SPSA Optimizer: Noisy Quantum Landscapes

Shot Noise & Step Size

Shot noise σ 0.15

Step size a₀ 0.35

Perturbation c₀ 0.30

Landscape ruggedness 1.20

Live Readouts

Iteration k

Circuit evals

Current cost C(θ)

0.00

Gap to ground state

0.00

How it works

Variational quantum algorithms (VQE, QAOA) minimize an energy-like cost C(θ) measured on real quantum hardware. Each measurement is a finite sample of shots, so every evaluation of C is noisy — a plain finite-difference gradient needs 2 evaluations per parameter, and noise corrupts each one.

SPSA (Simultaneous Perturbation Stochastic Approximation) instead perturbs every parameter at once along one random ± direction Δₖ, using just 2 noisy evaluations per step no matter how many parameters exist:

Δ_k,i ∈ {-1,+1}  (random, all i at once)
y± = C(θ_k ± c_k·Δ_k) + noise(σ)
ĝ_k,i = (y+ − y−) / (2·c_k·Δ_k,i)
θ_k+1 = θ_k − a_k·ĝ_k

a_k = a₀ / (k+1+A)^0.602   A = 5 (stability)
c_k = c₀ / (k+1)^0.101

Shot noise σ — Gaussian noise added to every circuit evaluation, mimicking finite measurement shots on real hardware.
Step size a₀ / Perturbation c₀ — the standard SPSA gain-sequence constants; too large and the walker overshoots, too small and it stalls.
Landscape ruggedness — raises the frequency of one interference term, adding more local minima the optimizer can get trapped in.
The translucent surface is the true (noiseless) cost landscape; the gold marker is its global minimum found by brute-force grid search. The trail shows where SPSA's noisy gradient estimates actually walked — it can wobble even while heading downhill.

This 2-evaluation-per-step property is why SPSA (and cousins like Nakanishi-Fujii-Todo) is the default noisy-hardware optimizer in real VQE/QAOA runs, where every circuit evaluation costs real device time.

SPSA trail (fades with age) Global minimum Current position

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