Succinct Classical Arguments for Quantum Merlin-Arthur (QMA) Problems from Standard Cryptographic Assumptions

Core Concepts

The authors construct a succinct classical argument system for QMA, the quantum analogue of NP, from generic and standard cryptographic assumptions such as collapsing hash functions and a mild version of quantum homomorphic encryption. This avoids the need for the stronger assumption of post-quantum indistinguishability obfuscation required in prior work.

Abstract

The authors present a new approach to constructing succinct classical arguments for QMA problems, building on prior work on quantum verification and the compilation of quantum nonlocal games into cryptographic argument systems.
Key highlights:

The authors avoid the use of post-quantum indistinguishability obfuscation, which was required in previous work, by instead relying on weaker cryptographic primitives such as collapsing hash functions and a mild version of quantum homomorphic encryption.
They start with a question-succinct two-prover protocol for QMA and then compile it into a succinct single-prover argument system using the KLVY transformation.
The analysis of the compiled protocol involves new techniques from approximate representation theory, including a version of the Gowers-Hatami theorem that supports non-uniform distributions.
The authors also develop a succinct version of the Pauli braiding test, building on the work of de la Salle, and show how to analyze it in the compiled setting.
The final protocol achieves constant completeness-soundness gap and polylogarithmic communication complexity, all from standard cryptographic assumptions.

Stats

None.

Quotes

None.

Key Insights Distilled From

Succinct arguments for QMA from standard assumptions via compiled nonlocal games

by Tony Metger,... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19754.pdf

Deeper Inquiries

What other cryptographic primitives beyond collapsing hash functions and quantum homomorphic encryption could potentially be used to construct succinct classical arguments for QMA from standard assumptions

In addition to collapsing hash functions and quantum homomorphic encryption, other cryptographic primitives that could potentially be used to construct succinct classical arguments for QMA from standard assumptions include:

Zero-knowledge proofs: These protocols allow a prover to convince a verifier of the validity of a statement without revealing any information beyond the validity of the statement itself. Zero-knowledge proofs could be used to ensure the privacy and integrity of the interactions between the prover and verifier in the argument system.
Fully homomorphic encryption: This type of encryption scheme allows for computations to be performed on encrypted data without decrypting it. By leveraging fully homomorphic encryption, one could potentially design a protocol where computations on quantum states can be verified without revealing the states themselves.
Multi-party computation: This cryptographic technique enables multiple parties to jointly compute a function over their inputs while keeping those inputs private. By incorporating multi-party computation protocols, it may be possible to distribute the computation and verification tasks among multiple parties in a secure and efficient manner.

How might the techniques developed in this work, such as the new version of the Gowers-Hatami theorem and the analysis of succinct Pauli braiding, find applications in other areas of quantum complexity and cryptography

The techniques developed in this work, such as the new version of the Gowers-Hatami theorem and the analysis of succinct Pauli braiding, have the potential to find applications in various areas of quantum complexity and cryptography:

Quantum Error Correction: The insights gained from the analysis of self-testing protocols and the development of new theorems could be applied to the field of quantum error correction. By understanding how to verify the correctness of quantum operations and states efficiently, researchers can improve the reliability and fault tolerance of quantum computing systems.
Quantum Cryptography: The cryptographic primitives and protocols designed in this work could be adapted for use in quantum key distribution and secure quantum communication. Techniques for verifying quantum states and operations could enhance the security and privacy of quantum cryptographic systems.
Quantum Machine Learning: The concepts of self-testing and verification of quantum operations are crucial in the field of quantum machine learning. By ensuring the correctness of quantum algorithms and models, researchers can build more robust and reliable quantum machine learning systems.

Are there other quantum complexity classes beyond QMA for which succinct classical arguments can be constructed from standard assumptions using similar approaches

There are other quantum complexity classes beyond QMA for which succinct classical arguments can potentially be constructed from standard assumptions using similar approaches:

QIP (Quantum Interactive Polynomial-Time): This complexity class deals with problems that can be efficiently solved by a quantum computer interacting with a classical verifier. By adapting the techniques developed for QMA verification, it may be possible to construct succinct classical arguments for problems in QIP.
BQP (Bounded-Error Quantum Polynomial-Time): BQP represents the set of problems that a quantum computer can solve efficiently with a bounded probability of error. Similar methodologies could be applied to design succinct classical arguments for problems in BQP, ensuring the correctness of quantum computations.
QMA(2) (Quantum Merlin-Arthur with Two Provers): This extension of QMA involves two quantum provers trying to convince a classical verifier of the validity of a quantum state. By extending the protocols and techniques developed for QMA, succinct classical arguments could be constructed for problems in QMA(2).