Quantum words, explained simply.

The basic unit of quantum information. Like a bit it can give 0 or 1 when measured, but before measurement it can be in a superposition — a weighted combination of both, described by two complex numbers.

A state that is a combination of |0⟩ and |1⟩ at the same time, written α|0⟩ + β|1⟩. It is not “secretly 0 or 1”: the two parts can interfere with each other, which is what quantum algorithms exploit.

The complex number in front of each basis state (α and β in α|0⟩ + β|1⟩). Its squared magnitude is a probability; its angle is the phase.

The rule that turns amplitudes into probabilities: the chance of measuring 0 is |α|² and the chance of measuring 1 is |β|². The two always add up to 1.

The angle of a complex amplitude. The relative phase between |0⟩ and |1⟩ does not change measurement probabilities directly, but it changes how the qubit reacts to later gates. On the Bloch sphere it is the longitude φ.

Multiplying the whole state by the same phase factor e^{iγ}. It has no observable effect at all, which is why two numbers α and β really carry only two angles of information.

A unit sphere on which every pure state of one qubit is a single point. |0⟩ is the north pole, |1⟩ the south pole, equal superpositions sit on the equator, and gates act as rotations.

The “bra-ket” notation physicists use for quantum states. A ket like |ψ⟩ is a state vector; |0⟩ and |1⟩ are the two basis states of a qubit.

One of the reference states every other state is built from. For one qubit in the computational basis these are |0⟩ and |1⟩.

Reading out a qubit. It returns a classical 0 or 1 with Born-rule probabilities and leaves the qubit in the state it reported, so the original superposition is gone.

The change of state caused by measurement: after reading 0 the qubit is |0⟩, after reading 1 it is |1⟩. On the Bloch sphere the arrow snaps to a pole.

Repetitions of the same experiment. A single measurement gives one random bit; running many shots reveals the probabilities.

An operation that changes a qubit’s state. Mathematically it is a unitary matrix; on the Bloch sphere a single-qubit gate is a rotation around some axis.

Three basic gates that rotate the Bloch sphere by 180° around the X, Y or Z axis. X is the quantum NOT, Z flips the phase, Y does both.

The gate H, a 180° rotation around the axis halfway between X and Z. It turns |0⟩ into the equal superposition |+⟩ = (|0⟩ + |1⟩)/√2 and back again.

Phase gates: S rotates 90° around the Z axis and T rotates 45°. They change the phase of |1⟩ without changing the probabilities.

A matrix U with U†U = I. Unitary operations keep the total probability equal to 1 and can always be undone, which is why quantum gates are reversible.

A diagram of a quantum program: horizontal wires are qubits, boxes on them are gates applied from left to right, and meters at the end are measurements.

A two-qubit gate: it flips the target qubit if and only if the control qubit is 1. Together with the Hadamard gate it creates entanglement.

A correlation between qubits so strong that the group has a definite state while the individual qubits do not. Measuring one immediately tells you about the other.

One of four maximally entangled two-qubit states, for example (|00⟩ + |11⟩)/√2: each qubit alone is a perfect coin flip, but the two always agree.

Amplitudes can add up or cancel out, just like waves. Quantum algorithms arrange interference so wrong answers cancel and the right one is amplified.