Chapter 4 — Key Takeaways (Multi-Qubit Programming)

The two-qubit reference. Part IV is built from this page plus Chapter 3's gate table.

Tensor product and ordering

$$|\psi\rangle = |q_{n-1}\rangle \otimes \cdots \otimes |q_1\rangle \otimes |q_0\rangle$$

Qubit 0 is LAST in the tensor product and RIGHTMOST in the bitstring. Reverse of most textbooks.

np.kron(zero, one) == Statevector(QuantumCircuit(2).x(0)).data     # both [0,1,0,0]

Use probabilities_dict() whenever the index needs a meaning. Reserve raw arrays for numerical work. Test with asymmetric states — symmetric ones hide endianness bugs.

CNOT

qc.cx(control, target)      # flips target iff control is 1

cx(0,1)                    cx(1,0)  ← the textbook matrix
[[1 0 0 0]                 [[1 0 0 0]
 [0 0 0 1]                  [0 1 0 0]
 [0 0 1 0]                  [0 0 0 1]
 [0 1 0 0]]                 [0 0 1 0]]

Both are correct CNOTs; the difference is endianness.

On a superposition CNOT acts on both branches, producing a state that does not factorize. That non-factorizability is entanglement.

The four Bell states

qc.x(1)   if Psi        # then:
qc.h(0)
qc.z(0)   if minus
qc.cx(0, 1)

$\Phi$ vs $\Psi$ shows in the counts. The $\pm$ does not.

Bell measurement = run the preparation backward (cx(0,1) then h(0)) before measuring → the four states map to 00, 01, 10, 11 deterministically. The primitive behind teleportation and superdense coding (Ch. 9).

★ Testing for entanglement

from qiskit.quantum_info import Statevector, partial_trace, entropy
rho0 = partial_trace(Statevector(qc), [1])       # keep qubit 0
ent = float(entropy(rho0))                        # 0 = product, 1 = maximal

A qubit of a Bell state sits at the CENTER of the Bloch sphere — completely undetermined on its own, while the pair is in a completely definite, zero-entropy state. All the information is in the correlation and none is in the parts. It is not "in superposition."

⚠️ Counts never prove entanglement

An impostor with zero entanglement gives identical Z-basis counts:

qc.h(0); qc.measure(0, 0)
with qc.if_test((qc.clbits[0], 1)):
    qc.x(1)

More shots cannot help — same distribution, measured more precisely.

The witness (the real standard):

$$W = \langle Z\otimes Z\rangle + \langle X\otimes X\rangle, \qquad W \le 1 \text{ for every separable state}, \quad W = 2 \text{ for } |\Phi^+\rangle$$

Gate costs — memorize these

Add 3 CNOTs per SWAP the router inserts for non-adjacent qubits. You rarely write swap — the transpiler inserts it, invisibly, and it is the largest hidden cost in quantum programming.

CZ is symmetric (no control/target). $H_{(1)}\,\mathrm{CX}(0,1)\,H_{(1)} = \mathrm{CZ}$, because $HXH = Z$.

GHZ vs W

ghz: h(0); cx(0,1); cx(1,2); ...          # (|00…0⟩ + |11…1⟩)/√2

Not interconvertible by local operations: genuinely different kinds of tripartite entanglement. GHZ's fragility makes it a good benchmark (Ch. 30).

The exponential wall, refined

Product states cost $2n$; general states cost $2^n$. The cost is entanglement, in proportion to how much there is — which is exactly what matrix product state simulation exploits (Ch. 11).

An algorithm that does not generate substantial entanglement is classically simulable and cannot give a quantum advantage.

📉 Hardware: layout decides correctness

GHZ correct-fraction on a device-derived model, 4096 shots:

The cliff is a broken physical qubit (readout error 0.257, ECR error 1.0), not scaling. Levels 0–1 pick qubits by position; levels 2–3 pick them by measured error rate.

Preflight, always:

isa.layout.final_index_layout()          # RECORD THIS with every result
backend.target["measure"][(q,)].error    # > 0.1 → unusable
backend.target["ecr"][(a, b)].error      # > 0.05 → suspect; 1.0 → dead

Common pitfalls

• Reading sv.data[2] and assuming you know which state that is.
• Testing with symmetric states, which hide every endianness bug.
• Inferring entanglement from computational-basis counts.
• Taking more shots to fix a wrong measurement setting.
• Trusting optimization level 1 for a result you will report.
• Reporting a hardware result without the layout.
• Reading "correct fraction" as evidence of coherence — a classical mixture scores 100%.

Project piece added this chapter

vqelab/circuits.py v1 — two_qubit_ansatz() (rotation layer → entangling layer → rotation layer), plus entanglement_entropy(), is_entangled(), and reachable_entanglement().

Without the CNOT the qubits stay in a product state forever, and a product-state ansatz can only represent product-state solutions — which for a molecular ground state is exactly wrong. reachable_entanglement() is the first diagnostic to run when a VQE will not converge.