Depolarizing Noise

qc/noise

The simplest way to model an imperfect gate: after the gate, the state gets partially replaced by the fully mixed (maximally random) state.

$$\rho \mapsto (1-\lambda )\rho +\lambda \frac{I}{2}$$

Larger $\lambda$ = more randomization. It’s intentionally non-specific about why the gate is imperfect — no direction, no timing dependence, just “some of the information got scrambled.”

The repeated-X experiment

A clean way to see this noise accumulate: apply an even number $2n$ of x gates to $|0⟩$ and measure. Since $X^2=I$, an ideal circuit always returns exactly to $|0⟩$ regardless of $n$ — so any decay in $\mathrm{Pr}(0)$ as $n$ grows is a pure noise signal, not a logic error.

def repeated_x_circuit(n):
    qc = QuantumCircuit(1, 1)
    for _ in range(2 * n):
        qc.x(0)
    qc.measure(0, 0)
    return qc

Under this model, the survival probability decays exponentially toward $1/2$ (fully random) as $n$ grows:

$$\mathrm{Pr}(0)=\frac{1+(1-\lambda {)}^{2n}}{2}$$

This same repeated-gate structure is reused (with different measurement bases) to probe directional Pauli errors and time-dependent decay — it’s a general-purpose noise-characterization pattern, not just a depolarization-specific trick.

Related

• Density Matrix
• Pauli Error Model
• Thermal Relaxation
• Backend Properties
• Bit-flip and Phase-flip Sensitivity — same repeated-gate experiment, extended to detect which Pauli error is present

Self-Check

• Why does the repeated-X experiment isolate noise from logic errors — what makes $2n$ gates special?
• Could you explain in plain language what depolarizing noise does to a qubit?
• Why is depolarizing noise called “non-specific” compared to the Pauli error model?