# Can any 1D critical state be represented by a MERA tensor network?

My understanding of the Multiscale Entanglement Renormalisation Ansatz (MERA) is that it is designed to represent highly entangled, but low complexity states.

Is MERA capable of representing high complexity states? For example, could it represent the history ground state of a Feynman-Kitaev Hamiltonian which encodes some computation? Are we able to describe these states (in theory) using a MERA, but in practice finding the isometries and unitaries necessary is computationally intractable? What prevents us from being able to describe these states?

It depends what you mean by "high complexity states", and by "highly entangled".

MERA can describe states which have an entanglement scaling $$E\propto \log N$$, where $$N$$ is the length of the chain, with a fixed bond dimension $$\chi$$.

If you are talking about history state Hamiltonians for a QMA problem, or states such as those in arXiv:1408.1657, those have an entanglement which scales algebraically with the length of the chain, $$E\propto N^\alpha$$ -- this requires a large enough bond dimension $$\chi$$ to reproduce this entanglement, this is, $$\chi$$ will scale like $$\exp(N^\alpha)$$, and is thus inefficient.

Note that all of these states are ground states of local Hamiltonians, and in that sense are low (Kolmogorov) complexity.

• Thanks for the reply! Are there any tensor network techniques which can be used to represent history states efficiently (of the type for QMA problems)? Also, is there a reference for the entropy of QMA history states having entropy scaling as $N^\alpha$? – user138901 May 14 at 15:56
• @user138901 Regarding the history state, it is a superposition of configurations at a time $t$ in the computation. Just run a circuit which creates a highly entangled state. (Entanglement=size of circuit, size of system = size of circuit * total time = poly(size of circuit)). Regarding the first question, there should be tensor network techniques for that, by considering the circuit for the state at time $t$ as a tensor network, and then superposing the different times (tensor network for a W state). But why would you want that? – Norbert Schuch May 14 at 17:09
• I'm trying to find a renormalization scheme for a Hamiltonian encoding a QMA computation (or similar). I was wondering if one could represent the ground state history state as a MERA, then apply a renormalization scheme in the usual way (using the raising superoperators). However, if we need a superposition of MERAs (or another tensor network) to do this,would this renomalization scheme still work? Are there other renormalization schemes that might work for this? I'd only need it for a 1D Hamiltonian. – user138901 May 15 at 10:00
• @user138901 What does "renormalization scheme for a Hamiltonian encoding a QMA computation" mean? What would be its properties? What would it do to the history state? Ad hoc this does not seem to make much sense to me. – Norbert Schuch May 15 at 10:35
• Ideally the RG scheme would be a real space scheme -- similar to blocking -- that would preserve the ground state energy (or the low energy subspace) of the Hamiltonian while reducing the number of spins/particles in the system. Since the ground state energy is preserved, then determining the ground state of this new, renormalized Hamiltonian is still QMA-hard. Something like that. It seemed like MERA was promising since, if we could describe the ground state history state as a MERA, then MERA provides a natural real space renormalization scheme with similar properties. – user138901 May 15 at 12:37