Measurement and Collapse

qc/basics qc/math

Measuring a qubit $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ in the standard (Z) basis gives outcome 0 with probability $|\alpha {|}^2$ and outcome 1 with probability $|\beta {|}^2$ — the Born rule. Immediately after, the state is no longer $\alpha |0⟩+\beta |1⟩$: it’s exactly $|0⟩$ or exactly $|1⟩$, matching whatever outcome occurred. The superposition is destroyed by the act of measuring — this is collapse, and it’s irreversible.

Why this makes quantum programs inherently probabilistic

A single run of a circuit on hardware gives you exactly one classical bit per qubit — never $\alpha$ and $\beta$ directly. To learn anything about the underlying amplitudes (or estimate an expectation value, see SparsePauliOp), you run the same circuit many times (“shots”) and build up statistics over the outcomes. This is why Qiskit’s `Sampler` returns a distribution of counts, not a single deterministic answer, and why quantities used throughout this vault (fidelity, expectation values) are estimates, not exact numbers, once real hardware is involved.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit_aer.primitives import SamplerV2
 
qc = QuantumCircuit(1, 1)
qc.h(0)
 
# Simulator only: peek at exact amplitudes before measurement (impossible on real hardware)
Statevector(qc).probabilities()   # array([0.5, 0.5]) — the ground truth
 
qc.measure(0, 0)
result = SamplerV2().run([qc], shots=1000).result()
result[0].data.c.get_counts()     # e.g. {'0': 512, '1': 488} — the statistical estimate of that ground truth

Related

• What is a Qubit
• Superposition
• Backend Properties — readout error: measurement itself isn’t perfectly faithful on real hardware
• TREX — the mitigation technique specifically for readout error
• The Primitives Family, Simulators — Statevector vs Shot-Based

Self-Check

• Why can’t you just “read off” $\alpha$ and $\beta$ from a single run on real hardware?
• What’s the difference between what Statevector(qc).probabilities() gives you and what SamplerV2 gives you, and why?
• Why does collapse being irreversible matter for how quantum programs have to be structured?