Learnability and Complexity of Quantum Samples

TL;DR

This paper investigates the learnability of quantum data using various generative models, demonstrating exponential complexity growth in learning parameters and highlighting the effectiveness of LSTM in capturing quantum distributions.

Contribution

It provides the first comprehensive analysis of the complexity of learning quantum samples with different generative models, including theoretical proofs and numerical experiments.

Findings

LSTM outperforms other models in learning quantum samples

Learning complexity grows exponentially with the number of qubits

A connection between learnability and model complexity is established

Abstract

Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through quantum sample learning: given samples from an n-qubit computation, can we learn the underlying quantum distribution using models with training parameters that scale polynomial in n under a fixed training time? We study four kinds of generative models: Deep Boltzmann machine (DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM) and Autoregressive GAN, on learning quantum data set generated by deep random circuits. We demonstrate the leading performance of LSTM in learning quantum samples, and thus the autoregressive structure present in the underlying quantum distribution from random quantum circuits. Both numerical experiments and a theoretical proof in the case of the DBM show exponentially growing complexity of learning-agent parameters required for achieving a fixed accuracy as n increases. Finally, we establish a connection between learnability and the complexity of generative models by benchmarking learnability against different sets of samples drawn from probability distributions of variable degrees of complexities in their quantum and classical representations.