Generative Quantum Learning of Joint Probability Distribution Functions

Abstract

Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this work, we design quantum machine learning algorithms to model copulas. We show that any copula can be naturally mapped to a multipartite maximally entangled state. A variational ansatz we christen as a `qopula' creates arbitrary correlations between variables while maintaining the copula structure starting from a set of Bell pairs for two variables, or GHZ states for multiple variables. As an application, we train a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables for historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to 8 qubits and show that our results outperform those obtained through equivalent classical generative learning. Further, we present theoretical arguments for exponential advantage in our model's expressivity over classical models based on communication and computational complexity arguments.

arXiv comment19 pages, 11 figures. v2: published version

References in corpus (27)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• Quantum algorithm for solving linear systems of equations
• Barren plateaus in quantum neural network training landscapes
• Quantum support vector machine for big data classification
• Quantum Convolutional Neural Networks
• Characterizing Quantum Supremacy in Near-Term Devices
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Evaluating analytic gradients on quantum hardware
• Quantum generative adversarial learning
• A rigorous and robust quantum speed-up in supervised machine learning
• Quantum Data Fitting
• Benchmarking an 11-qubit quantum computer
• Quantum generative adversarial networks
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• A generative modeling approach for benchmarking and training shallow quantum circuits
• Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
• Differentiable Learning of Quantum Circuit Born Machine
• Experimental Quantum Generative Adversarial Networks for Image Generation
• Information-theoretic bounds on quantum advantage in machine learning
• The cost of exactly simulating quantum entanglement with classical communication
• Training of Quantum Circuits on a Hybrid Quantum Computer
• Quantum generative adversarial learning in a superconducting quantum circuit
• The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
• Basic circuit compilation techniques for an ion-trap quantum machine
• Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
• Enhancing Generative Models via Quantum Correlations

Cited by in corpus (20)

• Quantum computing for finance
• Application-Oriented Performance Benchmarks for Quantum Computing
• Orbital-optimized pair-correlated electron simulations on trapped-ion quantum computers
• Generative quantum machine learning via denoising diffusion probabilistic models
• Quantum State Preparation of Normal Distributions using Matrix Product States
• A performance characterization of quantum generative models
• Learning shallow quantum circuits
• Rapid training of quantum recurrent neural networks
• Conditional Born machine for Monte Carlo event generation
• Exploring Unsupervised Anomaly Detection with Quantum Boltzmann Machines in Fraud Detection
• Quantum Financial Modeling on Noisy Intermediate-Scale Quantum Hardware: Random Walks using Approximate Quantum Counting
• Entanglement-induced provable and robust quantum learning advantages
• Quantum generative adversarial learning in photonics
• Benchmarking Quantum Generative Learning: A Study on Scalability and Noise Resilience using QUARK
• Protocols for Trainable and Differentiable Quantum Generative Modelling
• Application-Oriented Benchmarking of Quantum Generative Learning Using QUARK
• The State Preparation of Multivariate Normal Distributions using Tree Tensor Network
• Contextual Quantum Neural Networks for Stock Price Prediction
• Cross-platform hardware benchmark of style-based quantum GANs for data augmentation on superconducting and trapped-ion processors
• Quantum Chebyshev Probabilistic Models for Fragmentation Functions