Reusing Subcircuits

Real circuits quickly grow too large to read as one flat stream of gates. The standard remedy is factoring: carve the circuit into small, self-contained subcircuits and apply each one wherever it is needed. Subcircuits make large circuits easier to write, easier to audit, and easier to reuse across different programs — the same engineering principle that turns spaghetti code into well-named functions.

What a subcircuit is

A subcircuit is any fixed sequence of gates that performs a coherent, nameable operation on a specific set of qubits. It has no special syntax at the hardware level; it is simply a group of gate calls that you identify as a unit. In higher-level frameworks you would package it as a function or a named gate; in the flat API used here you apply it by repeating the same gate-call pattern on different qubits.

The Bell-pair preparation is a canonical example: apply $H$ to one qubit, then a CNOT with that qubit as the control and a second qubit as the target. This two-gate unit entangles the pair into

|\Phi^+\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr).

The derivation is straightforward: starting from $|00\rangle$ , the Hadamard produces $\tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)|0\rangle$ , and the CNOT flips the second qubit whenever the first is $|1\rangle$ , yielding $\tfrac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ .

Reuse without copying errors

Once you identify a subcircuit, applying it a second time means repeating the same gate calls with different qubit indices. No new logic is introduced — you just shift the "address" of the pattern. On a 4-qubit register with pairs $(0,1)$ and $(2,3)$ , the Bell subcircuit runs on $(0,1)$ first, then identically on $(2,3)$ . The final state is the tensor product of two independent Bell pairs:

|\Phi^+\rangle_{01} \otimes |\Phi^+\rangle_{23} = \frac{1}{2}\bigl(|0000\rangle + |0011\rangle + |1100\rangle + |1111\rangle\bigr).

Reading a factored circuit

In a circuit diagram, a subcircuit shows up as a repeated visual motif — the same shape of gates appearing at multiple horizontal positions or on multiple groups of wires. Recognising the motif is the first step to reading a large circuit quickly: instead of parsing every gate individually, you identify the building blocks and reason about their combined effect.

Try it

A 4-qubit circuit should prepare two independent Bell pairs by applying the Bell subcircuit to qubits $(0,1)$ and then again to qubits $(2,3)$ . The grader checks the full unitary matrix, so an empty circuit will not pass.

Run your code to see the quantum state.

After running, inspect the statevector: you should see four equal-amplitude basis states — $|0000\rangle$ , $|0011\rangle$ , $|1100\rangle$ , and $|1111\rangle$ — each with amplitude $\tfrac{1}{2}$ , confirming that both pairs are entangled but independent of each other.