12.5 An intro to QC programming

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §12.5–12.5.1. Original sketches — do not paste textbook Hadamard.py / CirqSwap.py dumps. Continues 12.4b. Labs: 12.5b.

Qubits and gates become circuits (programs). Cirq is a Python library for writing, simulating, and optimizing those circuits. Graded work on this site uses pure-Python amplitude / bit locks (no Cirq in the sandbox). Run the sketches below in a local conda/venv when you want the real simulator.

Install (local)

python -m pip install --upgrade pip
python -m pip install cirq
# or: conda install -c conda-forge cirq

First circuit: Hadamard on $|0\rangle$

# hadamard_sketch.py — structure only (run locally with Cirq)
import cirq

circuit = cirq.Circuit()
q = cirq.GridQubit(0, 0)
circuit.append(cirq.H(q))
sim = cirq.Simulator()
print(circuit)
print(sim.simulate(circuit))  # ≈ 0.707|0⟩ + 0.707|1⟩

Matches

$$H|0\rangle=\tfrac1{\sqrt2}(|0\rangle+|1\rangle).$$

Two Hadamards = identity

$H^2=I$. Append a second cirq.H(a) on a NamedQubit('a'); the statevector returns to $|0\rangle$.

X then H → $|-\rangle$

X maps $|0\rangle\to|1\rangle$; then H yields $\tfrac1{\sqrt2}(|0\rangle-|1\rangle)$, an eigenstate of $X$ with eigenvalue $-1$.

Measure: X, Z, H + shots

Measurement is not unitary. Append cirq.measure(a, key='result'), then sim.run(circuit, repetitions=1000) and plot a histogram. Ideal amplitudes after $X\,Z\,H$ on $|0\rangle$ are equal magnitude ($\pm 1/\sqrt2$), so counts of $0$ and $1$ each sit near $500$ (noise → slight tilt — Figure 12.3).

Two qubits: LineQubit, SWAP, CNOT

q0, q1 = cirq.LineQubit.range(2)
circuit.append(cirq.X(q0))   # |10⟩ if start |00⟩
circuit.append(cirq.Z(q1))   # phase only on |1⟩ of q1
# expected state |10⟩

SWAP exchanges the two lines: after $X$ on q0 you get $|01\rangle$.

CNOT(control=q0, target=q1): with $|10\rangle$ input, flips the target → $|11\rangle$.

Three qubits: Toffoli

q0, q1, q2 = cirq.LineQubit.range(3)
circuit.append(cirq.X(q0))
circuit.append(cirq.Toffoli(q0, q1, q2))  # flips q2 only if q0=q1=1

With only q0 flipped, controls are not both $1$, so the target stays put (e.g. $|100\rangle$ or $|101\rangle$ depending on q2’s prep). Flip both controls to see $T|110\rangle=|111\rangle$.

12.5.1 Half and full adders

Half-adder (sum + carry)

Classical: sum $=q0\oplus q1$, carry $=q0\cdot q1$. Quantum recipe on three lines (q2 starts at $|0\rangle$):

1. Toffoli(q0, q1, q2) → write carry into q2
2. CNOT(q0, q1) → write sum into q1

Lab: wire the half-adder

Check truth table: $1+1$ → sum $0$, carry $1$; $1+0$ and $0+1$ → sum $1$, carry $0$.

Full adder

A full adder folds in a carry-in. One Cirq-shaped skeleton (four qubits): cascade Toffoli / CNOT so that after prep of q0,q1,Cin the sum and Cout match classical binary addition. Fill the table in labs for all four input pairs (Cin fixed or varied).

q0	q1	Cin	Sum	Cout
$\|0\rangle$	$\|0\rangle$	…	…	…
$\|0\rangle$	$\|1\rangle$	…	…	…
$\|1\rangle$	$\|0\rangle$	…	…	…
$\|1\rangle$	$\|1\rangle$	…	…	…

Takeaway

Goal	Cirq idea
Superposition	H on GridQubit / NamedQubit
$H^2=I$	two H on same line
Statistics	measure + run(..., repetitions=)
Entangle / flip	CNOT, SWAP, Toffoli
Add	Toffoli carry + CNOT sum

Quiz

Cirq / adders

1 / 3

Two Hadamard gates on |0⟩ yield…

Pick an answer to see if you are right.

Next: 12.5b Cirq labs · 12.6 IBM Quantum.