ | 1866902035056558080 |
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| author | Eltgroth, Matthew |
| author_facet | Eltgroth, Matthew |
| contents | <p>We introduce the quantum realization procedure H_{\mathrm{QFT}}, which packages the Betti and étale realizations of a motive into a single Hilbert space and implements a computational analogue of the comparison isomorphism at polynomial cost via the quantum Fourier transform.</p> <p>The central result is the κ-obstruction theorem, a purely categorical statement: any exact functor from an abelian category to a semisimple target category annihilates all higher extension groups \mathrm{Ext}^n for n \geq 1. This applies in particular to quantum computation, where measurement maps quantum states to classical outputs in a semisimple category (finite-dimensional vector spaces).</p> <p>We show that the complementarity index \kappa of the Algorithmic Motives (AM) framework corresponds canonically to a nontrivial extension class arising from the derived inverse limit \varprojlim^1, which embeds into \mathrm{Ext}^1 of the motivic category. When \kappa = 0, the obstruction is fully captured by morphisms and can be resolved via the comparison isomorphism, yielding polynomial-time quantum algorithms (e.g., Shor, Biasse–Song). When \kappa > 0, the residual obstruction lies in \mathrm{Ext}^1, which is annihilated by any realization functor and therefore cannot be accessed by any computational model whose output lands in a semisimple category.</p> <p>This establishes a structural limitation on quantum computation: quantum speedup is possible precisely when the relevant motivic obstruction is visible at the level of realizations, and impossible when it resides in extension data invisible to all such functors.</p> <p>The result is independent of physical assumptions and depends only on standard homological algebra and the semisimplicity of the output category.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19273083 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Quantum Realization Functor and the κ-Obstruction Theorem: Why Quantum Computation is Bounded by the Algebraic Geometry of Motives Eltgroth, Matthew Quantum computation; motives; Tannakian categories; Ext groups; homological algebra; quantum Fourier transform; complexity theory; arithmetic geometry; Nori motives; category theory <p>We introduce the quantum realization procedure H_{\mathrm{QFT}}, which packages the Betti and étale realizations of a motive into a single Hilbert space and implements a computational analogue of the comparison isomorphism at polynomial cost via the quantum Fourier transform.</p> <p>The central result is the κ-obstruction theorem, a purely categorical statement: any exact functor from an abelian category to a semisimple target category annihilates all higher extension groups \mathrm{Ext}^n for n \geq 1. This applies in particular to quantum computation, where measurement maps quantum states to classical outputs in a semisimple category (finite-dimensional vector spaces).</p> <p>We show that the complementarity index \kappa of the Algorithmic Motives (AM) framework corresponds canonically to a nontrivial extension class arising from the derived inverse limit \varprojlim^1, which embeds into \mathrm{Ext}^1 of the motivic category. When \kappa = 0, the obstruction is fully captured by morphisms and can be resolved via the comparison isomorphism, yielding polynomial-time quantum algorithms (e.g., Shor, Biasse–Song). When \kappa > 0, the residual obstruction lies in \mathrm{Ext}^1, which is annihilated by any realization functor and therefore cannot be accessed by any computational model whose output lands in a semisimple category.</p> <p>This establishes a structural limitation on quantum computation: quantum speedup is possible precisely when the relevant motivic obstruction is visible at the level of realizations, and impossible when it resides in extension data invisible to all such functors.</p> <p>The result is independent of physical assumptions and depends only on standard homological algebra and the semisimplicity of the output category.</p> |
| title | The Quantum Realization Functor and the κ-Obstruction Theorem: Why Quantum Computation is Bounded by the Algebraic Geometry of Motives |
| topic | Quantum computation; motives; Tannakian categories; Ext groups; homological algebra; quantum Fourier transform; complexity theory; arithmetic geometry; Nori motives; category theory |
| url | https://doi.org/10.5281/zenodo.19273083 |