What is the theory behind PEA?#
Probabilistic error amplification (PEA) [59] combines probabilistic sampling of noise-amplified circuits with an extrapolation step similar to zero-noise extrapolation (ZNE).
Noise-amplified representations#
Consider an ideal circuit made of operations \(\mathcal G_i\). PEA constructs, for each scale factor \(s\), a noise-amplified representation in which each ideal operation is expressed as a linear combination of implementable noisy operations \(\mathcal O_{i,\alpha}^{(s)}\) corresponding to a noise level scaled by \(s\):
The coefficients \(\eta_{i,\alpha}^{(s)}\) form a quasi-probability representation with one-norm \(\gamma_i^{(s)} = \sum_\alpha |\eta_{i,\alpha}^{(s)}|\).
Monte Carlo estimation at each scale factor#
For a fixed scale factor \(s\), the overall quasi-probability representation of the circuit induces a probability distribution \(p(\vec{\alpha}) = |\eta_{\vec{\alpha}}^{(s)}|/\gamma^{(s)}\), where \(\gamma^{(s)} = \prod_i \gamma_i^{(s)}\) is the product of the gate-wise one-norms. Sampling a noisy circuit \(\Phi_{\vec{\alpha}}^{(s)}\) from this distribution and applying a sign \(\sigma_{\vec{\alpha}} = \mathrm{sign}(\eta_{\vec{\alpha}}^{(s)})\) yields an unbiased estimator of the expectation value at noise scale \(s\):
This is exactly what is computed when running mitiq.experimental.pea.pea.combine_results().
Extrapolation to the zero-noise limit#
After obtaining \(E^{(s)}\) for several scale factors \(s\), PEA applies a ZNE inference method to extrapolate to the zero-noise limit. The resulting value is the PEA estimate of the ideal expectation value. See What is the theory behind ZNE? for more details about the theory of extrapolation in ZNE.
Tip
Because the sampling overhead grows with \(\gamma^{(s)}\), PEA is most effective when the amplified representations remain reasonably low-norm (which means the ideal operations are close to the implementable ones) and the chosen scale factors are not too large.