bibkey: laneve2024multivariateqsp authors: L. Laneve, S. Wolf year: 2024 title: “On multivariate polynomials achievable with quantum signal processing” doi: 10.22331/q-2025-02-20-1641 url: https://arxiv.org/abs/2407.20823v2 claim: “Conjecture 8: if a three-level Protocol C segment A_m W̃ ⋯ A_1 W̃ maps a polynomial state of effective dimension ≤ 2 to one of effective dimension ≤ 2 while every intermediate state has effective dimension > 2, then the segment acts as a^k b^h · U on some pair of subspaces of dimension d ≥ 2; stated as the missing step towards Conjecture 7.” strata_touched:

• D5/S3/Quantum/Algebra/MultivariateQSPCollapseRefutation license: citation-only triage: anchor

Laneve and Wolf, multivariate polynomials achievable with quantum signal processing

L. Laneve and S. Wolf, On multivariate polynomials achievable with quantum signal processing, arXiv:2407.20823v2 (14 February 2025; quant-ph); Quantum 9, 1641 (2025), DOI 10.22331/q-2025-02-20-1641.

Verified locator

DOI: 10.22331/q-2025-02-20-1641. Primary version: https://arxiv.org/abs/2407.20823v2 (the latest arXiv version). The TeX source of v2 supplies Definition 1 (polynomial states), Protocol C and Definition 6 (effective dimension) in contents/tripartite-protocol.tex and the surrounding sections, and Conjecture 8 (thm:low-dim-identity-resolution, lines 130–138 of that file).

Source statements

Definition 1: “A polynomial state $|\gamma (z)⟩$ is a polynomial vector in $z\in {\mathbb{T}}^m$ that satisfies $⟨\gamma (z)|\gamma (z)⟩\equiv 1$.”

Protocol C: “Consider the following signal operator $\stackrel{=diag(1,a,b).\; We\; intertwine\; this\; operator\; with\; a\; sequence\; of\; processing\; operators\; Ak\in SU(3)\; such\; that\; AnW}{W}{A}_{n-1}\stackrel{\cdots W}{W}{A}_0|0⟩=|\gamma (a,b)⟩$.”

Definition 6: “A polynomial state $|\gamma (z)⟩={\sum }_k|{\gamma }_k⟩z^k$ has effective dimension $d$ if its coefficient vectors $|{\gamma }_k⟩$ span a subspace of dimension $d$.”

Conjecture 8: “Let \ket{\gamma(\vv{z})}, \ket{\gamma'(\vv{z})} be polynomial states of effective dimension $\le 2$ such that $\ket{\gamma(\vv{z})} = A_m \Tilde{W} A_{m-1} \cdots A_1 \Tilde{W} \ket{\gamma’(\vv{z})}$butanyintermediatestatehaseffectivedimension$> 2$. Then the operator A_m \Tilde{W} A_{m-1} \cdots A_1 \Tilde{W} acts as $a^k{b}^h\cdot U$ in some pair of subspaces \calH \rightarrow \calH' of dimension $d\ge 2$, for some $U\in SU(d)$, $k,h\in \mathbb{N}$.”

The text before it: “The final claim we could not prove in order to conclude Conjecture 7 is these situations are the only possible cases that prevent an immediate translation.”

Scope

The paper proves sufficient conditions for multivariate QSP and gives Protocol C; Conjectures 7 and 8 are stated as open.