Cryptography and Distribution Verification Achieves Identity Testing with Reduced Sample Complexity, Bypassing Exponential Support Size Limitations

The challenge of verifying whether a sample truly originates from a specific distribution lies at the heart of modern cryptography and statistical analysis, and researchers continually seek methods requiring fewer samples for accurate verification. Bruno Cavalar, Eli Goldin from New York University, Matthew Gray, and Taiga Hiroka from Hon-Hai Research Institute, alongside Tomoyuki Morimae from Yukawa Institute for Theoretical Physics, Kyoto University, now demonstrate significant progress in this area by exploring the relationship between efficient verification methods and the potential for quantum advantage. Their work bypasses established limitations on sample complexity, revealing conditions under which distributions can be efficiently verified, even when sampled from quantum sources. This research establishes a clear link between the existence of one-way functions and the verifiability of classically sampled distributions, and further demonstrates that, under certain conditions, sampling-based advantages can be verified with remarkable efficiency, potentially paving the way for more secure cryptographic protocols and improved statistical testing methods.

Gaussian Identity Testing via Cryptographic Protocols

One of the most fundamental problems in statistical analysis is determining whether a sample of data comes from a specific, known distribution. This work investigates this challenge, focusing on the case where the known distribution is a Gaussian distribution. Researchers developed novel cryptographic protocols to address this problem, combining techniques from cryptography and distribution verification to ensure data authenticity. This approach constructs a protocol that allows a prover to convincingly demonstrate to a verifier that their samples originate from the specified Gaussian distribution, while simultaneously preserving the prover’s privacy. The team demonstrates that this protocol achieves constant-time verification, a significant improvement over existing methods. Furthermore, the research establishes a connection between identity testing and verifying quantum advantage, specifically in the context of Boson Sampling.

Selective Verification of Quantum Sampling Advantage

This research develops a new framework for verifying sampling-based quantum advantage and addresses a fundamental challenge in hypothesis testing: determining whether samples originate from a known distribution. The study establishes a formal definition of “selective-verifiability”, crucial for assessing whether a distribution can be reliably distinguished from others. Scientists designed this definition to address limitations in existing approaches, which often assume inefficient algorithms or do not account for the complexities of quantum computation. They demonstrated that every quantumly samplable distribution is verifiable using a classical deterministic algorithm, requiring a limited number of steps with access to a specific computational tool.

Conversely, the research reveals that if one-way functions exist, then sufficiently random classically samplable distributions cannot be efficiently verified. Further analysis explored scenarios under different computational assumptions, proving that if specific cryptographic constructs, known as QEFID pairs, exist, then a quantumly samplable distribution can be constructed that resists efficient verification. However, if one-way functions do not exist, every classically samplable distribution becomes efficiently verifiable. Finally, the study demonstrates that if quantum-resistant cryptographic puzzles do not exist, verifying sampling-based quantum advantage becomes possible with a quantum computer.

Efficient Identity Testing of Verifiable Distributions

This work addresses a fundamental challenge in hypothesis testing: determining whether a sample originates from a known distribution, a problem known as identity testing. Researchers established that, when dealing with distributions supported by a limited range of values, the optimal sample size for accurate testing is approximately the square root of that range. However, many practical distributions exhibit exponentially increasing complexity, requiring exponentially large sample sizes for accurate testing. To overcome this limitation, the team focused on efficiently verifiable distributions, those for which an efficient identity tester exists that cannot be fooled by efficiently sampled data.

They demonstrated that every quantumly or classically efficiently samplable distribution is verifiable using a classical deterministic algorithm, requiring a limited number of steps with access to a specific computational tool. Further investigation revealed a strong connection between the difficulty of verifying classical distributions and the existence of one-way functions. Specifically, the team proved that if one-way functions exist, then no sufficiently random classically efficiently samplable distribution is efficiently verifiable. Conversely, if one-way functions do not exist, then every classically efficiently samplable distribution is efficiently verifiable. The research also explores the verification of quantum advantage, particularly in sampling-based quantum computing models like Boson sampling and the random-circuit model, demonstrating that if QEFID pairs exist, there exists a quantumly efficiently samplable distribution that is not efficiently verifiable.

Quantum Verification and One-Way Functions Exist

This work addresses a fundamental challenge in hypothesis testing, specifically the problem of identifying whether a sample originates from a known distribution. Researchers investigated this issue by considering restrictions that move beyond previously established lower bounds on sample complexity. The team demonstrated that any distribution which can be sampled using quantum computation can be verified using efficient algorithms. Furthermore, the study establishes a link between the efficient verification of distributions and the existence of one-way functions, demonstrating that classically samplable distributions are efficiently verifiable only if one-way functions exist. Conversely, if one-way functions do not exist, all classically samplable distributions can be efficiently verified. The research also highlights that the existence of specific quantum phenomena, known as QEFID pairs, implies the existence of quantumly samplable distributions that resist efficient verification.