12.11 Quantum-computing code sketches

Source: Structural ideas from Computational Physics (Landau, Páez, Bordeianu), §12.11 listings. Original sketches only — do not paste Entangle.py, QFT4.py, QFTn.py, OracleSim.py, OracleIBM.py, or Shor.py. Continues 12.10b. Labs: 12.11b.

Graded locks live in 12.1b–12.10b. This page maps how local notebook pieces fit. Qiskit / Cirq / IBM tokens stay on your machine.

Dipole Hamiltonian (NumPy, local)

Two-spin dipole piece $\propto XX+YY+ZZ-3ZZ$ on $\mathbb C^4$. Build the $4\times 4$ matrix, eig, then optionally rotate $H$ into the eigenbasis (should look diagonal up to numerics).

# dipole_h_sketch.py — original structure (local NumPy)
# XX = [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
# YY, ZZ similarly; H = XX + YY + ZZ - 3*ZZ
# es, ev = numpy.linalg.eig(H)
# Check Bell-like columns for degenerate subspace (see §12.3)

Pure-Python entry locks: 12.3b.

QFT₄ circuit (Qiskit, local)

Two qubits: H on qubit 1, controlled-phase $\pm\pi/2$, H on qubit 0, SWAP. Export the unitary with Operator and compare to the analytic $\mathrm{QFT}_4$ matrix (12.8).

# qft4_sketch.py — structure
# qc = QuantumCircuit(2)
# qc.h(1); qc.cp(±pi/2, 0, 1); qc.h(0); qc.swap(0, 1)
# print(Operator(qc).data)  # match §12.8 matrix to ~1e-3

$n$-qubit QFT

Loop: for $i = n-1,\ldots,0$: Hadamard on $i$, then controlled phases onto earlier qubits; optional bit-reversal SWAPs. Diff against the library QFT(n) gate — max abs entry error should be $\sim 0$ for small $n$.

# qft_n_sketch.py — structure
# for i in reversed(range(n)):
#     qc.h(i)
#     for j in range(i):
#         qc.cp(angle / 2**(i-j-1), j, i)
# swaps for bit reverse unless skip_swap

Grover: oracle + diffuser + driver

Oracle for marked $\omega\in\{0,\ldots,15\}$: $X$ on bits that should be $0$, multi-controlled $X$ with $H$ sandwich on the target (phase kick), undo $X$s.

Diffuser: $H^{\otimes n}$, $X^{\otimes n}$, multi-controlled $Z$ (via $H$–MCX–$H$), undo $X$ and $H$.

Driver: $H^{\otimes 4}$, then $R\approx\pi\sqrt{N}/4$ rounds of oracle($\omega$) + diffuser (use $N=16$, not $n=4$, in the square root), measure_all, simulator histogram. IBM path: transpile to a live backend — expect a noisier peak (12.9).

# grover_sketch.py — structure
# qc.h(range(4))
# R = ceil(pi * sqrt(16) / 4)   # ← N=16 items
# for _ in range(R):
#     qc.append(oracle(9), range(4))
#     qc.append(diffuser(4), range(4))
# qc.measure_all()
# # simulator vs IBM: same circuit, different backend

Lab: wire sketch pipeline

Shor: amod15 + QPE (local)

Modular-multiply gadgets for $a\in\{2,4,7,8,11,13,14\}$ via SWAPs / $X$s; phase-estimation register with controlled-$U^{2^j}$ and inverse QFT; classical Fraction.limit_denominator → period $r$; gcd(a^{r/2}\pm1,\,15).

# shor15_sketch.py — structure
# while not done:
#     a = random in 2..14; k = gcd(a, 15)
#     if k != 1: factor k
#     else: phase = qpe([amod15(a, 2**j) for j in range(t)])
#           r = continued_fraction_denominator(phase)
#           if r even: gcd(a**(r//2)±1, 15)

Never hard-code IBM API tokens in the repo.

Visualization note

Book listings mix Aer, plot_histogram, and provider package names that change. Keep the circuit math; update imports from current Qiskit / IBM docs. This course grades stdout locks, not PNG histograms.

Takeaway

Sketch	Section
Dipole $H$ + eig	12.3
QFT₄ / QFTₙ	12.8
Oracle + diffuser + IBM	12.9
amod15 + QPE	12.10

Quiz

QC sketches

1 / 3

Grover’s π√·/4 should use…

Pick an answer to see if you are right.

Next: 12.11b QC sketch labs.