Quantum circuit gate symbols (H, X, CNOT, measurement)

How to read it

IBM’s course Basics of Quantum Information puts the model in one line: wires represent qubits and gates represent operations on these qubits, and the flow of information goes from left to right. Single-qubit gates are squares with a letter naming the operation. A controlled gate is a filled-in circle on the control qubit, joined by a vertical line to the operation it controls; for the controlled-NOT the target is a circle resembling ⊕. Classical bits, which hold measurement results, are drawn as double lines. Circuits often start with every qubit set to |0⟩, a state written in Bra-ket notation (|ψ⟩).

Also searched as: CNOT symbol, Hadamard gate symbol, quantum circuit symbols, quantum gates.

What each symbol means

IBM’s Composer adds two operations that are not gates. The reset operation returns a qubit to |0⟩. A barrier tells the compiler not to combine gates across it, and IBM says it is also useful for visualizations; Qiskit’s documentation calls it a no-op in hardware.

Qiskit’s drawings and textbook ones

Qiskit, IBM’s software for writing circuits, names the controlled-NOT cx and, in its text drawings, puts an X in a box on the target where the textbook form has ⊕. Its SWAP is an X on each wire, joined by a line, and its controlled-Z a dot on each wire. Qiskit numbers the top wire qubit 0; IBM’s course notes that this is the rightmost position when the qubits are written as a string. Qiskit’s documentation for the controlled-NOT adds that many textbooks take the more significant qubit as the control, so a textbook matrix for the same gate is written the other way round.

Where the notation comes from

In their 1995 paper “Elementary gates for quantum computation”, written for Physical Review A, Adriano Barenco, Charles Bennett, David DiVincenzo, Peter Shor and five co-authors say they adopted a version of Richard Feynman’s notation, from his article “Quantum mechanical computers” (Optics News, February 1985), and that in all their diagrams time proceeds from left to right. They credit the circuit model itself, the quantum gate array in which each wire is a qubit, to David Deutsch’s “Quantum computational networks” (Proceedings of the Royal Society A, 1989). The paper showed that one-qubit gates and the two-bit exclusive-or gate, the controlled-NOT, are enough to build any unitary operation on any number of qubits.

Look-alikes

⊕ on its own is the exclusive-or (XOR) sign, and IBM’s course uses it for XOR in ordinary logic circuits. The reading carries over: Barenco and co-authors call the controlled-NOT the XOR gate. The state a circuit acts on is often written with Psi, as |ψ⟩.