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I am currently learning randomized compiling with this paper which seems to be the "main" reference on this topic (at least many papers refer to this one).

However, I do not understand how this technique can help "in practice": from my understanding the assumption on the noise model is very demanding, we have to assume that the noise occurs "after" (or "before") the gate. I would like to check wether or not I understand this point correctly.

Given a quantum channel $\mathcal{E}$, by applying correlated Paulis before and after the gate, I can make this quantum channel diagonal in the Pauli basis. This is called Pauli twirling and it is of central use in randomized compiling.

In practice, in the paper I cite, it seems that we need to assume that a noisy gate (completely positive, trace preserving) $\mathcal{G}$ that attempts to implement a perfect unitary $\mathcal{U}$ is modelled as:

$$\mathcal{G}=\mathcal{N} \circ \mathcal{U}$$

where $\mathcal{N}$ is some noise introduced by the gate.

Mathematically, I totally agree we can always define $\mathcal{N}$ in such a way (knowing $\mathcal{G}$ and $\mathcal{U}$, I can say $\mathcal{N} \equiv \mathcal{G} \circ \mathcal{U}^{\dagger}$). However it does not mean that physically, "in the lab", the evolution behave as a "perfect" unitary followed, after, by a noise channel. Realistically speaking, both happen at the same time.

In the paper I cite, the assumption that we have a unitary evolution followed by a noise acting after seems to be critical. For instance, in the legend of figure 2 it is written:

To tailor the noise into stochastic noise, we insert random twirling gates before the noise and the corresponding correction gates immediately after the noise.

How can we do this insertion in practice in the lab as the noise and the unitary evolution almost always occur at the same time for any realistic noise model?

It is possible I misunderstood the point of the paper.

Marco Fellous-Asiani
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1 Answers1

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Randomized compiling does not depend on whether you model the errors in the gate as occurring before, after, or during the gate. The important part is that you sandwich the target gate (which includes both the ideal unitary and the error process) by random Paulis and their inversions. In other words, $\mathcal{G} = \mathcal{N} \circ \mathcal{U}$ is mapped to

$P^c \circ \mathcal{G} \circ P = P^\dagger \circ \mathcal{N} \circ \mathcal{U} \circ P$,

where $P$ is a tensor product of $n$-qubit Paulis sampled at random, and $P^c$ is the set of Paulis which undo $P$ when commuted through $\mathcal{G}$. In the limit of a large number of randomly sampled Paulis, the error process $\mathcal{N}$ is tailored into a stochastic Pauli channel (i.e., it is diagonal in the Pauli basis). Therefore, the main restriction of the method is that the gate you are twirling must be a Clifford. This is because Cliffords maps Paulis into other Paulis, so it is trivial to compute the inversion Pauli operators. If $\mathcal{G}$ is non-Clifford, then you can only twirl using the subset of Paulis that commutes with $\mathcal{G}$; so it will end up being an incomplete twirl.

While the choice of modeling errors as occurring before, after, or during $\mathcal{G}$ does not matter for randomized compiling, it does have some interesting physical implications when one wants to model the error generator of a process (see, for example, Section II in https://arxiv.org/abs/2103.01928).

NegativeEntropyDemon
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