  The tunneling decay event of a metastable state in a fully connected quantum
spin model can be simulated efficiently by path integral quantum Monte Carlo
(QMC) [Isakov $et~al.$, Phys. Rev. Lett. ${\bf 117}$, 180402 (2016).]. This is
because the exponential scaling with the number of spins of the
thermally-assisted quantum tunneling rate and the Kramers escape rate of QMC
are identical [Jiang $et~al.$, Phys. Rev. A ${\bf 95}$, 012322 (2017).], a
result of a dominant instantonic tunneling path. In Ref. [1], it was also
conjectured that the escape rate in open-boundary QMC is quadratically larger
than that of conventional periodic-boundary QMC, therefore, open-boundary QMC
might be used as a powerful tool to solve combinatorial optimization problems.
The intuition behind this conjecture is that the action of the instanton in
open-boundary QMC is a half of that in periodic-boundary QMC. Here, we show
that this simple intuition---although very useful in interpreting some
numerical results---deviates from the actual situation in several ways. Using a
fully connected quantum spin model, we derive a set of conditions on the
positions and momenta of the endpoints of the instanton, which remove the extra
degrees of freedom due to open boundaries. In comparison, the half-instanton
conjecture incorrectly sets the momenta at the endpoints to zero. We also found
that the instantons in open-boundary QMC correspond to quantum tunneling events
in the symmetric subspace (maximum total angular momentum) at all temperatures,
whereas the instantons in periodic-boundary QMC typically lie in subspaces with
lower total angular momenta at finite temperatures. This leads to a lesser than
quadratic speedup at finite temperatures. We also outline the generalization of
the instantonic tunneling method to many-qubit systems without permutation
symmetry using spin-coherent-state path integrals.
