Bit-flip and Phase-flip Sensitivity

qc/noise

The same measurement can be “blind” to certain errors and highly sensitive to others — it all depends on which basis you measure in relative to what the error actually does.

The core intuition

• A bit flip swaps the roles of $|0⟩$ and $|1⟩$ — this is exactly what $X$ does.
• A phase flip flips the sign of the $|1⟩$ component without touching the $|0⟩$/$|1⟩$ labels — this is exactly what $Z$ does: $Z|0⟩=|0⟩$, $Z|1⟩=-|1⟩$.

A Z-basis measurement only cares about $|\alpha {|}^2$ vs $|\beta {|}^2$ — a sign flip on $\beta$ changes nothing observable there. That’s why a pure $Z$ error is invisible to a computational-basis measurement.

But $|+⟩$ and $|-⟩$ differ only by that same sign:

$$|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩),|-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)$$

So $Z|+⟩=|-⟩$ — the same sign flip that was invisible in the Z basis becomes a fully distinguishable outcome in the X basis. Measuring in a different basis turns invisible phase information into visible which-outcome information.

Two complementary experiments (built on the repeated-X circuit)

Experiment A — bit-flip test: start at $|0⟩$, apply $2n$ x gates, measure in Z-basis. Sensitive to $X$/$Y$ errors, blind to pure $Z$.

Experiment B — phase-flip test: prepare $|+⟩$ (via H), apply $2n$ x gates, then apply another H before measuring (this rotates the X-basis question into a Z-basis measurement). Sensitive to $Z$/$Y$ errors, blind to pure $X$ (since $X|+⟩=|+⟩$).

def repeated_x_meas_x_circuit(n):
    qc = QuantumCircuit(1, 1)
    qc.h(0)
    for _ in range(2 * n):
        qc.x(0)
    qc.h(0)          # rotate back before measuring — the "Hadamard sandwich" trick
    qc.measure(0, 0)
    return qc

The “Hadamard sandwich” — one H to enter a different basis, one H at the end to rotate back before a standard measurement — is a general technique for measuring in any basis, not just this experiment.

Related

• Pauli Error Model
• Depolarizing Noise
• GHZ States

Self-Check

• Could you explain why a pure Z error is invisible to a Z-basis measurement?
• What does the “Hadamard sandwich” trick actually accomplish, and why does it work?
• Why does a phase flip become visible in the X basis when it was invisible in the Z basis?