1. Conceptual Framework: Wavefunction Collapse
Quantum measurement is the interface between quantum states and classical-readable information.
- The Mechanism of Collapse: Qubits evolve as probability amplitudes until measurement, which forces collapse into a single classical outcome.
- The Irreversible Threshold: Measurement is destructive. After collapse to $0$ or $1$, the original superposition amplitudes and phases are not retained in the measured classical record.
- Ideal vs. Experimental Realities: Ideal theory matches exact probabilities, but real QPUs include noise from decoherence, gate error, and readout imperfections.
2. Mathematical Context: Born's Rule
Born's rule converts amplitudes into measurement probabilities.
Defining the Probability Vector
$$P(\text{measurement result}) = |c_i|^2$$
For complex amplitudes:
$$|c_i|^2 = c_i^* c_i$$
3. Comprehensive Circuit Mapping Example
Consider a two-qubit workflow: apply $H$ to qubit 0, apply $X$ to qubit 1, then fully measure in the computational $Z$-basis.
Step 1: Track the System State Vector
Before measurement:
$$|\psi\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|10\rangle$$
Step 2: Compute Outcome Probabilities via Born's Rule
$$P(|00\rangle) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2} \quad (50\%)$$
$$P(|01\rangle) = |0|^2 = 0 \quad (0\%)$$
$$P(|10\rangle) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2} \quad (50\%)$$
$$P(|11\rangle) = |0|^2 = 0 \quad (0\%)$$
4. Summary Matrix: Two-Qubit Full Readout Map
| Target Computational Basis State |
Amplitude Coefficient ($c_i$) |
Analytical Probability ($|c_i|^2$) |
Realized Readout Percentage |
Physical Wavefunction Behavior |
| $|00\rangle$ |
$\frac{1}{\sqrt{2}}$ |
$\frac{1}{2}$ |
50% |
Occupied branch after collapse sampling |
| $|01\rangle$ |
$0$ |
$0$ |
0% |
No branch occupancy |
| $|10\rangle$ |
$\frac{1}{\sqrt{2}}$ |
$\frac{1}{2}$ |
50% |
Occupied branch after collapse sampling |
| $|11\rangle$ |
$0$ |
$0$ |
0% |
No branch occupancy |
Operational interpretation: In an ideal noise-free run of 1,000 shots, outcomes concentrate near 500 counts of $00$ and 500 counts of $10$, with no $01$ or $11$ events.