Circuit depth versus energy in topologically ordered systems
- URL: http://arxiv.org/abs/2210.06796v1
- Date: Thu, 13 Oct 2022 07:21:30 GMT
- Title: Circuit depth versus energy in topologically ordered systems
- Authors: Arkin Tikku and Isaac H. Kim
- Abstract summary: We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We prove a nontrivial circuit-depth lower bound for preparing a low-energy
state of a locally interacting quantum many-body system in two dimensions,
assuming the circuit is geometrically local. For preparing any state which has
an energy density of at most $\epsilon$ with respect to Kitaev's toric code
Hamiltonian on a two dimensional lattice $\Lambda$, we prove a lower bound of
$\Omega\left(\min\left(1/\epsilon^{\frac{1-\alpha}{2}},
\sqrt{|\Lambda|}\right)\right)$ for any $\alpha >0$. We discuss two
implications. First, our bound implies that the lowest energy density
obtainable from a large class of existing variational circuits (e.g.,
Hamiltonian variational ansatz) cannot, in general, decay exponentially with
the circuit depth. Second, if long-range entanglement is present in the ground
state, this can lead to a nontrivial circuit-depth lower bound even at nonzero
energy density. Unlike previous approaches to prove circuit-depth lower bounds
for preparing low energy states, our proof technique does not rely on the
ground state to be degenerate.
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