Expressivity of Quantum Neural Networks

TL;DR

This paper investigates the conditions under which deep quantum neural networks can accurately approximate target functions, revealing that the input data's Hilbert space dimension relative to the circuit's space is crucial for expressivity.

Contribution

It establishes a quantum analogue of the universal approximation theorem, identifying key conditions for quantum neural network expressivity and proposing methods to enhance it.

Findings

Accurate approximation requires input Hilbert space to be less than half of the circuit's Hilbert space.

Symmetric datasets can naturally satisfy the expressivity conditions.

Adding an ancillary qubit can restore expressivity when conditions are not met.

Abstract

In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with simple but typical physical situations that the target functions are physical observables, and then we extend our discussion to situations that the learning targets are not directly physical observables, but can be expressed as physical observables in an enlarged Hilbert space with multiple replicas, such as the Loshimidt echo and the Renyi entropy. The main finding is that an accurate approximation is possible only when the input wave functions in the dataset do not exhaust the entire Hilbert space that the quantum circuit acts on, and more precisely, the Hilbert space dimension of the former has to be less than half of the Hilbert space dimension of the latter. In some cases, this requirement can be satisfied automatically because of the intrinsic properties of the dataset, for instance, when the input wave function has to be symmetric between different replicas. And if this requirement cannot be satisfied by the dataset, we show that the expressivity capabilities can be restored by adding one ancillary qubit where the wave function is always fixed at input. Our studies point toward establishing a quantum neural network analogy of the universal approximation theorem that lays the foundation for expressivity of classical neural networks.