Duality approach to quantum annealing of the 3-XORSAT problem
- URL: http://arxiv.org/abs/2106.06344v1
- Date: Fri, 11 Jun 2021 12:30:08 GMT
- Title: Duality approach to quantum annealing of the 3-XORSAT problem
- Authors: Raimel Medina, Maksym Serbyn
- Abstract summary: We study the performance of quantum algorithms for models with a unique ground state on simple hypergraphs.
The degeneracy of classical ground state manifold translates into the emergence of an extensive number of $Z$ symmetries.
The duality developed in this work provides a practical tool for studies of quantum models with classically degenerate energy manifold.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Classical models with complex energy landscapes represent a perspective
avenue for the near-term application of quantum simulators. Until now, many
theoretical works studied the performance of quantum algorithms for models with
a unique ground state. However, when the classical problem is in a so-called
clustering phase, the ground state manifold is highly degenerate. As an
example, we consider a 3-XORSAT model defined on simple hypergraphs. The
degeneracy of classical ground state manifold translates into the emergence of
an extensive number of $Z_2$ symmetries, which remain intact even in the
presence of a quantum transverse magnetic field. We establish a general duality
approach that restricts the quantum problem to a given sector of conserved
$Z_2$ charges and use it to study how the outcome of the quantum adiabatic
algorithm depends on the hypergraph geometry. We show that the tree hypergraph
which corresponds to a classically solvable instance of the 3-XORSAT problem
features a constant gap, whereas the closed hypergraph encounters a
second-order phase transition with a gap vanishing as a power-law in the
problem size. The duality developed in this work provides a practical tool for
studies of quantum models with classically degenerate energy manifold and
reveals potential connections between glasses and gauge theories.
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