Or dare we say... Engineering? 😬

Thanks Zoltan! You should petition the museum to add some hyperbolic tilings as well, there's plenty of material in our papers. 😁

It would be a lost opportunity if they didn't call it the Ministry of Magic (state distillation).

For more details, you'll have to read our paper! As always, many thanks for the support of Berlin Quantum for our work at @freieuniversitaet.bsky.social.
arxiv.org/abs/2103.02634

This suggests a deep relationship between equilibration strength and entanglement phases in many-body quantum systems! The main idea: More entanglement = stronger equilibration.

For the condensed-matter theorists among you, our work also leads to an interesting conjecture: RTNs on different geometries describe different phases of entanglement scaling. We show that D_eff follows a sharp hierarchy between area- and volume-law phases.

This means that random tensor networks know a lot more about holographic dynamics than we expected, and may be able to hold more insights into (holographic) quantum gravity.

And surprisingly, the result matches gravitational degree-of-freedom counting in holography: If we "fuse" tensors together, i.e., replace part of the bulk geometry by a "black hole", D_eff always *increases*. Just as in gravity, where a black hole is the highest-entropy state!

This brings us to holography: For holographic RTNs, we can now compute the minimum effective dimension D_eff that describes late-time dynamics! From the geometry and bond dimension of the RTN alone, we can determine how complex its dynamics must be.

Now here's the kicker: For random ensembles, we can strictly lower-bound D_eff *without knowing H*! In a sense, the randomness cancels out its exact eigenstate structure. This is a trick we learned from Haferkamp et al., who used it on random MPS:
arxiv.org/abs/2103.02634

The key quantity to describe the strength of equilibration is the "effective dimension" D_eff, which basically counts how many (energy) states are needed to describe late-time dynamics.

Here's how it works: In a quantum system, expectation values of observables fluctuate. At late times, even a pure state will *equilibrate*, meaning that local expectation values will fluctuate within a fixed window. This happens for all Hamiltonians H with "non-degenerate gaps".

In our paper, we bring in ideas from quantum statistical mechanics to show that the opposite is true: Thanks to the randomness in RTNs, we can probe late-time dynamics without knowing the explicit Hamiltonian! The key concept that enables this is called *equilibration*.

That makes choosing a Hamiltonian that performs time evolution on the boundary difficult: Any choice, e.g. motivated from AdS/CFT arguments, would time-evolve different RTN samples differently. Thus, it seemed that randomness made time evolution impossible to describe!

This sparked hundreds of follow-up papers - many of which refined the original proposal - but there was one limitation: Random tensor networks (RTNs) produce an *ensemble* of states, with every random sample looking quite different locally.

Some background: In a seminal paper from 2016, Hayden et al. showed that tensor networks with locally random tensors, if put on a hyperbolic geometry, reproduce quantum states that very closely resemble boundary states of the AdS/CFT duality.
arxiv.org/abs/1601.01694