Adiabatic Dynamics of Entanglement

Abstract

We show that, during adiabatic evolution generated by a time-dependent Hamiltonian, any changes in entanglement can be attributed to a succession of avoided crossings in the Hamiltonian's instantaneous energy spectrum, at which its eigenvalues swap their associated 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 ruggedness of the energy landscape, which directly affects the hardness of a computational problem, to the extent to which entanglement must be redistributed among subsystems. This enables an analysis of computational complexity and quantum advantage in terms of the required entanglement redistribution. 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.

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