OE-4.3 Quantum Logic Gates & Circuits

What this is and why it exists

Every quantum gate is a unitary matrix. So "applying a gate" is matrix multiplication, and nothing more mysterious than that.

Saying that plainly at the start is worth doing. The vocabulary around this subject invites people to expect something stranger than the mathematics is.

One result here is famous and also the most misread thing in the whole course. Teleportation moves a state, not matter, and it needs an ordinary classical message to complete. Nothing travels faster than light.

The vocabulary

• Gate — a unitary matrix applied to a state.
• Pauli gates — the three standard single-qubit flips and rotations.
• Hadamard gate — turns a basis state into an equal superposition.
• Phase shift — changes the relative phase between components.
• Controlled gate — acts on one qubit depending on another.
• Entanglement — a joint state that cannot be written as two separate states.
• Bell states — the four standard maximally entangled two-qubit states.
• Teleportation — transferring a state using entanglement plus a classical message.

The mental model

A gate is a unitary matrix. Unitary, from the first topic, means lengths are preserved — so a state stays normalised, and every quantum operation is reversible. That reversibility is a real constraint on what can be built, and it is why quantum circuits look different from classical ones.

The Hadamard gate is the workhorse. It manufactures superposition from a basis state: give it one basis state and it returns an equal combination of both. Almost every algorithm opens with it, applied across all the qubits at once, because that is how you get every input represented simultaneously.

Controlled gates act on one qubit depending on the value of another. The two-qubit controlled flip is the standard one, and the three-qubit version turns out to be enough to build any classical logic reversibly.

Entanglement has a precise definition worth holding on to. A joint state that cannot be written as a product of separate states for each qubit. Not "the qubits are connected", which is a metaphor. Cannot be factorised. The Bell states are the four standard maximally entangled pairs, and producing one is a two-gate exercise: a Hadamard followed by a controlled flip.

And then teleportation. Two parties share an entangled pair. One of them holds a third qubit in an unknown state. They measure their two qubits together and send the two classical bits of the result. The other applies a correction determined by those bits, and now holds the unknown state. The original is destroyed by the measurement.

That classical message is the whole reason no information travels faster than light. Without it the receiver holds something useless. It is worth being able to say that clearly. The popular account leaves it out, and the popular account is what most people have heard.

What you should now be able to explain or do

Say what a gate is and why every quantum operation is reversible. Apply the Pauli, Hadamard and phase-shift gates. Say what the Hadamard gate manufactures and why algorithms open with it. Use controlled gates. Define entanglement as non-factorisability rather than by metaphor. Build a Bell state in two gates. Describe teleportation including the classical message, and say why nothing exceeds light speed.

Check yourself