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# Finite time teleportation phase transition in random quantum circuits

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### Abstract

How long does it take to entangle two distant qubits in a quantum circuit evolved by generic unitary dynamics? We show that if the time evolution is followed by measurement of all but the two {\em infinitely} separated test qubits, then the entanglement between them can undergo a phase transition and become nonzero at a finite critical time $$t_c$$. The fidelity of teleporting a quantum state from an input qubit to an infinitely distant output qubit shows the same critical onset. Specifically, these finite time transitions occur in short-range interacting two-dimensional random unitary circuits and in sufficiently long-range interacting one-dimensional circuits. The phase transition is understood by mapping the random continuous-time evolution to a finite temperature thermal state of an effective spin Hamiltonian, where the inverse temperature equals the evolution time in the circuit. In this framework, the entanglement between two distant qubits at times $$t>t_c$$ corresponds to the emergence of long-range ferromagnetic spin correlations below the critical temperature. We verify these predictions using numerical simulation of Clifford circuits and propose potential realizations in existing platforms for quantum simulation.

### Author and article information

###### Journal
13 October 2021
###### Article
2110.06963
1a48fa1d-3097-4630-a303-4511b3758a94