Zero-Noise Extrapolation (ZNE)

Overview

Zero-Noise Extrapolation (ZNE) answers the question:

• “If my device were less noisy, what value would this circuit produce?”

ZNE is a mitigation method, not a hardware fix. It works by:

1. running the same logical circuit at multiple noise levels
2. observing how the measured quantity changes
3. extrapolating back to an estimate of the zero-noise value

ZNE is widely discussed because it is conceptually simple and broadly applicable to expectation-value style outputs.

Intuition

Noise scaling idea

If you could continuously “turn a noise knob,” then for a small enough circuit you might see:

• more noise → results drift away from the ideal

If that drift is smooth, you can fit a curve and estimate where the curve would hit at “zero noise.”

How do we get multiple noise levels?

In practice, you cannot always directly change hardware noise. Instead, you can amplify the effect of noise while keeping the ideal computation the same.

A common conceptual trick:

• replace a gate $U$ with $U U U$ (odd repetitions)

Ideally, $U^3$ is not equal to $U$ for arbitrary $U$, so real implementations are more careful. But the high-level idea is:

• increase circuit exposure to noise without changing the intended logical action

We keep this conceptual here—the point is to generate “same computation, different noise.”

Formal Description

We describe ZNE as a procedure on an observable or output statistic.

Target quantity

ZNE is most often used to estimate an expectation value like:

• $\langle O \rangle$ for some measurement operator $O$

You run the circuit many shots and compute an estimate of $\langle O \rangle$.

Steps of ZNE (conceptual)

1. Choose a set of noise scale factors, e.g. $\lambda \in \{1,2,3\}$.
2. For each $\lambda$, create a noise-scaled version of the circuit.
3. Execute each version and measure the quantity of interest, getting values like:
   • $E(1), E(2), E(3)$
4. Fit a simple curve through these points.
5. Extrapolate to $\lambda=0$ to estimate $E(0)$.

Conceptual curve fitting

A simple approach is to assume the measured value depends smoothly on noise scale and approximate it with:

• a line (first-order)
• a quadratic (second-order)

The exact choice depends on how much data you have and how stable the fit is.

When ZNE works and fails

ZNE tends to work better when:

• noise is not too large
• the circuit depth is moderate
• the noise scaling procedure behaves consistently
• the quantity you estimate changes smoothly with noise level

ZNE tends to fail or become unstable when:

• noise is strong and behavior is not smooth
• fits are dominated by shot noise
• noise scaling changes the circuit behavior in unintended ways
• errors are highly nonstationary or strongly correlated

Worked Example

Suppose your goal is an expectation value that should ideally be 0.80. You run ZNE at three noise scales and measure:

• $E(1)=0.74$
• $E(2)=0.69$
• $E(3)=0.64$

This looks roughly linear. A simple extrapolation back to “zero noise” might estimate something like:

• $E(0)\approx 0.79$

Interpretation:

• The corrected estimate is closer to the ideal.
• But you paid for it with extra experiments and you relied on a modeling assumption (linearity).

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Next we focus on a mitigation method that is extremely common and often low-hanging fruit: measurement error mitigation. It uses readout calibration to correct observed bitstring frequencies.