Quantum annealing by transverse ferromagnetic interaction
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
Quantum information processing has been drawing increasing interests
in broad areas of physics.
In particular, we are interested in an optimization scheme
of classical potential functions using quantum mechanics, called quantum
annealing.
The purpose of quantum annealing is to obtain an approximate
ground state of the random Ising model. To this end, we
introduce quantum fluctuations and control them with an appropriate
time schedule. At first, quantum fluctuations are set at sufficiently
large values so as to make the ground state simple and easily obtained.
Then quantum fluctuations are turned off gradually and
we make the quantum spin state evolve from the initial ground state.
If the adiabaticity of spin state is kept throughout the evolution,
the ground state of the classical Ising system is obtained when quantum
fluctuations vanish at last.
Conventionally, Ising models in the transverse field have been considered
in order to perform quantum annealing, supposing the transverse field
as a carrier of quantum fluctuations.
Several previous studies have discussed the advantage of quantum
annealing by transverse field over simulated annealing,
the classical counterpart of quantum annealing.
However, the fact that there is a room for the choice of fluctuations is a
prominent feature of quantum annealing that does not exist in
simulated annealing.
In the present study, we introduce the transverse ferromagnetic
interaction in addition to the transverse field.
In our presentation, we discuss the accuracy of quantum annealing
by transverse ferromagnetic interaction in comparison with
conventional quantum annealing and simulated annealing.
As an optimization problem, the random-field Ising model is employed.
By analyses of eigenenergies obtained by numerical exact diagonalization
of the random-filed Ising model in the
transverse field with and without transverse ferromagnetic interaction,
it is suggested that quantum annealing by transverse ferromagnetic
interaction should performs better than conventional quantum annealing
for ground states with the ferromagnetic order.
Moreover, on the basis of annealing procedure in the mean-field
approximation, we show that quantum annealing by ferromagnetic
interaction clearly yields states closer to the exact ground state
than conventional quantum annealing and simulated annealing for the
ferromagnetic ground state.
We remark here that
it has been reported that conventional annealing performs poorly
for ferromagnetic ground states compared to paramagnetic states.
Our result implies that such a deterioration of quantum annealing
is not an intrinsic feature but can be improved by choosing appropriate
quantum fluctuations.\\
1) S. Suzuki, H. Nishimori and M. Suzuki, arXiv:quant-ph/0702214,
Submitted to Phys.\ Rev.\ E.