Adiabatic Dynamics of Entanglement

TL;DR

This paper explores how entanglement dynamics during adiabatic evolution are linked to energy level crossings, affecting computational complexity and quantum advantage in quantum algorithms.

Contribution

It reveals the relationship between entanglement changes, avoided level crossings, and computational hardness in adiabatic quantum processes.

Findings

Entanglement changes are tied to avoided energy level crossings.

The speed of adiabatic evolution is constrained by the narrowness of these crossings.

Entanglement requirements relate to the ruggedness of the energy landscape.

Abstract

We show that, during adiabatic evolution, any changes in entanglement can be attributed to a succession of avoided energy level crossings at which eigenvalues swap their eigenvectors. These swaps mediate the generation and redistribution of entanglement in multipartite systems. The efficiency of this redistribution depends on the narrowness of the avoided level crossings and thus constrains the speed of adiabatic evolution. Moreover, we relate the amount of entanglement involved to the ruggedness of the energy landscape, which directly affects the hardness of a computational problem. This enables an analysis of computational complexity and quantum advantage from the point of view of entanglement requirements. Applied to adiabatic quantum computation, our findings directly relate the computation's speed to its utilization of entanglement as a resource. The same principles extend to gate-based discretized adiabatic quantum algorithms, including those for Hamiltonian simulation and combinatorial optimization.