# Three qubits
**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.

## Entanglement classes
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-state|GHZ states]] and the [[w-state|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.

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