Quantum Encoding of Structured Data with Matrix Product States

TL;DR

This paper demonstrates that matrix product states enable efficient quantum encoding of structured data, such as images and functions, with high fidelity using shallow circuits, overcoming exponential complexity of amplitude encoding.

Contribution

It introduces a method using matrix product states for efficient quantum state preparation of structured data with high accuracy and low circuit depth.

Findings

Achieves >99.99% accuracy in representing low-degree polynomial functions.

Successfully encodes a 128x128 medical image with >99.2% fidelity on 14 qubits.

Uses shallow quantum circuits with only 425 gates for high-fidelity image encoding.

Abstract

The amplitude encoding of an arbitrary $n$-qubit state vector requires $\Omega(2^n)$ gate operations, owing to the exponential dimension of the Hilbert space. We can, however, form dimensionality-reduced representations of quantum states using matrix product states (MPS). In this article, we illustrate that MPS techniques enable the preparation of quantum states representative of functions with complexity up to low-degree piecewise polynomials via shallow-depth quantum circuits with accuracy exceeding 99.99\%. We extend these results to the approximate amplitude encoding of pixel values. We showcase this approach by efficiently preparing a $128\times 128$ ChestMNIST medical image (https://medmnist.com/) on 14 qubits with fidelity exceeding 99.2\% on a circuit with a total depth of just 425 single-qubit rotation and CNOT gates.