arXiv:1006.1736v2
[cond-mat.stat-mech]
25 Jun 2010
A solvable model of quantum random optimization problems
Laura Foini,
1, 2
Guilhem Semerjian,
1
and Francesco Zamponi
1
1
LPTENS, CNRS UMR 8549, associ´
ee `
a l’UPMC Paris 06, 24 Rue Lhomond, 75005 Paris, France.
2
SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy
We study the quantum version of a simplified model of optimization problems, where quantum
fluctuations are introduced by a transverse field acting on the qubits. We find a complex low-energy
spectrum of the quantum Hamiltonian, characterized by an abrupt condensation transition and a
continuum of level crossings as a function of the transverse field. We expect this complex structure
to have deep consequences on the behavior of quantum algorithms attempting to find solutions to
these problems.
PACS numbers: 75.10.Nr; 03.67.Ac; 64.70.Tg
A large part of theoretical research in quantum com-
puting has been devoted to the development of algo-
rithms that could use quantum properties to achieve a
faster velocity in performing computational tasks with
respect to classical devices. A typical problem that is en-
countered in almost all branches of science is that of op-
timizing irregularly shaped cost functions
H
P
: two stan-
dard examples are
k
-SAT,
H
P
counting the number of
unsatisfied constraints on
k
boolean variables, and the
coloring of a graph with
q
colors (
q
-COL),
H
P
then be-
ing the number of monochromatic edges.
The decision
version (whether a solution, i.e.
a configuration with
H
P
= 0, exists) of both problems belong to the class of
NP-complete problems.
One is mostly interested in the scaling of the difficulty
of these problems when the number
N
of variables in-
volved becomes large. Besides the formal computational
complexity theory which classifies the difficulty of prob-
lems according to a worst-case criterion, their typical case
complexity is often studied through random ensemble of
instances, for instance assuming a flat probability dis-
tribution over the choice of
M
=
αN
constraints on
N
degrees of freedom.
Statistical mechanics tools have provided a very de-
tailed and intricate picture of the configuration space of
such typical problem Hamiltonians
H
P
[1].
A key con-
cept that emerged in this context is that of
clustering of
solutions
. The topology of the space of solutions changes
abruptly upon increasing the density of constraints
α
in
the following way:
i)
at a first threshold
α
d
it goes from
a single connected cluster to a set of essentially disjoint
clusters;
ii)
the number of clusters itself undergoes a tran-
sition at
α
c
> α
d
from a phase where it is exponential
in
N
, thus defining an entropy of clusters (the
complex-
ity
), to a phase where the vast majority of solutions are
contained in a finite number of clusters;
iii)
finally, the
total entropy of solutions vanishes at
α
s
> α
c
where the
problem does not admit a solution anymore (the ground
state energy becomes positive).
This sequence of tran-
sitions (sketched in Fig. 1) has a deep impact on the