The Hardness of Learning Quantum Circuits and its Cryptographic Applications
Bill Fefferman, Soumik Ghosh, Makrand Sinha, Henry Yuen
TL;DR
The paper builds a bridge between quantum learning theory and cryptography by positing two concrete hardness assumptions—the Computational No-Learning and Computational No-Cloning hypotheses for random quantum circuits—and developing evidence and limitations for these conjectures. It then deriving cryptographic primitives, notably one-way state generators, quantum digital signatures, and quantum bit commitments, from these assumptions, with parallel-repetition and Chernoff-type analyses used to amplify security. A major emphasis is placed on NISQ-friendliness, showing how threshold-repetition schemes and noise-tolerant construction can yield practical, near-term quantum cryptographic protocols that rely on quantum-learning hardness rather than classical one-way functions. The work also situates itself among related efforts, discusses barriers to provable guarantees, and suggests future directions in quantum pseudorandomness, learning vs. cloning, and realizations over noisy quantum networks. Overall, the results illuminate a concrete path to secure quantum cryptography grounded in quantum learning hardness and highlight the potential for end-to-end cryptographic protocols on near-term quantum hardware.
Abstract
We show that concrete hardness assumptions about learning or cloning the output state of a random quantum circuit can be used as the foundation for secure quantum cryptography. In particular, under these assumptions we construct secure one-way state generators (OWSGs), digital signature schemes, quantum bit commitments, and private key encryption schemes. We also discuss evidence for these hardness assumptions by analyzing the best-known quantum learning algorithms, as well as proving black-box lower bounds for cloning and learning given state preparation oracles. Our random circuit-based constructions provide concrete instantiations of quantum cryptographic primitives whose security do not depend on the existence of one-way functions. The use of random circuits in our constructions also opens the door to NISQ-friendly quantum cryptography. We discuss noise tolerant versions of our OWSG and digital signature constructions which can potentially be implementable on noisy quantum computers connected by a quantum network. On the other hand, they are still secure against noiseless quantum adversaries, raising the intriguing possibility of a useful implementation of an end-to-end cryptographic protocol on near-term quantum computers. Finally, our explorations suggest that the rich interconnections between learning theory and cryptography in classical theoretical computer science also extend to the quantum setting.