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From Wikipedia:

A qubit is a two-state quantum system [...]

There are two possible outcomes for the measurement of a qubit — usually $0$ and $1$, like a bit. The difference is that whereas the state of a bit is either $0$ or $1$, the state of a qubit can also be a superposition of both.

Is it true that quantum computing must be based on qubits, limiting it to only a two-state quantum system? Is it physically possible to build a $n$-state quantum-system (where $n>2$)?

This is 'repost' of a ("legitimate") question was deleted by the OP, here. I decided to repost it along with the answer which I started writing.

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Sanchayan Dutta
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Is quantum computing limited to a superposition of only two states?

In theory, it is not. Keep in mind that a qubit is a quantum analogue of the classical "bit" which has only two states $0$ and $1$. In principle, there is no limit to the dimension of the state space of a quantum system. There could even be an "infinite" dimensional separable Hilbert space (in short, separable means denumerable/countable with a one-one onto mapping to the natural numbers). For non-separable Hilbert spaces there are some complications involved. In the context of quantum information systems with a state-space dimension greater than $2$ are called "qudits".

And yes, there has been ongoing work to make physical implementations of higher dimensional quantum systems, like qutrits (with trapped ions), as mentioned by @Andrew O, on their currently deleted answer (only users having the privilege to view deleted posts can see it at present).

Relevant paper: Qutrit quantum computer with trapped ions - A. B. Klimov, R. Guzmán, J. C. Retamal, and C. Saavedra

Edit:

  • @glS mentions here that in some cases making higher-dimensional quantum systems can, in fact, be easier, which is an interesting fact I did not know earlier.

In the context of photonics, for example, it is relatively easy to generate states in high-dimensional Hilbert spaces, for example exploiting the orbital angular momentum of single photons. See for example 1607.05114 and the many references therein, or Fickler 2012, in which they experimentally demonstrate entanglement of states living in 600-dimensional Hilbert spaces.

It is also to be noted that the matter of non-separability is absolutely a non-issue for practical implementation of whatever protocol, and also that continuous variable quantum computation is a big subject in quantum computing

Sanchayan Dutta
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    However, practically producing a quantum system in a superposition of three or more systems is much harder: this is not true. In the context of photonics for example, it is relatively easy to generate states in high-dimensional Hilbert spaces, for example exploiting the orbital angular momentum of single photons. See for example 1607.05114 and the many references therein, or Fickler 2012, in which they experimentally demonstrate entanglement of states living in 600-dimensional Hilbert spaces. – glS Mar 29 '18 at 19:06
  • The Fickler article describes using "off-the-shelf telecommunications components". Great question BTW. – Rob Apr 05 '18 at 00:13
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Even if you assume that your quantum computer will be based on qubits, it can operate with an arbitrarily large number of states (if it has enough qubits): The combination of two qubits allow it to calculate with a total of four states, that of three qubits with eight states, etc.