Expressivity of Variational Quantum Machine Learning on the Boolean Cube

Abstract

Categorical data plays an important part in machine learning research and appears in a variety of applications. Models that can express large classes of real-valued functions on the Boolean cube are useful for problems involving discrete-valued data types, including those which are not Boolean. To this date, the commonly used schemes for embedding classical data into variational quantum machine learning models encode continuous values. Here we investigate quantum embeddings for encoding Boolean-valued data into parameterized quantum circuits used for machine learning tasks. We narrow down representability conditions for functions on the $n$-dimensional Boolean cube with respect to previously known results, using two quantum embeddings: a phase embedding and an embedding based on quantum random access codes. We show that for any real-valued function on the $n$-dimensional Boolean cube, there exists a variational linear quantum model based on a phase embedding using $n$ qubits that can represent it and an ensemble of such models using $d<n$ qubits that can express any function with degree at most $d$. Additionally, we prove that variational linear quantum models that use the quantum random access code embedding can express functions on the Boolean cube with degree $d\le \lceil \frac{n}{3}\rceil$ using $\lceil \frac{n}{3}\rceil$ qubits, and that an ensemble of such models can represent any function on the Boolean cube with degree $d\le \lceil \frac{n}{3}\rceil$. Furthermore, we discuss the potential benefits of each embedding and the impact of serial repetitions. Lastly, we demonstrate the use of the embeddings presented by performing numerical simulations and experiments on IBM quantum processors using the Qiskit machine learning framework.

References in corpus (55)

• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Supervised learning with quantum enhanced feature spaces
• Barren plateaus in quantum neural network training landscapes
• A Quantum Approximate Optimization Algorithm
• Quantum machine learning in feature Hilbert spaces
• Quantum Circuit Learning
• Kernel methods in machine learning
• Dynamical suppression of decoherence in two-state quantum systems
• Quantum Convolutional Neural Networks
• Parameterized quantum circuits as machine learning models
• Evaluating analytic gradients on quantum hardware
• The power of quantum neural networks
• From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Power of data in quantum machine learning
• Data re-uploading for a universal quantum classifier
• Variational ansatz-based quantum simulation of imaginary time evolution
• A rigorous and robust quantum speed-up in supervised machine learning
• Generalization in quantum machine learning from few training data
• Quantum Natural Gradient
• Theory of variational quantum simulation
• Training variational quantum algorithms is NP-hard
• Quantum Computing for Finance: State of the Art and Future Prospects
• Absence of Barren Plateaus in Quantum Convolutional Neural Networks
• Quantum algorithms for algebraic problems
• Quantum machine learning beyond kernel methods
• Structure optimization for parameterized quantum circuits
• Variational Quantum Linear Solver
• Theory of overparametrization in quantum neural networks
• Generalization in Quantum Machine Learning: a Quantum Information Perspective
• Entanglement Devised Barren Plateau Mitigation
• Capacity and quantum geometry of parametrized quantum circuits
• Efficient measure for the expressivity of variational quantum algorithms
• A Survey of Quantum Computing for Finance
• Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces
• Representation Learning via Quantum Neural Tangent Kernels
• On the statistical complexity of quantum circuits
• Characterizing the loss landscape of variational quantum circuits
• Monotonic Calibrated Interpolated Look-Up Tables
• Quantum Analytic Descent
• Quantum Algorithms for Learning and Testing Juntas
• A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver
• Quantum Machine Learning for Finance
• Toward Trainability of Quantum Neural Networks
• A Fine-Grained Spectral Perspective on Neural Networks
• Structural risk minimization for quantum linear classifiers
• Pseudo-dimension of quantum circuits
• Bandwidth Enables Generalization in Quantum Kernel Models
• A Convergence Theory for Over-parameterized Variational Quantum Eigensolvers
• Quantum Random Access Codes for Boolean Functions
• Avoiding local minima in Variational Quantum Algorithms with Neural Networks
• Approximate Solutions of Combinatorial Problems via Quantum Relaxations
• Two new results about quantum exact learning

Cited by in corpus (3)

• Understanding quantum machine learning also requires rethinking generalization
• The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze
• Quantum Deep Hedging