A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire

TL;DR

This paper introduces a broad duality principle linking quantum state manipulation to Fourier analysis, enabling new constructions of quantum money, lightning, and fire with improved security assumptions and broader applicability.

Contribution

It generalizes a quantum duality principle, leading to novel quantum money and lightning schemes, and formalizes the concept of quantum fire with a classical oracle construction.

Findings

Extended quantum money to non-Abelian groups

Constructed quantum lightning from non-Abelian groups without black-box assumptions

Proposed the first classical oracle-based quantum fire construction

Abstract

Aaronson, Atia, and Susskind (2020) established that efficiently mapping between quantum states $|\psi\rangle$ and $|\phi\rangle$ is computationally equivalent to distinguishing their superpositions $|\psi\rangle \pm |\phi\rangle$. We generalize this insight into a broader duality principle, wherein manipulating quantum states in one basis is equivalent to extracting their value in a complementary basis. This general duality principle states that the ability to implement a unitary representation of a group is computationally equivalent to the ability to perform a Fourier subspace extraction from its irreducible representations. Building on our duality principle, we present the following applications: * We extend the construction of publicly-key quantum money of Zhandry (2024) from Abelian group actions to a construction of quantum lightning from non-Abelian group actions, and eliminate Zhandry's reliance on a black-box model for justifying security. Instead, we prove a direct reduction to a computational assumption -- the pre-action security of cryptographic group actions. Our construction is realizable with symmetric group actions, including those implicit in the McEliece cryptosystem. * We provide an alternative quantum lightning construction from one-way homomorphisms, with security holding under certain conditions. This scheme shows equivalence among four security notions: quantum lightning security, worst-case and average-case cloning security, and security against preparing a canonical state. * We formalize the notion of quantum fire, states that are efficiently clonable, but not efficiently telegraphable. These states can be spread like fire, provided they are kept alive quantumly and do not decohere. The only previously known construction relied on a unitary quantum oracle, whereas we present the first candidate construction of quantum fire using a classical oracle.