Quantum density estimation with density matrices: Application to quantum anomaly detection

Abstract

Density estimation is a central task in statistics and machine learning. This problem aims to determine the underlying probability density function that best aligns with an observed data set. Some of its applications include statistical inference, unsupervised learning, and anomaly detection. Despite its relevance, few works have explored the application of quantum computing to density estimation. In this article, we present a novel quantum-classical density matrix density estimation model, called Q-DEMDE, based on the expected values of density matrices and a novel quantum embedding called quantum Fourier features. The method uses quantum hardware to build probability distributions of training data via mixed quantum states. As a core subroutine, we propose a new algorithm to estimate the expected value of a mixed density matrix from its spectral decomposition on a quantum computer. In addition, we present an application of the method for quantum-classical anomaly detection. We evaluated the density estimation model with quantum random and quantum adaptive Fourier features on different data sets on a quantum simulator and a real quantum computer. An important result of this work is to show that it is possible to perform density estimation and anomaly detection with high performance on present-day quantum computers.

arXiv comment27 pages, 13 figures

References in corpus (30)

• Quantum Computing in the NISQ era and beyond
• A variational eigenvalue solver on a quantum processor
• Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
• Supervised learning with quantum enhanced feature spaces
• On the Measurement of Qubits
• Quantum support vector machine for big data classification
• Quantum Circuit Learning
• Quantum principal component analysis
• Circuit-centric quantum classifiers
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Efficient quantum state tomography
• Synthesis of Quantum Logic Circuits
• A rigorous and robust quantum speed-up in supervised machine learning
• Quantum Circuits for General Multiqubit Gates
• Anomaly Detection with Density Estimation
• Learning the quantum algorithm for state overlap
• Quantum Circuits for Isometries
• Quantum machine learning for quantum anomaly detection
• Nonparametric Density Estimation for High-Dimensional Data - Algorithms and Applications
• Anomaly detection with variational quantum generative adversarial networks
• Quantum Anomaly Detection with Density Estimation and Multivariate Gaussian Distribution
• Quantum algorithms for anomaly detection using amplitude estimation
• Quantum Mixed State Compiling
• A novel approach to noisy gates for simulating quantum computers
• Learning with Density Matrices and Random Features
• Compact quantum kernel-based binary classifier
• Generalized Coherent States, Reproducing Kernels, and Quantum Support Vector Machines
• Supervised Learning with Quantum Measurements
• Quantum Measurement Classification with Qudits
• Optimisation-free Classification and Density Estimation with Quantum Circuits

Cited by in corpus (2)

• A near-term quantum simulation of the transverse field Ising model hints at Glassy Dynamics
• Quantum generative classification with mixed states