Quantum Probability and Applications IV

Yu. I. Bogdanov, Nadezhda A. Bogdanova, Vladimir F. Lukichev

Abstract Using an example of Pearson type IV distributions, we propose a procedure of completing the classical probability distribution to a quantum state. We obtain a wave function corresponding to Pearson type IV distributions and construct the corresponding set of basis functions. Then we demonstrate how the developed method applies to problems of statistical data analysis and quantum mechanics, and show the efficiency of our approach for the problem of approximating statistical distributions with heavy tails.

Luigi Accardi, Wilhelm von Waldenfels

Thomas Radtke, S. Fritzsche

Luigi Accardi, Wilhelm von Waldenfels

Tomoyuki Morimae, Yuki Shirakawa, Takashi Yamakawa

Quantum computational advantage refers to an existence of computational tasks that are easy for quantum computing but hard for classical one. Unconditionally showing quantum advantage is beyond our current understanding of complexity theory, and therefore some computational assumptions are needed. Which complexity assumption is necessary and sufficient for quantum advantage? In this paper, we show that inefficient-verifier proofs of quantumness (IV-PoQ) exist if and only if classically-secure one-way puzzles (OWPuzzs) exist. As far as we know, this is the first time that a complete cryptographic characterization of quantum advantage is obtained. IV-PoQ capture various types of quantum advantage previously studied, such as sampling-based quantum advantage and searching-based one. Previous work [Morimae and Yamakawa, Crypto 2024] showed that IV-PoQ can be constructed from OWFs, but a construction of IV-PoQ from weaker assumptions was left open. Our result solves the open problem. OWPuzzs are one of the most fundamental quantum cryptographic primitives implied by many quantum cryptographic primitives weaker than one-way functions (OWFs). The equivalence between IV-PoQ and classically-secure OWPuzzs therefore highlights that if there is no quantum advantage, then these fundamental primitives do not exist. The equivalence also means that quantum advantage is an example of the applications of OWPuzzs. Except for commitments, no application of OWPuzzs was known before. Our result shows that quantum advantage is another application of OWPuzzs, which solves the open question of [Chung, Goldin, and Gray, Crypto 2024]. Moreover, it is the first quantum-computation-classical-communication (QCCC) application of OWPuzzs.

Accardi, Luigi, Waldenfels, Wilhelm