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All quantum computers now are working with qubit which is zero or one.

Can we have quantum computer with multiple-state unit? Such as trit (trinary, 3 states), digit (10 states) instead of qubit (2 states)?

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Generally, you can have quantum computer with so-called qudits, i.e. qubits with two possible states are replaced by an entity having $d$ distinguished states. Example is so-called qutrit having three possible values ($|0\rangle$, $|1\rangle$ and $|2\rangle$).

Application of qutrit was devised for example in quantum RAM. See this paper Quantum random access memory for more information.

Although in theory you can have as many states as you want, in practise it can be problematic to build suchs device because of noise. Increasing the number of possible values ($d$), the noise also increases. See how much the noise "destroy" computation in quantum computers based on qubits.

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