Quantum Private Information Retrieval has linear communication complexity

TL;DR

This paper proves that quantum private information retrieval protocols require linear communication complexity, even under relaxed privacy conditions and against the weakest quantum adversaries, confirming the optimality of trivial solutions.

Contribution

It extends existing linear lower bounds on PIR to the quantum setting with approximate privacy and weak adversaries, showing sublinear protocols lack information-theoretic privacy.

Findings

Quantum PIR requires linear communication even with approximate privacy.

Le Gall's sublinear QPIR protocol is not information-theoretically private.

Lower bounds hold against the weakest quantum adversaries.

Abstract

In Private Information Retrieval (PIR), a client queries an n-bit database in order to retrieve an entry of her choice, while maintaining privacy of her query value. Chor, Goldreich, Kushilevitz, and Sudan showed that, in the information-theoretical setting, a linear amount of communication is required for classical PIR protocols (and thus that the trivial protocol is optimal). This linear lower bound was shown by Nayak to hold also in the quantum setting. Here, we extend Nayak's result by considering approximate privacy, and requiring security only against "specious" adversaries, which are, in analogy to classical honest-but-curious adversaries, the weakest reasonable quantum adversaries. We show that, even in this weakened scenario, Quantum Private Information Retrieval (QPIR) requires n qubits of communication. From this follows that Le Gall's recent QPIR protocol with sublinear communication complexity is not information-theoretically private, against the weakest reasonable cryptographic adversary.