Quantum decision trees with information entropy

TL;DR

This paper introduces a quantum decision tree algorithm that uses information gain to classify quantum states, demonstrating improved performance with physically-motivated observables but facing challenges with large, Haar-random states.

Contribution

The paper develops a quantum decision tree method based on information gain, avoiding neural networks, and applies it to classify quantum states and Hamiltonian ground states.

Findings

Effective classification of Haar-random quantum states with simulations.

Performance improves with physically-motivated observables.

Exponential suppression of information gain with system size.

Abstract

We present a classification algorithm for quantum states, inspired by decision-tree methods. To adapt the decision-tree framework to the probabilistic nature of quantum measurements, we utilize conditional probabilities to compute information gain, thereby optimizing the measurement scheme. For each measurement shot on an unknown quantum state, the algorithm selects the observable with the highest expected information gain, continuing until convergence. We demonstrate using the simulations that this algorithm effectively identifies quantum states sampled from the Haar random distribution. However, despite not relying on circuit-based quantum neural networks, the algorithm still encounters challenges akin to the barren plateau problem. In the leading order, we show that the information gain is proportional to the variance of the observable's expectation values over candidate states. As the system size increases, this variance, and consequently the information gain, are exponentially suppressed, which poses significant challenges for classifying general Haar-random quantum states. Finally, we apply the quantum decision tree to classify the ground states of various Hamiltonians using physically-motivated observables. On both simulators and quantum computers, the quantum decision tree yields better performances when compared to methods that are not information-optimized. This indicates that the measurement of physically-motivated observables can significantly improve the classification performance, guiding towards the future direction of this approach.