Quantum Logic Gates — How to Read Them

A one-page primer on reading quantum gates: unitaries, circuit symbols, the Bloch sphere, and why reversibility matters.

A gate is a unitary matrix

Every quantum gate on n qubits is a 2ⁿ×2ⁿ unitary matrix U with U†U = I. Unitarity guarantees the operation is reversible and preserves total probability — the defining constraint that separates quantum logic from classical, irreversible logic like AND and OR.

Reading a circuit

Time flows left to right; each horizontal line is a qubit. A box is a single-qubit gate; a filled dot joined to a box (or ⊕) is a control. The example below shows a Hadamard building superposition, then a CNOT entangling the two qubits into a Bell state.

Universality

A small set of gates can approximate any unitary to arbitrary accuracy. Two common universal sets are {Clifford + T} and {single-qubit rotations + CNOT}. This is why the gates in this section are enough, in principle, to run any quantum algorithm.