Uncloneable Quantum Advice

• URL: http://arxiv.org/abs/2309.05155v1
• Date: Sun, 10 Sep 2023 22:09:05 GMT
• Title: Uncloneable Quantum Advice
• Authors: Anne Broadbent, Martti Karvonen, S\'ebastien Lord
• Abstract summary: We show for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation. We show the unconditional existence of promise problems admitting uncloneable quantum advice, and the existence of languages with uncloneable advice.
• Score: 1.1970409518725493
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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