Constrained and Vanishing Expressivity of Quantum Fourier Models

Abstract

In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier Series which frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially when the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for Quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviours are observed in numerical simulations.

References in corpus (24)

• Variational Quantum Algorithms
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps
• Diagnosing Barren Plateaus with Tools from Quantum Optimal Control
• Barren Plateaus in Variational Quantum Computing
• Quantum machine learning beyond kernel methods
• Theory of overparametrization in quantum neural networks
• Equivalence of quantum barren plateaus to cost concentration and narrow gorges
• A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Does provable absence of barren plateaus imply classical simulability?
• Shadows of quantum machine learning
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Generalization despite overfitting in quantum machine learning models
• Classical surrogates for quantum learning models
• Exponential data encoding for quantum supervised learning
• Efficient classical algorithms for simulating symmetric quantum systems
• On fundamental aspects of quantum extreme learning machines
• Let Quantum Neural Networks Choose Their Own Frequencies
• Potential and limitations of random Fourier features for dequantizing quantum machine learning
• Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning
• Gradients and frequency profiles of quantum re-uploading models

Cited by in corpus (5)

• Barren Plateaus in Variational Quantum Computing
• Efficient quantum-enhanced classical simulation for patches of quantum landscapes
• Fourier Analysis of Variational Quantum Circuits for Supervised Learning
• On the Differential Topology of Expressivity of Parameterized Quantum Circuits
• When Quantum and Classical Models Disagree: Learning Beyond Minimum Norm Least Square