Exact quantum splitting and the structure of finite algebras
Abstract
For squarefree f∈Fq[x], Berlekamp's fixed-point algebra is B≅Fqr, where r is the number of irreducible factors. We give an exact quantum splitting procedure in a circuit model permitting efficiently computable, input-dependent single-qubit rotations. For a block with s≥2 factors, a quadratic-character test (odd q) or absolute-trace test (characteristic $2$) is nonconstant with an exactly known probability pq,s≥1/2. Its value depends only on q,s, not on the hidden factors. An auxiliary-qubit reduction to success probability 1/4 makes one Grover iteration exact. Thus factorization takes exactly r−1 splitting rounds and O(n3log q) quantum Fq-operations, with deterministic polynomial classical preprocessing and bookkeeping. The method also extracts primitive idempotents of arbitrary finite-dimensional commutative unital algebras given by structure constants in O(n4log q) quantum Fq-operations, staying over Fq throughout. Computing the radical and passing to the center extends it to the decomposition of A/Rad(A) into simple two-sided ideals, for an arbitrary finite-dimensional associative Fq-algebra A.
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BibTeX
@misc{imran2026exact,
title = {{Exact quantum splitting and the structure of finite algebras}},
author = {Muhammad Imran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30340},
eprint = {2608.30340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30340},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact quantum splitting and the structure of finite algebras
AU - Muhammad Imran
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2608.30340
UR - https://arxiv.org/abs/2608.30340
AB - For squarefree f∈F_q[x], Berlekamp's fixed-point algebra is B≅F_q^r, where r is the number of irreducible factors. We give an exact quantum splitting procedure in a circuit model permitting efficiently computable, input-dependent single-qubit rotations. For a block with s≥2 factors, a quadratic-character test (odd q) or absolute-trace test (characteristic $2$) is nonconstant with an exactly known probability p_q,s≥1/2. Its value depends only on q,s, not on the hidden factors. An auxiliary-qubit reduction to success probability 1/4 makes one Grover iteration exact. Thus factorization takes exactly r−1 splitting rounds and O(n^3log q) quantum F_q-operations, with deterministic polynomial classical preprocessing and bookkeeping. The method also extracts primitive idempotents of arbitrary finite-dimensional commutative unital algebras given by structure constants in O(n^4log q) quantum F_q-operations, staying over F_q throughout. Computing the radical and passing to the center extends it to the decomposition of A/Rad(A) into simple two-sided ideals, for an arbitrary finite-dimensional associative F_q-algebra A.
ER -