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Expectation-Value Nudge for ZNE: No Signal

Script: scripts/noise_mitigation_validation/expectation_nudge_zne.py. Negative result.

Question: can the Richardson nudge of the library's predictive ZNE be driven by the three measured expectation values alone (no tomography, any number of qubits)? Signal: (|y2 - y3| - |y1 - y2|) / (|y2 - y3| + |y1 - y2|), rectified.

Benchmark after Majumdar et al. (arXiv:2307.05203, Sect. IV, Fig. 6): 4-qubit brickwork circuits (random single-qubit gates, CZ), local noise after every gate with probability p scaled by the factor, factors 1, 2, 3, 8000 shots per factor, observable Z on qubit 0, exact density-matrix simulation (checked against full Kronecker products to 3e-17). 100 random circuits per row; RMSE against the exact zero-noise value.

Channel Depth p Richardson Linear Exponential (a = 0) Nudge Nudge shuffled Constant nudge Curvature test
depolarizing 4 0.005 0.0416 0.0151 0.0321 0.0404 0.0398 0.0369 0.0233
depolarizing 4 0.020 0.0466 0.0241 0.0308 0.0451 0.0457 0.0438 0.0326
depolarizing 12 0.005 0.0437 0.0265 1.9824 0.0414 0.0417 0.0382 0.0347
depolarizing 12 0.020 0.0875 0.1243 0.4225 0.0866 0.0879 0.0894 0.1034
amplitude damping 4 0.005 0.0466 0.0189 0.0212 0.0443 0.0446 0.0416 0.0299
amplitude damping 4 0.020 0.0456 0.0186 3.0381 0.0438 0.0444 0.0417 0.0273
amplitude damping 12 0.005 0.0390 0.0208 0.0502 0.0375 0.0379 0.0357 0.0250
amplitude damping 12 0.020 0.0513 0.0883 6.8648 0.0509 0.0515 0.0524 0.0693

Reading it

  • The nudge equals its shuffled control in every row: the signal from three expectation values carries no information about when to extrapolate less.
  • As Majumdar et al. report, the best extrapolator depends on the regime: linear wins at weak noise (up to 3× lower error than Richardson), Richardson wins at depth 12 with p = 0.02. The exponential fit is unstable in several rows, also as they describe (Appendix A).
  • A curvature test (linear if |y1 - 2 y2 + y3| is within two shot-noise standard deviations, Richardson otherwise), following their advice to check whether the data vary enough before choosing, lands between the two and is never the best: at 8000 shots the test has too little power to separate the regimes reliably.
  • Nothing here is ready for promotion.

Bounded exponential extrapolation in this benchmark

Same circuits; the library's bounded_exponential_extrapolate (Miranskyy, Sorrenti, Thind, Gravel, arXiv:2604.24475: zero-noise value constrained to [-1, 1], asymptote fitted) and the paper's variant with the asymptote fixed to 0 (their Table I), fitted in the script.

Channel Depth p Linear Richardson Bounded exp (library, 3 parameters) Bounded exp, asymptote 0
depolarizing 4 0.005 0.0151 0.0416 0.5720 0.1425
depolarizing 4 0.020 0.0241 0.0466 0.3814 0.0897
depolarizing 12 0.005 0.0265 0.0437 0.3458 0.0306
depolarizing 12 0.020 0.1243 0.0875 0.4366 0.2312
amplitude damping 4 0.005 0.0189 0.0466 0.5136 0.0206
amplitude damping 4 0.020 0.0186 0.0456 0.2995 0.1122
amplitude damping 12 0.005 0.0208 0.0390 0.3896 0.1020
amplitude damping 12 0.020 0.0883 0.0513 0.2893 0.2785
  • On the exact expectation values (no shot noise) the library function recovers the zero-noise value to RMSE 0.0001 (40 circuits, depolarizing, depth 4, p = 0.005). With 8000 shots it lands on the bound ±1 in 11 of those 40 circuits: three noise factors and three free parameters let the fit follow the shot noise, the instability Majumdar et al. describe for nearly λ-independent data (Appendix A). The bound keeps the result in range but does not make it accurate.
  • Fixing the asymptote to 0 removes one parameter and helps, but it is still worse than linear or Richardson in most rows, and 0 is the wrong asymptote for amplitude damping, which drives <Z> towards +1.
  • Miranskyy et al. also report that the benefit of bounding is model-dependent. For Dense-Evolution this is a limitation worth stating in bounded_exponential_extrapolate's documentation: with three noise factors and realistic shot counts it is not reliable.