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I know this question looks ridiculous.

But I simply want to know if possible to store/load a superposition state into classical register from qubit. Specifically two bit of classical register.

Consider 1-qubit $q$ and 2-bit $c$:

  • If $q=|0\rangle$, then $c=00$.
  • If $q=|1\rangle$, then $c=01$.
  • If $q=|+\rangle$, then $c=10$.
  • If $q=|-\rangle$, then $c=11$.

Basically left classical bit (MSB) will set to 1 if in superposition state, otherwise 0.

Right classical bit (LSB) was about binary state in qubit. But depends on context too, if in superposition state (MSB=1) then it represent previous value before became superposition state.

So is it possible to just using one qubit in this problem? I don't mind using many logic gates in single qubit.

What I mean with store and load is basically copy from/to classical register. Store with measurement, Load with using conditional X gate in classical bit, here is what I mean.

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As any pure quantum state of $n$-qubits can be described by a vector from vector space $\mathbb{C}^{2^n}$, you can of course store it classically. However, with increasing number of qubits, a memory requiremens increase exponentially. That is the reason why quantum computers are built, they cannot be simulated classically with reasonable (i.e. polynomial) computing resources.

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