Error Mitigation: Zero-Noise Extrapolation

Zero-Noise Extrapolation (ZNE)

ZNE is the most practical error mitigation technique for NISQ devices. Instead of correcting errors after the fact, it measures the effect of noise at several amplified levels and extrapolates back to the zero-noise limit.

The Problem ZNE Solves

On real hardware, every gate has a small error probability. A Bell state circuit (2 gates) has ~99.5% fidelity per gate, giving ~99% total. A VQE ansatz with 20 gates has ~82% fidelity. These errors bias the expectation value ⟨O⟩ away from its ideal value.

Error correction fixes this but requires thousands of physical qubits per logical qubit: infeasible on NISQ devices. Error mitigation reduces bias at a cost of more shots. ZNE is the most hardware-agnostic mitigation technique.

The ZNE Idea

Assume the expectation value degrades with noise level λ: E(λ) = E₀ + a₁λ + a₂λ² + ...

If we measure E(λ) at several values of λ (the same circuit at different effective noise levels), we can fit this polynomial and extrapolate to λ=0 to recover E₀.

Richardson Extrapolation

For m noise levels λ₁ < λ₂ < ... < λₘ with measured values E₁, E₂, ..., Eₘ, Richardson extrapolation uses Lagrange interpolation at λ=0:

E₀ = Σᵢ Eᵢ × Lᵢ(0)

where the Lagrange basis weights are: Lᵢ(0) = ∏_{j≠i} (0 − λⱼ) / (λᵢ − λⱼ)

Two-point (linear): E₀ = (λ₂E₁ − λ₁E₂) / (λ₂ − λ₁). Simple but only removes linear noise bias.

Three-point (quadratic): Also removes quadratic noise terms. The "right" number of points depends on the noise model.

Practical Limits