Oracle Separation between Noisy Quantum Polynomial Time and the Polynomial Hierarchy

TL;DR

The paper investigates oracle separations between noisy quantum circuits, modeled as BQP$_{\lambda}$ with depolarizing noise after each gate, and classical hierarchies. It first establishes that a constant-noise regime can achieve NP separation via a noisy Deutsch–Jozsa protocol, and then shows that a milder noise rate of $\lambda(n)=\Omega(\log n/n)$ suffices for PH separation using Square Forrelation constructs, all without error correction. Central to the results are adaptations of Deutsch–Jozsa and Forrelation/Real Forrelation frameworks under noise, and a careful analysis of Fourier spectra, majority voting, and concentration inequalities to bound success probabilities. The work also identifies a no-go boundary: constant-noise cannot yield PH separation via Square Forrelation, highlighting fundamental limits of these techniques. Overall, the results illuminate how even imperfect quantum devices, operating at constant depth and without fault-tolerance, can transcend classical complexity classes under oracle access and under specific noise profiles, contributing to our understanding of NISQ-era quantum advantage and its limitations.

Abstract

This work investigates the oracle separation between the physically motivated complexity class of noisy quantum circuits, inspired by definitions such as those presented by Chen, Cotler, Huang, and Li (2022). We establish that with a constant error rate, separation can be achieved in terms of NP. When the error rate is $Ω(\log n/n)$, we can extend this result to the separation of PH. Notably, our oracles, in all separations, do not necessitate error correction schemes or fault tolerance, as all quantum circuits are of constant depth. This indicates that even quantum computers with minor errors, without error correction, may surpass classical complexity classes under various scenarios and assumptions. We also explore various common noise settings and present new classical hardness results, generalizing those found in studies by Raz and Tal (2022) and Bassirian, Bouland, Fefferman, Gunn, and Tal (2021), which are of independent interest.

Figures (4)

• Figure 1: Quantum circuit for the Deutsch–Jozsa algorithm.
• Figure 2: DJ algorithm in the presence of local noise denotes that with probability $\lambda$, an depolarization channel of single-qubit operation is applied. We have labeled the layers of noise for ease of reference in the proof. Here $\textbf{H}$ represents the Hadamard gate and $\textbf{E}$ represents the depolarization noise. $\mathcal{O}_f$ indicates the oracle realization of $f.$
• Figure 3: Illustration of relationship between distribution
• Figure 4: From Classical Oracle to Phase Oracle , We can observe that this control qubit with probability $p_1$, will remain in the $\ket{-}$ state; otherwise, it becomes $\ket{+}$ where $p_1 = (1-\lambda)(1-\lambda/2) + \lambda/2$.

Theorems & Definitions (53)

• Definition 1.1: $\mathsf{BQP}_{\lambda}$ complexity class, informal
• Theorem 1.2: Informal
• Theorem 1.3: Informal
• Theorem 1.4
• Theorem 2.1
• Definition 2.2: Real Forrelation problem
• Theorem 2.3: Theorem 2 in aaronson2015forrelation
• Theorem 2.4: Oracle Separation of $\mathsf{BQP}$ and $\mathsf{PH}$ raz2022oracle
• Definition 2.5: SquaredForrelation bassirian2021certified
• Theorem 2.6: bassirian2021certified