Efficient Generation of Generic Entanglement

Abstract

We find that generic entanglement is physical, in the sense that it can be generated in polynomial time from two-qubit gates picked at random. We prove as the main result that such a process generates the average entanglement of the uniform (Haar) measure in at most $O(N^3)$ steps for $N$ qubits. This is despite an exponentially growing number of such gates being necessary for generating that measure fully on the state space. Numerics furthermore show a variation cut-off allowing one to associate a specific time with the achievement of the uniform measure entanglement distribution. Various extensions of this work are discussed. The results are relevant to entanglement theory and to protocols that assume generic entanglement can be achieved efficiently.