Multi-Qubit Hilbert Space

General 2ⁿ-Dimensional State

Any $n$-qubit quantum state is a linear combination of all computational basis states:

$$|\psi\rangle = \sum_{i=0}^{2^n - 1} \alpha_i |i\rangle, \quad \text{with } \sum_{i=0}^{2^n - 1} |\alpha_i|^2 = 1$$

This is represented as a normalized column vector in $\mathbb{C}^{2^n}$, where each amplitude $\alpha_i$ can encode probabilities, payoff values, or other structured data.

Qubit Basis and Vector Representation

A Hilbert space is a complex vector space that represents the state of quantum systems. For a single qubit, the Hilbert space is $\mathbb{C}^2$, with the computational basis:

$$|0\rangle = \begin{bmatrix}1 \\ 0\end{bmatrix}, \quad |1\rangle = \begin{bmatrix}0 \\ 1\end{bmatrix}$$

In the standard big-endian convention (most significant bit first), the binary basis state $|0\rangle$ corresponds to the first index, hence it is represented by $[1, 0]^T$. Similarly, $|1\rangle$ corresponds to the second index, represented by $[0, 1]^T$.

This mapping generalizes to multi-qubit systems. For example, with $n = 3$, the state space is $\mathbb{C}^{2^3} = \mathbb{C}^8$. The basis state $|001\rangle$ corresponds to index 1 in binary ordering and is represented as:

$$|001\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$$

Each basis state $|i\rangle$ maps to a one-hot vector of length $2^n$, with a 1 at position $i$ and 0 elsewhere.