Organizations: Department of Chemistry, University of Washington, Seattle, Washington 98195, USA · Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA · Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA
Quantum noise is expected to degrade quantum machine learning by driving circuits away from their noiseless implementations. Yet recent studies show moderate noise can reduce testing error, a behavior unexplained by weak-noise perturbative error accumulation or strong-noise trainability collapse. Here we develop a statistical learning theory connecting microscopic noise processes to macroscopic learning performance. At its heart is a noise-order purity parameter, derived from a surrogate model analysis, that predicts the noise-induced reduction in model complexity and the consequent reduction in the generalization gap. Noise simultaneously increases prediction bias. Their competition explains the intermediate-noise regime left open between these limits. It produces a finite-noise optimum whose location depends on the learning setup and can disappear in the large-sample limit. Numerical experiments validate these predictions. Noise programming can move a model towards this optimum. These results make the non-monotonic effect of noise predictable and provide a route to harness it.
Figures & tables
Figure 1: Non-monotonic response of the four-qubit regression model to physical noise. (a–d) Final training and testing mean-squared errors for four quantum noise models. Curves show the mean over ten independently trained runs and shaded bands show one standard deviation. (e) Generalization gap normalized by its value at zero noise. The dashed line shows the surrogate prediction introduced below.
Figure 2: Effective model complexity under physical noise. (a) Effective model complexity as a function of noise rate for the four noise models. Lines show the mean over ten runs and shaded regions show one standard deviation. (b) Effective model complexity normalized by its zero-noise value and plotted against s2(β,geff) . Error bars show one standard deviation; fitted values of geff are reported in Section B.2 .
Figure 3: Parameter-wise output-gradient norms under physical noise. (a–d) Root-mean-square output-gradient norm γi(β) versus parameter index i for representative noise rates. Each point is averaged over ten independent runs. In (b), the β=0.5 bit-flip channel symmetrically averages the identity and bit-flip branches in the final noisy layer, giving γi(0.5)=0 exactly; the corresponding curve is therefore absent on the logarithmic scale. (e) Output-gradient norm averaged over parameters and normalized by its zero-noise value. Roman numerals indicate three noise regimes.
Figure 4: Testing error across physical noise rates and noise-programming levels. Mean testing MSE after training at physical noise rate β and programming level k , corresponding to a parameter-perturbation strength σ=2−k . The row labelled “none” is the unprogrammed baseline. Each cell is averaged over ten runs.
Figure 5: Noise-induced regularization in molecular QML. Testing MSE is compared (a) between ntrain=80 and 160 on QM9-HA8 under bit-flip noise, (b) between bit-flip and depolarizing noise on QM9-HA8 with ntrain=160 , and (c) between QM9-HA8 and PCQM4Mv2-HA9 under bit-flip noise with ntrain=80 . (d) Training MSE for the four distinct settings in (a–c). (e) Generalization gap versus effective model complexity, with both quantities normalized by their values at β=0 within each run. (f) Effective model complexity normalized by its value at β=0 versus s2(β,geff) , where geff is fitted separately for each setting. The QM9-HA8 and PCQM4Mv2-HA9 models are evaluated after 300 and 100 epochs, respectively. Lines and shaded regions in (a–d) show the mean and one standard deviation over five runs; points and error bars in (e,f) show the corresponding mean and standard deviation.
Figure 6: Training-set-size dependence of the finite-noise response. (a) Training MSE, (b) testing MSE and (c) generalization gap under bit-flip noise for nested QM9-HA8 training sets with ntrain=80 , 160, 320 and 1000. The first three models are evaluated after 300 epochs and the ntrain=1000 model after 100 epochs. Lines and shaded regions show the mean and one standard deviation over five fixed runs.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Effective model complexity of the noise-order surrogate. (a) Surrogate effective model complexity as a function of noise rate. Lines show the mean over ten runs and shaded regions show one standard deviation. (b) Effective model complexity normalized by its zero-noise value and plotted against s2(β,geff) . Error bars show one standard deviation; fitted values of geff are reported in the text.
Depolarizing
Bit flip
Amplitude damping
Phase damping
Physical model
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Surrogate
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Appendix
Table 1: Fitted effective noise counts obtained from the ten-run mean normalized complexity curves in Figs. 2 and 7 . Both calculations use the soft-counting threshold 10−4 .
Figure 8: Ablation of optimization epochs. Training MSE, testing MSE and generalization gap are evaluated after 100, 200 and 300 epochs for QM9-HA8 with (a–c) n=80 and (d–f) n=160 training molecules. Lines show the mean over five fixed runs, and shaded regions show one sample standard deviation.
Figure 9: Ablation of more complex classical post-processing. (a–c) Training MSE, testing MSE and generalization gap for a linear readout and a nonlinear readout with 128 tanh hidden units. (d,e) Absolute effective model complexities computed using the full parameter vector and the quantum-circuit parameters only, respectively. Both models use the same 53-parameter quantum circuit, n=80 QM9-HA8 training molecules, bit-flip noise and 300-epoch endpoint. Lines and shaded regions show the mean and one standard deviation over five fixed runs.
Figure 10: Ablation of noise models in the molecular QML model. (a) Testing MSE, (b) training MSE, (c) generalization gap and (d) effective model complexity normalized by its zero-noise value for bit-flip, depolarizing, amplitude-damping and phase-damping noise. The damping channels are shown over 0≤β≤1 , while bit-flip and depolarizing noise are shown over 0≤β≤0.5 . All models use the same QM9-HA8 training set with n=160 and are evaluated after 300 epochs. Lines and shaded regions show the mean and one standard deviation over five fixed runs.
Figure 11: Measurement dependence under phase damping. (a) Training MSE, (b) testing MSE, (c) generalization gap and (d) effective model complexity normalized by its zero-noise value for full-probability measurements in the Z and X bases. (e) Normalized generalization gap versus normalized effective model complexity. (f) Normalized effective model complexity versus s2(β,geff) , with geff fitted separately for the two bases. Green circles and orange squares denote the Z - and X -basis measurements, respectively. Both models use the same QM9-HA8 training set with n=160 and are evaluated after 300 epochs. Lines and shaded regions in (a–d), and points and error bars in (e,f), show the mean and one standard deviation over five fixed runs.
Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Seoul 03722, Republic of Korea · Department of Quantum Information, Yonsei University, Incheon 21983, Republic of Korea · Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea +1
Department of Physical Chemistry, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain · EHU Quantum Center, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain · TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Spain +4