What is a Qubit?

qc/basics qc/math

A classical bit is always definitely 0 or 1. A qubit is a vector in a 2-dimensional complex vector space, written in ket notation:

$|\psi ⟩=\alpha |0⟩+\beta |1⟩,\alpha ,\beta \in \mathbb{C},|\alpha {|}^2+|\beta {|}^2=1$

where $|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$ — kets are just column vectors, and $\alpha ,\beta$ are its amplitudes.

Key insight: “0 and 1 at once” is a misleading pop-science shorthand. A qubit isn’t secretly holding two classical values for free — it holds one pair of complex amplitudes that determine probabilities when you measure it (see Measurement and Collapse), and those amplitudes can interfere — add or cancel — in ways plain probabilities never can. That interference, not “parallel storage,” is the actual resource quantum algorithms exploit. See Superposition for what that means in practice.

from qiskit.quantum_info import Statevector
 
psi = Statevector([0.6, 0.8])   # alpha=0.6, beta=0.8 — must satisfy |alpha|^2+|beta|^2=1
psi.probabilities()             # array([0.36, 0.64]) — P(0), P(1)

The normalization constraint $|\alpha {|}^2+|\beta {|}^2=1$ is why arbitrary pairs of numbers don’t make a valid qubit state — Qiskit enforces it.

Bras and inner products

A ket $|\psi ⟩$ is a column vector; its bra $⟨\psi |$ is the conjugate-transpose row vector. Pairing a bra with a ket gives the inner product: $⟨a|b⟩=1$ if $a=b$, else $0$, for computational basis states — the formal statement that $|0⟩$ and $|1⟩$ are orthonormal. More generally $⟨\varphi |\psi ⟩$ gives the overlap (amplitude) between two states. The reverse pairing, a ket times a bra like $|0⟩⟨1|$, is an outer product — a matrix, not a number — and is exactly how operators like the Pauli matrices can be built up from basis states.

Related

• What is a Quantum Computer
• Superposition
• Bloch Sphere — the geometric picture of this state
• Measurement and Collapse
• Tensor Products and Multi-Qubit States — how single-qubit states like this one combine into many-qubit states
• Why Gates Are Unitary

Self-Check

• Why is “a qubit is a 0 and a 1 at the same time” a misleading way to describe $\alpha |0⟩+\beta |1⟩$?
• What does the normalization constraint $|\alpha {|}^2+|\beta {|}^2=1$ actually guarantee, physically?
• What does Measurement and Collapse let you extract from $\alpha$ and $\beta$, and what can it never give you directly?
• What’s the difference between a bra and a ket, and what does pairing them (in each order) give you?