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A nondeterministic TM may be able to do anything a QTM can do, possibly with exponential slowdown (though I've not seen a proof of this claim) but a clasical TM can not.

A nondeterministic TM can be simulated by a classical TM by computing all the branches of the computation.
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It's clear how a QTM can be programmed so, but how do you do it with a classical TM?

Your point is taken, though I argue since its not an algorithm which solves any problem, you're not really adding any additional computational power. Any algorithm which uses randomness can be simulated by a classical TM by computing every possible outcome.