One qubit as a Universal Approximant

Abstract

A single-qubit circuit can approximate any bounded complex function stored in the degrees of freedom defining its quantum gates. The single-qubit approximant presented in this work is operated through a series of gates that take as their parameterization the independent variable of the target function and an additional set of adjustable parameters. The independent variable is re-uploaded in every gate while the parameters are optimized for each target function. The output state of this quantum circuit becomes more accurate as the number of re-uploadings of the independent variable increases, i. e., as more layers of gates parameterized with the independent variable are applied. In this work, we provide two different proofs of this claim related to both the Fourier series and the Universal Approximation Theorem for Neural Networks, and we benchmark both methods against their classical counterparts. We further implement a single-qubit approximant in a real superconducting qubit device, demonstrating how the ability to describe a set of functions improves with the depth of the quantum circuit. This work shows the robustness of the re-uploading technique on Quantum Machine Learning.

10 pages + 6 (appendix); 7 figures + 4 (appendix) Changes made for publication. The text has been changed to improve clarity. A stronger relationship between the model in this paper and Machine Learning was stated. Some content has been moved to appendix. Acknowledgements added

References in corpus (5)

• Simple pulses for elimination of leakage in weakly nonlinear qubits
• Robust randomized benchmarking of quantum processes
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Demonstrating a Driven Reset Protocol of a Superconducting Qubit
• Determining the proton content with a quantum computer

Cited by in corpus (33)

• Quantum machine learning beyond kernel methods
• Automatic design of quantum feature maps
• Style-based quantum generative adversarial networks for Monte Carlo events
• Learning quantum states and unitaries of bounded gate complexity
• Fock State-enhanced Expressivity of Quantum Machine Learning Models
• Multidimensional Fourier series with quantum circuits
• Quantum reservoir computing in finite dimensions
• Single-qubit universal classifier implemented on an ion-trap quantum device
• Fragmented imaginary-time evolution for early-stage quantum signal processors
• Reduction of finite sampling noise in quantum neural networks
• Fault-tolerant quantum algorithms for quantum molecular systems: A survey
• Neural quantum kernels: training quantum kernels with quantum neural networks
• Hybrid quantum learning with data re-uploading on a small-scale superconducting quantum simulator
• Magnetic penetration depth of Aluminum thin films
• Parameterized quantum circuits as universal generative models for continuous multivariate distributions
• Entanglement-induced provable and robust quantum learning advantages
• Direct implementation of a perceptron in superconducting circuit quantum hardware
• Qudit Machine Learning
• Satellite image classification with neural quantum kernels
• Symmetry-invariant quantum machine learning force fields
• Superconducting nitridized-aluminum thin films
• Multi-variable integration with a variational quantum circuit
• A didactic approach to quantum machine learning with a single qubit
• Gradients and frequency profiles of quantum re-uploading models
• Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs
• Quantum machine learning for multiclass classification beyond kernel methods
• Approximation and Generalization Capacities of Parametrized Quantum Circuits for Functions in Sobolev Spaces
• Quantum circuits for partial differential equations in Fourier space
• Expressivity of deterministic quantum computation with one qubit
• Single-Qudit Quantum Neural Networks for Multiclass Classification
• Universal approximation of continuous functions with minimal quantum circuits
• Quantum-Enhanced Neural Exchange-Correlation Functionals
• Multivariate unbounded quantum regression via log-ratio probabilities mitigating barren plateaus