Sampling a Haar Random State Conditioned on Having Low Entanglement Entropy

Question

After applying exponentially, or even polynomially many random local gates to a fixed input state, the resulting distribution of the output state $\lvert\psi_{out}\rangle$ will (be very close) to Haar random. This gives a procedural way of generating Haar-random states, given the ability to apply gates from a fixed gate set: generate some random circuit and take the output.

My question: is there a similar procedure for generating a (close to) Haar-random state conditioned on the state having low entanglement entropy (say constant)?

One attempt to generate low-entanglement states could be to apply a random-constant depth circuit instead of, say, a polynomial depth circuit, but this would not suffice for Haar-randomness. I have seen the concept of a random matrix product state here, but don't have the background to determine to what extent this setup coincides with the Haar-random definition above.

Answer 1

It is important to be precise what you mean by Haar-random here. Consider a Hilbert space $\mathcal{H}$ of dimension $d$. If you want to integrate over the Haar measure on $U(d)$, then the average entanglement entropy will not be constant in $d$.

If you are interested in a random matrix ensemble with bounded entanglement entropy, a matrix product state (MPS) ansatz as in your reference is a reasonable choice. If you fix the bond dimension $\chi$ you have a constant entanglement entropy by construction. Drawing $n$ Haar-random unitaries for the MPS, however, refers to the Haar measure on $U(\chi)^{\otimes n}$, which is a smaller space, in general.