Tripartite entanglement and matrix inversion quantum algorithm

TL;DR

This paper analyzes the evolution of tripartite entanglement throughout the HHL quantum algorithm for matrix inversion, revealing that entanglement is generated and then largely destroyed during the process.

Contribution

It provides a detailed step-by-step analysis of tripartite entanglement dynamics in the HHL algorithm, highlighting that entanglement is not maximized and is mostly annihilated by the end.

Findings

Tripartite entanglement is generated during the quantum phase estimation step.

Entanglement is diminished during the rotation step.

Complete annihilation of entanglement occurs at the final inverse-QPE step.

Abstract

The role of entanglement is discussed in the Harrow-Hassidim-Lloyd (HHL) algorithm. We compute all tripartite entanglement at every steps of the HHL algorithm. The tripartite entanglement is generated in the first quantum phase estimation (QPE) step. However, it turns out that amount of the generated entanglement is not maximal except very rare cases. In the second rotation step some tripartite entanglement is annihilated. Thus, the net tripartite entanglement is diminished. At the final inverse-QPE step the matrix inversion task is completed at the price of complete annihilation of the entanglement. An implication of this result is discussed.