Three-qubit system is a quantum system consisting of three qubits. The state space is the tensor product $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \cong \mathbb{C}^8$, spanned by eight computational basis states. A general three-qubit state requires eight complex probability amplitudes.

$$\lvert\psi\rangle = \sum_{x \in \{0,1\}^3} c_x\lvert x\rangle = c_{000}\lvert 000\rangle + c_{001}\lvert 001\rangle + c_{010}\lvert 010\rangle + c_{011}\lvert 011\rangle + \cdots + c_{111}\lvert 111\rangle$$

The normalization condition requires $\sum_{x}|c_x|^2 = 1$. The exponential growth of the state space with the number of qubits — $2^n$ amplitudes for $n$ qubits — is the fundamental reason quantum computers can potentially outperform classical ones for certain problems.

Three-qubit entanglement is richer than two-qubit entanglement. Whereas any two-qubit state is either separable or entangled (with the Bell states as the maximally entangled examples), three-qubit systems have multiple inequivalent classes of entanglement. The two canonical classes of genuinely tripartite entanglement are the GHZ states and the W states. These two classes are inequivalent under local operations and classical communication (LOCC), meaning neither can be converted to the other by any local operations alone.

Gates that act on three qubits are represented by $8 \times 8$ unitary matrices. The most important three-qubit gate is the Toffoli gate (doubly-controlled NOT), which is classically universal and is used in fault-tolerant quantum computation.