{"paper":{"title":"Quantum Algorithms for Multivariable Polynomial Transformations: From Efficient Synthesis to Quantum Channel Transformations","arxiv_id":"2610.08714","primary_cat":"quant-ph","submitted_at":"2026-10-06T17:22:36+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.36,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"gap","text":"Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.29,"role_probabilities":{"gap":1.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"contribution","text":"We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.27,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":3,"p":0.82,"id":"s3","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.82,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.18,"interpretation":0.0}},{"i":4,"p":0.89,"id":"s4","own":true,"role":"result","text":"The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.89,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":5,"p":0.92,"id":"s5","own":true,"role":"method","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.92,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":6,"p":0.99,"id":"s6","own":true,"role":"method","text":"Factoring this certificate yields the contractive realization underlying the query algorithm.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":7,"p":0.9,"id":"s7","own":true,"role":"contribution","text":"Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.91,"main_claim_p":0.19,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.04,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.9,"interpretation":0.06}},{"i":8,"p":0.81,"id":"s8","own":true,"role":"contribution","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.17,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.81,"interpretation":0.0}},{"i":9,"p":0.63,"id":"s9","own":true,"role":"contribution","text":"Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.37,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.63,"interpretation":0.0}},{"i":10,"p":0.9,"id":"s10","own":false,"role":"interpretation","text":"Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.","hedge":{"p":0.91,"label":"hedged_possibility"},"own_p":0.41,"main_claim_p":0.57,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.1,"contribution":0.0,"interpretation":0.9}}]},"compiled":{"main_claim":{"p":0.74,"text":"Collectively, these results point to a broader program: multivariable approximation 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