  Quantum phases of naturally-occurring systems exhibit distinctive collective
phenomena as manifestation of their many-body correlations, in contrast to our
persistent technological challenge to engineer at will such strong correlations
artificially. Here we show theoretically that quantum correlations exhibited in
the two-dimensional valence bond solid phase of a quantum antiferromagnet,
modeled by Affleck, Kennedy, Lieb, and Tasaki as a precursor of spin liquids
and topological orders, are sufficiently complex yet structured enough to
simulate universal quantum computation when every single spin can be measured
individually. This unveils that an intrinsic complexity of naturally-occurring
2D quantum systems -- which has been a long-standing challenge for traditional
computers -- could be tamed as a computationally valuable resource, even if we
are limited not to create newly entanglement during computation. Our
constructive protocol leverages a novel way to herald the correlations suitable
for deterministic quantum computation through a random sampling, and may be
extensible to other ground states of various 2D valence bond phases beyond the
AKLT state.
