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Quantum Algorithms for Multivariable Polynomial Transformations: From Efficient Synthesis to Quantum Channel Transformations — arXiv:2610.08714

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.08714
License (as archived): CC0-1.0
Archived: 2026-10-07T15:29:13.112Z
SHA-256 of the archived text: a057607ec2b0a22c86b35a5d03856366f054463ff74c974b17e5c38695d86061
Stable URL of this page: https://metraipolis.com/documento/dc_2d89ca2ef614

How to cite

Quantum Algorithms for Multivariable Polynomial Transformations: From Efficient Synthesis to Quantum Channel Transformations — arXiv:2610.08714 — arXiv (en), https://arxiv.org/abs/2610.08714 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2d89ca2ef614 (Metraipolis archive dc_2d89ca2ef614, sha256 a057607ec2b0a22c86b35a5d03856366f054463ff74c974b17e5c38695d86061)

BibTeX

@misc{dc2d89ca2ef614,
  title = {Quantum Algorithms for Multivariable Polynomial Transformations: From Efficient Synthesis to Quantum Channel Transformations — arXiv:2610.08714},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.08714},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2d89ca2ef614; sha256 a057607ec2b0a22c86b35a5d03856366f054463ff74c974b17e5c38695d86061},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.08714. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Polynomial transformations are basic primitives in quantum algorithms: quantum signal processing and singular value transformation compile univariate polynomials into circuits with query complexity essentially set by degree. Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory. We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain. Given a compact finite-state description of a degree-$D$ polynomial $P$ and $0<\tau<1$, we synthesize $P$ with $(1+\tau)$-optimal normalization using $O(D/\sqrt{\tau})$ queries, and exactly $D$ under row-block access, matching a degree lower bound. The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy. The key to our construction is a finite algorithmic Schur--Agler theorem: the coefficient recurrence defines a polynomial-dimensional continuation space supporting a complete semidefinite certificate for contractivity on the prescribed domain. Factoring this certificate yields the contractive realization underlying the query algorithm. Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations. Given coherent Kraus access $K=\{K_a\}_a$, we synthesize any finite jointly contractive family of noncommutative polynomial maps $K\mapsto\{F_b(K)\}_b$ as completely positive operations, allowing coherent interference among Kraus histories. Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs. Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.

Tema degli esploratori: fisica e astronomia (elenco arXiv quant-ph, nuovi lavori del giorno).
Categorie arXiv: Quantum Physics (quant-ph).
Pagina del lavoro: https://arxiv.org/abs/2610.08714 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T15:00:59.451Z dalla pagina https://arxiv.org/list/quant-ph/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).