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A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire (2411.00529v1)

Published 1 Nov 2024 in quant-ph and cs.CR

Abstract: Aaronson, Atia, and Susskind established that swapping quantum states $|\psi\rangle$ and $|\phi\rangle$ is computationally equivalent to distinguishing their superpositions $|\psi\rangle\pm|\phi\rangle$. We extend this to a general duality principle: manipulating quantum states in one basis is equivalent to extracting values in a complementary basis. Formally, for any group, implementing a unitary representation is equivalent to Fourier subspace extraction from its irreducible representations. Building on this duality principle, we present the applications: * Quantum money, representing verifiable but unclonable quantum states, and its stronger variant, quantum lightning, have resisted secure plain-model constructions. While (public-key) quantum money has been constructed securely only from the strong assumption of quantum-secure iO, quantum lightning has lacked such a construction, with past attempts using broken assumptions. We present the first secure quantum lightning construction based on a plausible cryptographic assumption by extending Zhandry's construction from Abelian to non-Abelian group actions, eliminating reliance on a black-box model. Our construction is realizable with symmetric group actions, including those implicit in the McEliece cryptosystem. * We give 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. * Quantum fire describes states that are clonable but not telegraphable: they cannot be efficiently encoded classically. These states "spread" like fire, but are viable only in coherent quantum form. The only prior construction required a unitary oracle; we propose the first candidate in the plain model.

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Summary

  • The paper introduces a duality principle linking unitary representations and Fourier subspace extraction, expanding foundational results in quantum computation.
  • It applies group representation theory to construct quantum money and lightning, using non-Abelian group actions to overcome limitations of previous models.
  • The research pioneers the concept of 'quantum fire,' defining states that are efficiently clonable but resistant to telegraphic transfer, thus advancing secure quantum state management.

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

The paper "A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire" explores a profound duality within quantum computation, drawing connections between group representations and quantum cryptographic constructs. This duality extends a foundational result by Aaronson, Atia, and Susskind, elucidating the computational equivalence between manipulating quantum states in different bases. The authors generalize this to a broader duality principle for groups, establishing that implementing a unitary representation is equivalent to Fourier subspace extraction corresponding to irreducible representations.

Key Contributions and Applications

  1. Quantum Money and Lightning: The paper presents significant advancements in constructing quantum money and its stricter variant, quantum lightning. These concepts leverage the no-cloning theorem to ensure states are verifiable but unclonable. The manuscript delineates a framework for quantum lightning grounded in concrete cryptographic assumptions, overcoming previous insurmountable barriers in this space.
  2. Enhancing Quantum Money through Group Actions: The duality insights lead to a compelling construction of quantum money using non-Abelian group actions, particularly highlighting advancements over Abelian group-based methods. This transition eradicates prior reliance on idealized models and provides real-world instantiations, notably utilizing group actions from the symmetric group involved in the McEliece cryptosystem.
  3. Quantum Fire and Cloning vs. Telegraphing: Introducing the notion of "quantum fire," the authors discuss states that are efficiently clonable but non-telegraphable, addressing quantum state's inherent intangible properties that prevent classical encoding while allowing replication. The manuscript provides the first plausible construction of quantum fire within a non-oracle model.

Implications and Theoretical Insights

  • Quantum Complexity and Cryptography:

This work has profound implications for quantum cryptographic primitives, offering pathways to secure constructs devoid of overly strong assumptions like indistinguishability obfuscation (iO). The duality principle herein is anticipated to influence quantum complexity theory by establishing new equivalences and simplifying reduction proofs.

  • Practical Cryptographic Schemes:

By applying group action representations, this research advances quantum money's feasibility, presenting verifiable and unclonable currency systems that can transform how digital transactions acknowledge quantum principles.

  • Generalized Framework:

Notably, the framework transcends quantum money to accommodate instances of cryptographic constructs previously not imaginable through singular group action perspectives. This generalization is potent for theorists seeking to extrapolate quantum mechanical phenomena to practical cryptographic applications without rendering classical security assumptions irrelevant.

Future Prospectives

  • Empirical Evaluation:

Real-world adaptation would necessitate simulations and experimental validations on quantum systems capable of exemplifying the discussed duality, particularly focusing on scalability and implementation across varied quantum architecture.

  • Exploration of Non-Abelian Properties:

Delving deeper into non-Abelian groups could uncover new classes of cryptographic functions, further decoupling quantum constructs from classical dependencies, and potential exploration could redefine complexity boundaries.

  • Quantum Networking and State Communication:

The concept of quantum fire suggests novel mechanisms for handling quantum states in distributed networks, proposing decentralized handling of state cloning and distribution while maintaining state integrity.

In conclusion, this paper not only advances theoretical constructs but also sets the foundation for next-generation quantum cryptographic developments. The applications to quantum money, lightning, and fire underscore the depth of the proposed duality principle, marking a significant stride toward practical quantum computing paradigms. The robustness, backed by non-trivial cryptographic assumptions, bolsters confidence in these constructs being pivotal elements in both theoretical exploration and applied quantum computing.

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