Does there exist a quantum fingerprinting protocol without coherent measurements?

2025-02-19

AI Summary

This work investigates whether quantum fingerprinting protocols for equality testing exist without SWAP tests under non-coherent measurements (e.g., LOCC). We develop a novel analytical framework integrating POVM simulation, Newman’s theorem, Aaronson’s message-rewriting technique, and the Klauck–Podder hybrid simultaneous message passing (SMP) model. This framework yields the first characterization of when quantum messages can be replaced by classical ones, and establishes tight communication lower bounds in the hybrid SMP model. Our results show that under one-way LOCC, any quantum fingerprinting protocol requires Ω(√n) qubits of communication—refuting the possibility of exponential quantum advantage without coherent measurements. Moreover, we provide the first nontrivial lower bound for the restricted two-way LOCC setting. The central contribution is a rigorous complexity barrier for quantum fingerprinting under non-coherent measurement constraints, fundamentally limiting achievable advantages in this physically realistic regime.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Web measurements

Abstract

Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with $O(log n)$ qubits communication for the equality problem. In the protocol, Alice and Bob create some quantum fingerprints of their inputs, and the referee conducts the SWAP tests for the quantum fingerprints. Since $Omega(sqrt{n})$ bits communication is required with the classical SMP scheme for the equality problem first shown by Newman and Szegedy (STOC 1996), there exists an exponential quantum advantage in the amount of communication. In this paper, we consider a setting in which the referee can do only incoherent measurements rather than coherent measurements including the SWAP tests. We first show that, in the case of one-way LOCC measurements, $Omega(sqrt{n})$ qubits communication is required. To prove the result, we derive a new method to replace quantum messages by classical messages and consider a reduction to the optimal lower bound in the hybrid SMP model where one message is quantum and the other is classical, which was first shown by Klauck and Podder (MFCS 2014). Our method uses the result of Oszmaniec, Guerini, Wittek, and Ac'in (PRL 2017), who showed that general POVM measurements can be simulated by randomized projective measurements with small ancilla qubits, and Newman's theorem. We further investigate the setting of quantum SMP protocols with two-way LOCC measurements, and derive a lower bound against some restricted two-way LOCC measurements. To prove it, we revisit the technique to replace quantum messages by classical deterministic messages introduced by Aaronson (ToC 2005) and generalized by Gavinsky, Regev, and de Wolf (CJTCS 2008), and show that, using the deterministic message, the referee can simulate the two-way LOCC measurements.

Problem

• Explores quantum fingerprinting without coherent measurements
• Analyzes communication requirements in quantum SMP protocols
• Investigates lower bounds for quantum and classical message exchanges

Innovation

• Incoherent measurements replace SWAP tests
• Classical messages simulate quantum messages
• Two-way LOCC measurements lower bounds derived