Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today’s quantum devices.

A fundamental issue for noise characterization is that not everything about quantum noise is learnable due to the existence of gauge freedom.

The question of what information is self-consistently learnable has been unclear even for a single CNOT gate. Here, we give a precise characterization of the learnability of Pauli noise associated with Clifford gates using graph theoretical tools, showing the learnable information corresponds exactly to the cycle space of the pattern transfer graph of the gate set. Our results reveal the optimality of cycle benchmarking in the sense that it can extract all learnable information about Pauli noise. We experimentally demonstrate Pauli noise characterization of IBM’s CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we show that attempts to extract unlearnable information by ignoring state preparation noise yield unphysical estimates, which can be used to lower bound the state preparation noise independently of the measurement.

[Preprint: https://arxiv.org/abs/2206.06362]

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