### abstract ###
AIMX	This paper studies quantum annealing (QA) for clustering, which can be seen as an extension of simulated annealing (SA)
AIMX	We derive a QA algorithm for clustering and propose an annealing schedule, which is crucial in practice
OWNX	Experiments show the proposed QA algorithm finds better clustering assignments than SA
OWNX	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  SYMBOL
MISC	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 CITATION proposed
MISC	In statistical mechanics, quantum annealing (QA) has been proposed as a novel alternative to SA  CITATION
MISC	QA adds another dimension,  SYMBOL , to SA for annealing, see Fig
MISC	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  SYMBOL
AIMX	We propose quantum effect  SYMBOL , 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  SYMBOL  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
AIMX	Thus, we approximate QA by the Suzuki-Trotter (ST) expansion  CITATION  to derive a tractable sampler, which is the QA-ST sampler
BASE	QA-ST looks like parallel  SYMBOL  SAs with interaction  SYMBOL  (see Fig )
BASE	At the beginning of the annealing process, QA-ST is almost the same as  SYMBOL  SAs
OWNX	Hence, QA-ST finds  SYMBOL  (local) optima independently
OWNX	As the annealing process continues, interaction  SYMBOL  in Fig becomes stronger to move  SYMBOL  states closer
OWNX	QA-ST at the end picks up the state with the lowest energy in  SYMBOL  states as the final solution
OWNX	QA-ST with the proposed quantum effect  SYMBOL  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  SYMBOL  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  SYMBOL  and  SYMBOL  are local optima, the interaction  SYMBOL  in Fig allows  SYMBOL  and  SYMBOL  to search for a better clustering assignment between  SYMBOL  and  SYMBOL
OWNX	Quantum effect  SYMBOL  defines the distance metric of clustering assignments
OWNX	In this case, the proposed  SYMBOL  locates  SYMBOL  between  SYMBOL  and  SYMBOL
OWNX	Thus, the interaction  SYMBOL  gives good chance to go to  SYMBOL  because  SYMBOL  makes  SYMBOL  and  SYMBOL  closer (see Fig )
OWNX	The proposed algorithm actually finds  SYMBOL  from  SYMBOL  and  SYMBOL
OWNX	Fig is just an example
MISC	However, a similar situation often occurs in clustering
CONT	Clustering algorithms in most cases give ``almost'' globally optimal solutions like  SYMBOL  and  SYMBOL , where the majority of data points are well-clustered, but some of them are not
CONT	Thus, a better clustering assignment can be constructed by picking up well-clustered data points from many sub-optimal clustering assignments
CONT	Note an assignment constructed in such a way is located between the sub-optimal ones by the proposed quantum effect  SYMBOL  so that QA-ST can find a better assignment between sub-optimal ones
