Zero-Noise Extrapolation Lab

Hardware Noise Model

Per-layer error rate p 6.0%

Circuit depth (layers) 8

Target rotation θ 60°

Extrapolation

Live Readout

ZNE estimate ⟨Z⟩(0)

0.00

Raw (λ=1, unmitigated)

0.00

Ideal value cos θ (ground truth)

0.00

Error reduction vs. raw measurement

λ=1.0 → 0.00

How it works

A noisy quantum processor doesn't return the ideal expectation value ⟨Z⟩_ideal = cos θ of the target rotation — depolarizing-style noise accumulated over the circuit's depth layers shrinks it toward zero:

⟨Z⟩(λ) ≈ cos θ · e^(−γλ),   γ = 2 · depth · p

where λ = 1 is the circuit run normally. Zero-noise extrapolation (ZNE) deliberately runs the same circuit at amplified noise levels — physically via "unitary folding" (each gate G is replaced by G·G⁻¹·G, tripling its noise for λ=3, halfway for λ=2, etc.) — producing measurements at λ = 1, 1.5, 2, 2.5, 3. A polynomial is fit through these noisier-than-normal points and extrapolated backward to the unphysical point λ = 0, which estimates what the circuit would have measured with no noise at all:

Richardson (order-2): E(0) ≈ Σᵢ cᵢ·E(λᵢ),  Σcᵢ=1,  Σcᵢλᵢᵏ=0 for k=1..n−1

p, depth, θ — set the hidden hardware noise rate, circuit size and the true rotation the circuit is trying to measure.
Linear / Quadratic fit — the extrapolation model: quadratic (Richardson order 2) tracks the true exponential curve much better than a straight line when noise is large.
Resample — draws a fresh statistical shot-noise realization, exactly like re-running the circuit for another batch of measurement shots.

Real-world relevance: ZNE is a production technique on IBM, Google and IonQ hardware today — it needs no extra qubits or error-correcting code, only extra (noisier) circuit runs, which is why it's the leading error-mitigation method for the current NISQ (noisy intermediate-scale quantum) era.

Measured points (with shot noise) Extrapolated ZNE estimate (λ=0) Ideal value (ground truth)

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