Quantum Circuits

A quantum circuit is a recipe: horizontal wires carry qubits through time (left to right), and gates are the boxes you drop onto those wires. Reading one is like reading sheet music — each column is a moment, and the symbols tell you which operation happens to which qubit.

Build a circuit below by adding gates, then step through it. The diagram lights up gate by gate while the state vector and probabilities update underneath. The default H then CNOT is the standard way to create an entangled Bell pair.

Reading the diagram

• Each wire is one qubit, labeled q0, q1, …, starting in $|0\rangle$.
• A boxed letter (H, X, Z) is a single-qubit gate applied at that moment.
• CNOT is drawn as a filled dot (the control) connected to a $\oplus$ (the target). When the control is $1$, the target flips.
• The M at the right edge is measurement, which collapses the state to a classical bitstring.

Time flows left to right, so the circuit $|0\rangle \xrightarrow{H} \xrightarrow{Z}$ means “apply $H$, then $Z$” — order matters, because gates generally do not commute.

Composing single-qubit gates

Stacking gates on one wire just multiplies their effects. Try H then Z then H on q0: the surrounding Hadamards turn the otherwise invisible phase flip into a visible bit flip, ending in $|1\rangle$. This $HZH = X$ identity is a first taste of how interference is engineered by sequencing gates.

Two-qubit states

With two qubits the state vector has four amplitudes, ordered $|00\rangle, |01\rangle, |10\rangle, |11\rangle$:

$|\psi\rangle = c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle$

Apply H on q0 and the amplitude spreads across $|00\rangle$ and $|10\rangle$. Add CNOT and those become $|00\rangle$ and $|11\rangle$ — a state you cannot factor into “q0 does this, q1 does that.” That non-factorable result is entanglement, produced by a circuit just two gates long.

Building intuition

A few patterns worth trying in the builder:

1. X q1 alone → the state moves to $|01\rangle$ (remember q0 is the left digit).
2. H q0, H q1 → all four outcomes equally likely (uniform superposition).
3. H q0, CNOT → the Bell state: only $|00\rangle$ and $|11\rangle$ survive.

Notice the probabilities always sum to 1 at every step — each gate is unitary, so it shuffles amplitude around without ever creating or destroying it.

Takeaways

• Circuit diagrams read left to right; wires are qubits and boxes are gates.
• CNOT (control dot + target $\oplus$) is the workhorse two-qubit gate that builds entanglement.
• Composing gates multiplies their effects; order matters and every step preserves total probability.

References

• Nielsen & Chuang, Quantum Computation and Quantum Information — §1.3 (quantum gates and circuits)
• Qiskit Textbook — Multiple qubits and entangled states