Optimize self join on millions of rows

I have a table which is a link table from objects in my SQL Server 2012 database, (annonsid, annonsid2). This table is used to create chains of triangle or even rectangles to see who can swap with who.

This is the query I use on the table Matching_IDs which has 1,5 million rows in it, producing 14 million possible chains using this query:

SELECT COUNT(*)
FROM Matching_IDs AS m
  INNER JOIN Matching_IDs AS m2
     ON m.annonsid2 = m2.annonsid
  INNER JOIN Matching_IDs AS m3
     ON m2.annonsid2 = m3.annonsid
       AND m.annonsid = m3.annonsid2

I must improve performance to take maybe 1 second or less, Is there a faster way to do this? The query takes about 1 minute on my computer. I normally use a WHERE m.annonsid=x, but it takes just the same amount of time, cause it has to go through all possible combinations anyway.

Update: the latest query plan

|--Compute Scalar(DEFINE:([Expr1006]=CONVERT_IMPLICIT(int,[globalagg1011],0)))
   |--Stream Aggregate(DEFINE:([globalagg1011]=SUM([partialagg1010])))
        |--Parallelism(Gather Streams)
             |--Stream Aggregate(DEFINE:([partialagg1010]=Count(*)))
                  |--Hash Match(Inner Join, HASH:([m2].[annonsid2], [m2].[annonsid])=([m3].[annonsid], [m].[annonsid2]), RESIDUAL:([MyDatabase].[dbo].[Matching_IDs].[annonsid2] as [m2].[annonsid2]=[MyDatabase].[dbo].[Matching_IDs].[annonsid] as [m3].[annonsid] AND [MyDatabase].[dbo].[Matching_IDs].[annonsid2] as [m].[annonsid2]=[MyDatabase].[dbo].[Matching_IDs].[annonsid] as [m2].[annonsid]))
                       |--Parallelism(Repartition Streams, Hash Partitioning, PARTITION COLUMNS:([m2].[annonsid2], [m2].[annonsid]))
                       |    |--Index Scan(OBJECT:([MyDatabase].[dbo].[Matching_IDs].[NonClusteredIndex-20121229-133207] AS [m2]))
                       |--Parallelism(Repartition Streams, Hash Partitioning, PARTITION COLUMNS:([m3].[annonsid],