Classical Verification of Quantum Learning Advantages with Noises

TL;DR

This paper demonstrates that classical verification of quantum learning advantages remains feasible in noisy scenarios by introducing an error rectification algorithm that effectively reconstructs noise-free results from noisy quantum Fourier sampling, enabling practical verification on current noisy quantum devices.

Contribution

The paper introduces an efficient classical error correction algorithm for noisy quantum Fourier sampling and proves its effectiveness in verifying quantum learning advantages under realistic noise conditions.

Findings

The error rectification algorithm restores heavy Fourier coefficients with logarithmic sample complexity.

The approach enables efficient agnostic parity learning on noisy quantum devices.

Classical verification of quantum learning advantages is feasible with current noisy quantum technology.

Abstract

Classical verification of quantum learning allows classical clients to reliably leverage quantum computing advantages by interacting with untrusted quantum servers. Yet, current quantum devices available in practice suffers from a variety of noises and whether existed classical verification protocols carry over to noisy scenarios remains unclear. Here, we propose an efficient classical error rectification algorithm to reconstruct the noise-free results given by the quantum Fourier sampling circuit with practical constant-level noises. In particular, we prove that the error rectification algorithm can restore the heavy Fourier coefficients by using a small number of noisy samples that scales logarithmically with the problem size. We apply this algorithm to the agnostic parity learning task with uniform input marginal and prove that this task can be accomplished in an efficient way on noisy quantum devices with our algorithm. In addition, we prove that a classical client with access to the random example oracle can verify the agnostic parity learning results from the noisy quantum prover in an efficient way, under the condition that the Fourier coefficients are sparse. Our results demonstrate the feasibility of classical verification of quantum learning advantages with noises, which provide a valuable guide for both theoretical studies and practical applications with current noisy intermediate scale quantum devices.