Anomaly Detection & Density Estimation with Density Matrices

Here we present our project for the Qiskit Hackathon Global 2021

The preprint of the work with additional experiments can be found at https://arxiv.org/abs/2201.10006

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Our Group:

Quantum Mind UN

Diego Useche - [email protected]

Oscar Alberto Bustos B. - [email protected]

The problem:

Anomaly Detection:

For Qiskit Fall Fest Hackathon 2021, we proposed a quantum protocol to compute the expected value of any nxn density matrix. In this project, we improve this quantum circuit to develop a new method for anomaly detection. The method works by estimating the probability density function (pdf) of normal data and classifying as anomalies new data which lies below a probability threshold in the pdf.

Can we build a quantum circuit to perform Anomaly Detection?

Our Approach:

Anomaly Detection based on Density Estimation:

• Use normal data to induce a probability density function (pdf) (Density Estimation)
• Use the pdf to estimate the density of new samples. If the density is below a threshold, then classify it as anomaly.

Quantum Density Estimation:

• Combine Density Matrices with Random Fourier Features to perform Density Estimation.
• Propose a quantum circuit to estimate expected values of density matrices.

Main contributions:

• We developed a new method for Anomaly Detection in quantum computers.
• We implemented the method in a quantum circuit and showed that it’s a feasible approach.
• Most quantum machine learning algorithms are based on pure states. Our algorithm can work with mixed states and pure states.
• We develop a new method to calculate the expected value of a density matrix.
• We used this method to approximate a probability density function (pdf) in a quantum computer.