Protocols for Trainable and Differentiable Quantum Generative Modelling

Abstract

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, simplified latent space training provides models that are automatically differentiable, and we show how samples from distributions propagated by stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

arXiv comment16 pages, 9 figures; first version, to be updated soon

References in corpus (59)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Quantum computational advantage using photons
• Quantum Teleportation is a Universal Computational Primitive
• Quantum machine learning in feature Hilbert spaces
• Quantum Circuit Learning
• Characterizing Quantum Supremacy in Near-Term Devices
• Strong quantum computational advantage using a superconducting quantum processor
• Parameterized quantum circuits as machine learning models
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Evaluating analytic gradients on quantum hardware
• Quantum Computational Supremacy
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• Quantum generative adversarial learning
• Quantum advantage in learning from experiments
• Quantum generative adversarial networks
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Training variational quantum algorithms is NP-hard
• A generative modeling approach for benchmarking and training shallow quantum circuits
• Quantum Computing for Finance: State of the Art and Future Prospects
• Layerwise learning for quantum neural networks
• Variational quantum algorithms for nonlinear problems
• Information Scrambling in Computationally Complex Quantum Circuits
• Differentiable Learning of Quantum Circuit Born Machine
• Experimental Quantum Generative Adversarial Networks for Image Generation
• Quantum computational finance: Monte Carlo pricing of financial derivatives
• Unsupervised Generative Modeling Using Matrix Product States
• Option Pricing using Quantum Computers
• Solving nonlinear differential equations with differentiable quantum circuits
• Theory of overparametrization in quantum neural networks
• High-precision quantum algorithms for partial differential equations
• Quantum Algorithm for Simulating the Wave Equation
• The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
• Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
• Quantum spectral methods for differential equations
• A Threshold for Quantum Advantage in Derivative Pricing
• Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
• Learning and Inference on Generative Adversarial Quantum Circuits
• The Bivariate Normal Copula
• Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces
• Adversarial quantum circuit learning for pure state approximation
• On the practical usefulness of the Hardware Efficient Ansatz
• Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
• Evolutional Deep Neural Network
• Towards Pricing Financial Derivatives with an IBM Quantum Computer
• Generalized quantum circuit differentiation rules
• Variational quantum simulations of stochastic differential equations
• Generative Quantum Learning of Joint Probability Distribution Functions
• A Framework for Demonstrating Practical Quantum Advantage: Racing Quantum against Classical Generative Models
• A variational quantum algorithm for the Feynman-Kac formula
• Dual-Parameterized Quantum Circuit GAN Model in High Energy Physics
• Efficient Online Quantum Generative Adversarial Learning Algorithms with Applications
• Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations
• Efficient Bipartite Entanglement Detection Scheme with a Quantum Adversarial Solver
• Efficient Hamiltonian Simulation for Solving Option Price Dynamics
• Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series
• F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits

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