### abstract ###
AIMX	this paper studies quantum annealing  qa  for clustering  which can be seen as an extension of simulated annealing  sa 
OWNX	we derive a qa algorithm for clustering and propose an annealing schedule  which is crucial in practice
CONT	experiments show the proposed qa algorithm finds better clustering assignments than sa
CONT	furthermore  qa is as easy as sa to implement
### introduction ###
MISC	clustering is one of the most popular methods in data mining
MISC	typically  clustering problems are formulated as optimization problems  which are solved by algorithms  for example the em algorithm or convex relaxation
MISC	however  clustering is typically np hard
MISC	the simulated annealing  sa   CITATION  is a promising candidate
MISC	CITATION  proved sa was able to find the global optimum with a slow cooling schedule of temperature
CONT	although their schedule is in practice too slow for clustering of a large amount of data  it is well known that sa still finds a reasonably good solution even with a faster schedule than what  citeauthor geman NUMBER   proposed
CONT	in statistical mechanics  quantum annealing  qa  has been proposed as a novel alternative to sa  CITATION
CONT	qa adds another dimension     to sa for annealing  see fig
BASE	thus  it can be seen as an extension of sa
MISC	qa has succeeded in specific problems  e g the ising model in statistical mechanics  and it is still unclear that qa works better than sa in general
OWNX	we do not actually think qa intuitively helps clustering  but we apply qa to clustering just as procedure to derive an algorithm
MISC	a derived qa algorithm depends on the definition of quantum effect
OWNX	we propose quantum effect    which leads to a search strategy fit to clustering
AIMX	our contribution is 1) to propose a qa based optimization algorithm for clustering in particular quantum effect for clustering and a good annealing schedule  which is crucial for applications 2) and to experimentally show the proposed algorithm optimizes clustering assignments better than sa
OWNX	we also show the proposed algorithm is as easy as sa to implement 
OWNX 	the algorithm we propose is a markov chain monte carlo  mcmc  sampler  which we call qa st sampler
OWNX	as we explain later  a naive qa sampler is intractable even with mcmc
OWNX	thus  we approximate qa by the suzuki trotter  st  expansion  CITATION  to derive a tractable sampler  which is the qa st sampler
OWNX	qa st looks like parallel   sas with interaction    see fig  
OWNX	at the beginning of the annealing process  qa st is almost the same as   sas
OWNX	hence  qa st finds    local  optima independently
OWNX	as the annealing process continues  interaction   in fig becomes stronger to move   states closer
OWNX	qa st at the end picks up the state with the lowest energy in   states as the final solution
OWNX	qa st with the proposed quantum effect   works well for clustering
OWNX	fig is an example where data points are grouped into four clusters
OWNX	SYMBOL and SYMBOL are locally optimal and   is globally optimal
OWNX	suppose SYMBOL is equal to two and SYMBOL and SYMBOL in fig correspond to SYMBOL and SYMBOL in fig
OWNX	although   and   are local optima  the interaction   in fig allows   and   to search for a better clustering assignment between   and
OWNX	quantum effect   defines the distance metric of clustering assignments
OWNX	in this case  the proposed   locates   between   and
OWNX	thus  the interaction   gives good chance to go to   because   makes   and   closer  see fig  
OWNX	the proposed algorithm actually finds   from   and
OWNX	fig is just an example
MISC	however  a similar situation often occurs in clustering
MISC	clustering algorithms in most cases give   almost   globally optimal solutions like   and    where the majority of data points are well clustered  but some of them are not
MISC	thus  a better clustering assignment can be constructed by picking up well clustered data points from many sub optimal clustering assignments
MISC	note an assignment constructed in such a way is located between the sub optimal ones by the proposed quantum effect   so that qa st can find a better assignment between sub optimal ones
