Uncloneable Quantum Advice

TL;DR

This paper introduces the concept of uncloneable quantum advice, demonstrating its existence and implications for quantum complexity classes, cryptography, and the no-cloning principle, with potential applications in quantum cryptography and complexity theory.

Contribution

It establishes the first results on uncloneable quantum advice, linking it to complexity classes, cryptographic applications, and the no-cloning principle in a novel way.

Findings

Existence of promise problems with uncloneable quantum advice.

Languages with uncloneable advice assuming quantum copy-protection feasibility.

Generalization of state complexity classes to include cloning complexity.

Abstract

The famous no-cloning principle has been shown recently to enable a number of uncloneable functionalities. Here we address for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation, but that is natively unkeyed: quantum advice. Remarkably, this is an application of the no-cloning principle in a context where the quantum states of interest are not chosen by a random process. We show the unconditional existence of promise problems admitting uncloneable quantum advice, and the existence of languages with uncloneable advice, assuming the feasibility of quantum copy-protecting certain functions. Along the way, we note that state complexity classes, introduced by Rosenthal and Yuen (ITCS 2022) - which concern the computational difficulty of synthesizing sequences of quantum states - can be naturally generalized to obtain state cloning complexity classes. We make initial observations on these classes, notably obtaining a result analogous to the existence of undecidable problems. Our proof technique establishes the existence of ingenerable sequences of finite bit strings - essentially meaning that they cannot be generated by any uniform circuit family. We then prove a generic result showing that the difficulty of accomplishing a computational task on uniformly random inputs implies its difficulty on any fixed, ingenerable sequence. We use this result to derandomize quantum cryptographic games that relate to cloning, and then incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable advice. Applying this two-step process to a monogamy-of-entanglement game yields a promise problem with uncloneable advice, and applying it to the quantum copy-protection of pseudorandom functions with super-logarithmic output lengths yields a language with uncloneable advice.