Comment on "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle"

TL;DR

This paper critiques and revises the theoretical foundations of Multivariable Quantum Signal Processing, identifying inconsistencies, providing counterexamples, and proposing revised necessary conditions to improve the framework.

Contribution

It identifies errors in the original M-QSP theorem, presents counterexamples, and proposes revised necessary conditions for the theory's validity.

Findings

Original Theorem 2.3 contains inconsistencies.

Counterexample invalidates Conjecture 2.1.

Revised necessary conditions are proposed, but are not sufficient.

Abstract

Multivariable Quantum Signal Processing (M-QSP) [1] is expected to provide an efficient means to handle polynomial transformations of multiple variables simultaneously. However, we noticed several inconsistencies in the main Theorem 2.3 and its proof in Ref. [1]. Moreover, a counterexample for Conjecture 2.1 in Ref. [1], which is used as an assumption in the proof of Theorem 2.3, is presented at Quantum Information Processing 2023 [2], meaning the requirement of the conjecture should be included as a condition in Theorem 2.3. Here we note our observations and propose the revised necessary conditions of M-QSP. We also show that these necessary conditions cannot be sufficient conditions, and thus some additional condition on top of these revisions is essentially required for complete M-QSP Theorem.