# Quantum Compression of Tensor Network States

## TL;DR

This paper introduces quantum algorithms for efficiently compressing tensor network states, establishing bounds on memory requirements and providing polynomial-time algorithms for matrix product states.

## Contribution

It presents a new quantum compression method for tensor network states, with tight bounds on memory and an efficient algorithm for matrix product states.

## Key findings

- Memory bound determined by minimum cut of flow network
- Upper bound is tight when edge dimensions are powers of the same integer
- Algorithm runs in polynomial time for matrix product states

## Abstract

We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by the minimum cut of a suitable flow network, and is related to the flow of information from the manifold of parameters that specify the states to the physical systems in which the states are embodied. For given network topology and given edge dimensions, our upper bound is tight when all edge dimensions are powers of the same integer. When this condition is not met, the bound is optimal up to a multiplicative factor smaller than 1.585. We then provide a compression algorithm for general state families, and show that the algorithm runs in polynomial time for matrix product states.