A classical computer composed of '0' or '1' transistors stores $2^n$ states.
Is it true that a quantum computer composed of '0' or '1' or '0 & 1' qubits stores $3^n$ states?
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No, it is not true. A quantum computer stores the same $2^n$ states that the classical computer stores. The difference is that the quantum computer stores a linear superposition of those states, where the classical computer can only store one of those states at a time. What you refer to as '0 & 1' qubits are actually linear superpositions of the two basis qubits 0 and 1.