# Probabilistic error cancellation: how are we sure (i) the inverse noise *mathematically* exists, (ii) its expansion contains implementable operations

I am currently learning the basics of probabilistic error cancellation.

The idea, in summary, is the following:

We want to implement a quantum circuit having a noiseless (unitary) implementation $$\mathcal{U}$$, but, because of noise, we can only implement $$\mathcal{E}=\mathcal{N} \circ \mathcal{U}$$, where I assume $$\mathcal{N}$$ is a CPTP (Completely Positive Trace Preserving) operation that introduces noise in the computation (the noiseless case corresponds to $$\mathcal{N}=\mathbb{I}$$).

The idea behind probabilistic error cancellation is to try to find an inverse to $$\mathcal{N}$$ and to apply it. Ideally we then wish to implement the circuit: ( * )

$$\mathcal{N}^{-1} \circ \mathcal{N} \circ \mathcal{U}=\mathcal{U}$$

Unfortunately, this inverse (if it exists!) will (i) not always correspond to a physical map (i.e. we loose the positivity condition for instance), (ii) will not always be implementable by the hardware.

Hence, the "trick" is to perform the following decomposition:

$$\mathcal{N}^{-1}=\sum_{n} \alpha_n \mathcal{B}_n$$

where the $$\{\alpha_n\}$$ are some coefficients and $$\{\mathcal{B}_n\}$$ is a family of physical maps (i.e. CPTP) that I can exactly implement on the hardware. Hence, the $$\{\mathcal{B}_n\}$$ is typically a family of noisy maps (because my hardware is noisy).

My questions:

In order to be applicable, we need:

• $$\mathcal{N}^{-1}$$ exists mathematically (not all maps have an inverse).
• We can express $$\mathcal{N}^{-1}$$ as a linear combination of noisy operations that the hardware can actually do.

If I understood correctly the general idea, how are we sure that these conditions will be satisfied? Am I exactly expressing the limit in the applicability of the method?

( * ) I took an example where the noise map of the entire circuit can easily be found which is too complicated in general. However the idea can be generalized by introduced the noise map of each gate and writing a big sum (I skip these details here).

• Do you mean we don't know the analytical form of $\mathcal{N}$? Commented Mar 11, 2023 at 12:12
• @narip In the context of my question, we can assume we know perfectly the (analytical) description of $\mathcal{N}$. Thanks! Commented Mar 11, 2023 at 12:13
• Then for the first one, since $\mathcal{N}$ is linear, we can easily check if it's invertible by writing it as a matrix. Commented Mar 11, 2023 at 12:15
• @narip yes I agree. But my question is more to know if these two conditions are in practice two actual limitations of the method. In the papers I read these limitations were never explicitly written which I found weird (it seemed like we can always do P.E.C). Maybe there are good reasons to believe that these conditions are always satisfied. If so I would like to know these reasons. Commented Mar 11, 2023 at 12:18