An optimal oracle separation of classical and quantum hybrid schemes

TL;DR

This paper establishes an optimal oracle separation between quantum depth d and d+1 for classical-quantum hybrid schemes, improving previous results by showing a tighter separation with a new problem construction.

Contribution

It introduces a new oracle separation result that is optimal, demonstrating a more precise distinction between quantum depths in hybrid computational models.

Findings

Existence of an oracle separating quantum depth d and d+1

Construction of a new problem based on d-Bijective Shuffling Simon's Problem

Improved understanding of the power of quantum depth in hybrid schemes

Abstract

Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle $\mathcal{O}$ such that $\mathsf{BQP}^\mathcal{O} \neq (\mathsf{BPP^{BQNC}})^\mathcal{O} \cup (\mathsf{BQNC^{BPP}})^\mathcal{O}$. In fact, Chia et al. proved a stronger statement: for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $2d+1$, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power. In this paper, we show that for any depth parameter $d$, there exists an oracle that separates quantum depth $d$ and $d+1$, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider $d$-Bijective Shuffling Simon's Problem (which is a variant of $d$-Shuffling Simon's Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.