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The disadvantage of the ME method is that the number of states that have
to be considered becomes often very large. Consequently the matrix operations
involved in calculating the exponential operator are time consuming and
the approximations do not converge quickly. Numerical instabilities can easily
appear. By filtering out the largest eigenvalues the Krylov subspace method
rectifies much of the numerical problems.
However the a priori identification of important states remains. The
MC method on the other hand is a very stable and robust method, that has one
major drawback which is the resolution of rare tunnel events, for instance
co-tunneling, which take place in presence of much more frequent events.
In the past variance reducing methods [95] were developed that
can help to alleviate this problem. However, in the case of SET circuits
the tunnel rates may vary over a range of many magnitudes, so that most
variance reducing methods would not trigger, since no rare state is ever
reached. We tackled this problem with a new algorithm that combines the
MC method with a partial direct calculation of state probabilities
(see Section 3.6).
Other good features of the MC method are an easy trade-off
between accuracy and simulation time. Thus one can achieve quick
approximate results of very large circuits, which would otherwise not be
feasible to compute. Furthermore the MC method offers a detail richer
microscopic model for the tunnel process, since in real SET circuits
electrons tunnel from island to island as simulated by the MC method.

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*Christoph Wasshuber*