Anomaly Detection Based on Quantum Autoencoder

Quantum autoencoder (QAE) is a key tool in quantum machine learning and can be applied to areas such as dimensionality reduction and feature extraction [1]. Anomaly detection identifies deviations from normal patterns, which is crucial yet challenging due to the need to detect risks while minimizing false positives [2]. We propose a QAE-based anomaly detection scheme using an optical quantum neural network. Figure 1 (A) shows the basic structure of the QAE: the encoder compresses the input quantum state into a low-dimensional latent space, and the decoder reconstructs the original quantum state from the latent space. QAE-based anomaly detection works by learning the low-dimensional features of normal data and minimizing its reconstruction error. Normal data can be well reconstructed, while anomalous data exhibits a larger reconstruction error. When the reconstruction error of test data exceeds a set threshold, it is classified as anomalous. We constructed the QAE using an optical quantum neural network consisting of 50:50 beam splitters and controllable phase shifters [3]. Due to the unitary matrix property, only the encoder needs to be trained, and the decoder is simply the conjugate transpose of the encoder. To compress an N-dimensional quantum state <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\rho$</tex> into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${K}$</tex> dimensions, efficient compression can be achieved by minimizing the expectation value of the projection onto the trash state, given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C(\theta,\phi)=Tr\left(PU(\theta,\phi)\rho U^+(\theta,\phi)\right)$</tex> [4]. Here, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P=(I_{K}\oplus\vert 0\rangle^{\otimes N-K}\langle 0\vert ^{\otimes(N-K)})$</tex> represents the projection onto the trash state, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$I_{K}$</tex> is the K-dimensional identity matrix, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\vert 0\rangle^{\otimes(N-K)}\text{is}$</tex> the trash state, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Tr$</tex> denotes the trace, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$U(\theta,\phi)$</tex> is the encoder with adjustable parameters, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$U\dagger(\theta_,\phi)$</tex> is the decoder. We only need to train the encoder during the training process, and the decoder is simply its conjugate transpose.