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Efficient Generation of Generic Entanglement
R. Oliveira, O. C. O. Dahlsten, M. B. Plenio
TL;DR
The paper asks whether generic entanglement can be reached efficiently despite the complexity of multipartite entanglement and concerns that randomization may take infeasibly long. It analyzes random circuits of two-qubit gates and proves polynomial-time convergence of expected entanglement to its asymptotic value, with implications for protocols that assume generic entanglement can be generated efficiently.
Problem
The paper asks whether generic entanglement properties can be obtained in polynomial time using one- and two-qubit gates, given the complexity of multipartite entanglement and concerns that randomization may take infeasibly long.
Source: p. 1Method
The authors analyze circuits made from randomly chosen two-qubit gates, tracking the expected entanglement of states evolved from an arbitrary initial state.
Source: p. 2Results
Polynomially many random two-qubit gates suffice for the expected entanglement to converge to its asymptotic value at any fixed accuracy, independently of the initial state.
Source: p. 2Takeaways & Limitations
The result allows efficient random circuits to replace inefficient Haar-random unitary generation in a protocol argument, and is relevant to protocols that assume efficient generation of generic entanglement.
Source: pp. 2, 4
Abstract
from arXiv · showhide
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.