Uncloneable Quantum Advice

TL;DR

This work addresses 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, through unconditional constructions for promise problems admitting uncloneable quantum advice.

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.

Figures (6)

• Figure 1: An annotated and informal representation of the space of sequences of quantum states contrasting the difficulty to generate and to clone a given sequence. Ingenerable sequences, which are novel to this work, are discussed in more details in \ref{['sc:ingen-review']}.
• Figure 2: A schematic representation of the $\tilde{C}_n$ circuits constructed in the proof of \ref{['th:money-derandomized']}. The wires are labelled, when possible, with the states they are expected to carry in the context of the proof. Every wire represents $n$ qubits, except the initial and final wires which represent $2n$ and $1$ qubits, respectively.
• Figure 3: A schematic representation of the $\tilde{C}_\lambda$ circuit constructed in the proof of \ref{['th:tfkw-derandomized']}. The wires are labelled, when possible, with the states they are expected to carry in the context of the proof. Every wire represents $\lambda$ qubits, except the initial wire, final wire, and those between the $A_\lambda$, $B_\lambda$, and $C_\lambda$ circuits. These represent, respectively, $2\lambda$, $1$, $b_\lambda$, and $c_\lambda$ qubits.
• Figure 4: The circuits $C_\nu$ for $\nu = 3,4$ used in the proof of \ref{['th:promise']} to demonstrate correctness.
• Figure 5: An illustration of an algorithm which determines if a given string $w \in \{0,1\}^*$ is in the set $L^g_0$ or the set $L^g_1$ where these are as given in \ref{['df:language']}. The single bit produced, at the right of the diagram, determines to which set $w$ belongs. In the case where the criterion $\abs{w} > (d + c)(\abs{w})$ is met, we parse $w$ as a triplet of strings $(x,y,z)$, each of the unique length ensuring that $(g_{\abs{w}}(x) \cdot y) \oplus (z \cdot z)$ is well defined.

Theorems & Definitions (91)

• Lemma 1
• Definition 2
• Definition 3
• Theorem 4
• proof
• Theorem 5
• proof : Proof sketch.
• Definition 6
• Lemma 7
• proof