Are uncloneable proof and advice states strictly necessary?

TL;DR

This paper introduces and studies the necessity of uncloneable quantum proofs and advice, defining stronger classes that require proofs or advice states to be inherently uncloneable, and demonstrates their existence and separation results in quantum complexity theory.

Contribution

It defines strictly uncloneable quantum complexity classes and proves their existence and separations in the quantum oracle model and unconditionally.

Findings

Existence of languages in strictly uncloneable QMA and BQP/qpoly in the quantum oracle model.

Quantum oracle separation between QMA and cloneableQMA.

Unconditional inclusion of a language in strictly uncloneable FEQP/qpoly.

Abstract

Yes, we show that they are. We initiate the study of languages that necessarily need uncloneable quantum proofs and advice. We define strictly uncloneable versions of the classes QMA, BQP/qpoly and FEQP/qpoly (which is the class of relational problems solvable exactly with polynomial-sized quantum advice). Strictly uncloneable QMA is defined to be the class of languages in QMA that only have uncloneable proofs, i.e., given any family of candidate proof states, a polynomial-time cloning algorithm cannot act on it to produce states that are jointly usable by $k$ separate polynomial-time verifiers, for arbitrary polynomial $k$. This is a stronger notion of uncloneable proofs and advice than those considered in previous works, which only required the existence of a single family of proof or advice states that are uncloneable. We show that in the quantum oracle model, there exist languages in strictly uncloneable QMA and strictly uncloneable BQP/qpoly. The language in strictly uncloneable QMA also gives a quantum oracle separation between QMA and the class cloneableQMA introduced by Nehoran and Zhandry (2024). We also show without using any oracles that the language, used by Aaronson, Buhrman and Kretschmer (2024) to separate FEQP/qpoly and FBQP/poly, is in strictly uncloneable FEQP/qpoly.