Efficient distributed inner product estimation via Pauli sampling

TL;DR

This paper introduces an efficient Pauli sampling-based protocol for cross-platform quantum state verification, significantly reducing sample complexity for states with low magic and entanglement, while establishing super-polynomial lower bounds for more complex states.

Contribution

The paper presents a novel, efficient protocol for quantum state verification using Pauli sampling, applicable to states with low magic and entanglement, and provides complexity bounds for more complex states.

Findings

Efficient protocol for pure states with low magic and entanglement

Super-polynomial lower bounds for states with high magic and entanglement

Weakened requirements for states with real amplitudes

Abstract

Cross-platform verification is the task of comparing the output states produced by different physical platforms using solely local quantum operations and classical communication. While protocols have previously been suggested for this task, their exponential sample complexity renders them unpractical even for intermediate-scale quantum systems. In this work, we propose a novel protocol for this task based on Pauli sampling, a subroutine which generates Paulis distributed according to their weight in the expansion of a quantum state in the Pauli basis. We show that our protocols for both Pauli sampling and cross-platform verification are efficient for pure states with low magic and entanglement (i.e., of the order $O(\log n)$). Conversely, we show super-polynomial lower bounds on the complexity of both tasks for states with $\omega(\log n)$ magic and entanglement. Interestingly, when considering states with real amplitudes the requirements of our protocol for cross-platform verification can be significantly weakened.