The 2ⁿ State Vector

Classical computers store $n$ bits and that gives $n$ independent pieces of information — add a bit, add a slot. A quantum register works very differently: $n$ qubits require a state vector with $2^n$ complex amplitudes. This exponential growth is both the source of quantum computing's potential power and the reason classical simulation becomes infeasible quickly.

One qubit, two amplitudes

A single qubit lives in a two-dimensional complex Hilbert space. Any pure state can be written

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1.

The two numbers $\alpha$ and $\beta$ are the amplitudes: complex numbers whose squared magnitudes give measurement probabilities via the Born rule.

Two qubits, four amplitudes

When we combine two qubits the joint state lives in the tensor-product space $\mathbb{C}^2 \otimes \mathbb{C}^2 \cong \mathbb{C}^4$ . The computational basis is now $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$ and the general state is

|\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle,

with four amplitudes satisfying $\sum_{x \in \{0,1\}^2} |\alpha_x|^2 = 1$ .

Notice the index $x$ runs over all 2-bit strings. For two qubits that is $2^2 = 4$ strings.

$n$ qubits: $2^n$ amplitudes

The pattern continues. For an $n$ -qubit register the computational basis is all $n$ -bit strings $x \in \{0,1\}^n$ , so there are exactly $2^n$ basis states and $2^n$ amplitudes:

|\psi\rangle = \sum_{x \in \{0,1\}^n} \alpha_x |x\rangle, \qquad \sum_x |\alpha_x|^2 = 1.

Each amplitude $\alpha_x \in \mathbb{C}$ encodes how much probability amplitude "sits on" the basis state $|x\rangle$ . The exponential arises directly from the tensor-product structure: each new qubit introduces a factor of 2 because the new degree of freedom can independently be in $|0\rangle$ or $|1\rangle$ in superposition with everything else.

Why this matters for algorithms

An $n$ -qubit gate applies a $2^n \times 2^n$ unitary matrix to the state vector. A quantum algorithm can, in a single step, manipulate all $2^n$ amplitudes simultaneously — a kind of massive parallelism. However, measurement collapses the state to a single $n$ -bit string, so the art of quantum algorithm design is arranging interference so that the useful outcomes have large amplitudes by the time we measure.

Classical versus quantum information content

It is tempting to say $n$ qubits "store" $2^n$ values. This is misleading: you can only extract $n$ classical bits per measurement. The exponential structure lives in the pre-measurement superposition and is accessible only through interference. Algorithms like Grover's and the Quantum Fourier Transform exploit this structure without trying to read out all $2^n$ amplitudes directly.

Try it

Return the number of complex amplitudes required to describe a 3-qubit state vector.

Run your code to see the quantum state.