Probabilistic Error Cancellation
Inverting the hardware noise channel by sampling from a quasi-probability representation of its inverse.
Inverting the noise channel
If the noisy hardware applies a known noise channel N to each ideal gate, the ideal result could be recovered by applying N inverse. But N inverse is generally not a physical (completely positive) operation, so it cannot be run directly. Probabilistic error cancellation (PEC) writes N inverse as a quasi-probability combination of physical operations that the device can actually perform.
Quasi-probability decomposition
Express the inverse channel as N^{-1} = sum_i q_i O_i, where the O_i are implementable operations (gates, Paulis, measurements) and the coefficients q_i are real but can be negative, summing to one. The negative coefficients are why this is quasi-probability. To estimate an observable, sample operation O_i with probability |q_i|/gamma, multiply each measurement outcome by sign(q_i)*gamma, and average, where gamma = sum_i |q_i| is the total negativity.
The procedure
- Characterize the noise channel for each gate (via gate set tomography or simpler models).
- Compute the quasi-probability decomposition of the inverse of each channel.
- For each circuit run, randomly insert the sampled correction operations.
- Reweight and average the results with the appropriate signs to get an unbiased estimate.
The sampling cost
PEC gives an unbiased estimator, its key advantage over the biased extrapolation of ZNE. The price is variance: the number of samples needed grows like gamma^2, and gamma itself grows exponentially with circuit depth and error rate. This makes PEC accurate but expensive, feasible only for circuits where the accumulated negativity stays bounded.
Comparison and use
Relative to zero-noise extrapolation, PEC removes bias at the cost of higher and more sharply growing sampling overhead, and it requires accurate noise characterization that ZNE does not. In practice the two are complementary: ZNE for cheap approximate correction, PEC when an unbiased estimate is worth the shots. Both are central methods in the error mitigation toolkit.