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What is the idea behind of quantum simulator aimed to study properties of matter, such as using quantum dots to study the exotic quantum states?

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Although we typically think of the laws of quantum mechanics as governing phenomena only at the smallest length scales, there are situations in which we can measure quantum effects at reasonable experimental sizes. Superconductivity is a great example of this insofar as a macroscopic piece of superconducting material can have a quantum coherence length of the same scale as itself. Crystalline nanoparticles, such as those of CdSe which are grown with little difficulty in chemistry laboratories across the world, are another great example since their macroscopic properties (e.g., color) are intimately tied up with quantum effects due to their small size. These are not simulators, per se, but systems in which quantum effects manifest themselves at a human scale.

True quantum simulators are experimental systems which are constructed to follow the same mathematical laws as a particular quantum phenomenon that defies (or at least frustrates) experimental investigation. These simulators display non-classical behavior themselves, but can be modified so that their time evolution mimics that of some other quantum system. Examples include optical quantum simualtors and the quantum dots you describe. As noted by Richard Feynman, such a simualtor can act as a quantum computer and (as was determined later) vice versa.

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