Dimension reduction with structure-aware quantum circuits for hybrid machine learning

TL;DR

This paper introduces a structure-aware quantum circuit approach for data compression in hybrid machine learning, leveraging tensor network decompositions to achieve efficient $k$-rank approximations and exponential input compression.

Contribution

It proposes a novel quantum circuit design based on tensor decompositions for data reduction, integrated with classical neural networks for hybrid learning.

Findings

Quantum circuits can effectively approximate reduced-form data representations.

The method achieves exponential compression of input data.

Experimental results confirm successful data compression and approximation.

Abstract

Schmidt decomposition of a vector can be understood as writing the singular value decomposition (SVD) in vector form. A vector can be written as a linear combination of tensor product of two dimensional vectors by recursively applying Schmidt decompositions via SVD to all subsystems. Given a vector expressed as a linear combination of tensor products, using only the $k$ principal terms yields a $k$-rank approximation of the vector. Therefore, writing a vector in this reduced form allows to retain most important parts of the vector while removing small noises from it, analogous to SVD-based denoising. In this paper, we show that quantum circuits designed based on a value $k$ (determined from the tensor network decomposition of the mean vector of the training sample) can approximate the reduced-form representations of entire datasets. We then employ this circuit ansatz with a classical neural network head to construct a hybrid machine learning model. Since the output of the quantum circuit for an $2^n$ dimensional vector is an $n$ dimensional probability vector, this provides an exponential compression of the input and potentially can reduce the number of learnable parameters for training large-scale models. We use datasets provided in the Python scikit-learn module for the experiments. The results confirm the quantum circuit is able to compress data successfully to provide effective $k$-rank approximations to the classical processing component.