### abstract ###
OWNX 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
MISC 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  SYMBOL  SAs with interaction  SYMBOL  (see Fig )
OWNX 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
MISC Fig is an example where data points are grouped into four clusters
MISC SYMBOL and  SYMBOL are locally optimal and  SYMBOL  is globally optimal
MISC Suppose  SYMBOL  is equal to two and  SYMBOL  and  SYMBOL  in Fig correspond to  SYMBOL  and  SYMBOL  in Fig
MISC 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
MISC Quantum effect  SYMBOL  defines the distance metric of clustering assignments
MISC In this case, the proposed  SYMBOL  locates  SYMBOL  between  SYMBOL  and  SYMBOL
MISC 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
MISC 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  SYMBOL  and  SYMBOL , 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
OWNX 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
