Dual thread cores have nothing to do with quantum mechanics (other than how they affect physics in general). It's just duplicating some resources that can't be shared and adding an identifier bit to the others that can be shared and taking advantage of instruction level parallelism mechanisms to execute instructions from two threads at once.

While some quantum mechanics effects are used in computing (eg flash memory uses quantum tunneling to write values into bits), quantum effects are more of a limiting factor for bleeding edge classical computing. Things are packed so closely together that it's difficult to prevent undesired quantum tunneling, which breaks the rules we normally associate with electrical circuits, like current preferring to flow through a conductive connection instead of through insulation protecting a nearby conductor.

Though, that said, the fact that we continue to make progress at the bleeding edge despite that implies that these quantum effects are becoming better understood and designs are at least compensating for them, if not outright taking advantage of them. But that's at a lower level than bigger features like hyperthreading.

Computers that use quantum effects at a higher level are called quantum computers.