# About 2D graph state and branching MERA

In my former post I asked if a 2D graph state on a 2D lattice can be represented by branching MERA. I got an answer that it seems this is true.

Then I have to following deductions

(1) 2D graph state on a 2D lattice is universal for measurement based quantum computation, so that all quantum computations can be achieved by a 2D graph state with local measurements.

(2) Branching MERA state can be classically simulated in the meaning that local observables on such a state can be computed efficiently.

(3) If (1)(2) are true, then the local measurement on the 2D graph state can be classically simulated.

(4) From (1)(2)(3), it seems that all quantum computations can be classically simulated.

This of course not true. What's wrong with my deduction?

• Could you please devise the simulation algorithm you mention in (3)? – Norbert Schuch Oct 29 '18 at 10:14
• A local measurement is a function on the local density matrix. To compute the local observables on a branching MERA state, we must at least have the capability to compute the local density matrix. So (2) means that local density matrix can be computed efficiently. If the local density matrix can be efficiently computed, then the local measurement can be classically computed. Something wrong here? – XXDD Oct 29 '18 at 10:19