Lower Bounds on Learning Pauli Channels with Individual Measurements
This work addresses the challenge of characterizing quantum noise for scaling quantum technologies, providing theoretical limits that are incremental but crucial for optimizing learning procedures.
The paper tackles the problem of learning Pauli channels, a key noise model in quantum technologies, by establishing fundamental lower bounds on sample complexity in diamond norm, showing optimality of recent methods and providing bounds like Ω(2^{3n}ε^{-2}) for non-adaptive algorithms.
Understanding the noise affecting a quantum device is of fundamental importance for scaling quantum technologies. A particularly important class of noise models is that of Pauli channels, as randomized compiling techniques can effectively bring any quantum channel to this form and are significantly more structured than general quantum channels. In this paper, we show fundamental lower bounds on the sample complexity for learning Pauli channels in diamond norm. We consider strategies that may not use auxiliary systems entangled with the input to the unknown channel and have to perform a measurement before reusing the channel. For non-adaptive algorithms, we show a lower bound of $Ω(2^{3n}\varepsilon^{-2})$ to learn an $n$-qubit Pauli channel. In particular, this shows that the recently introduced learning procedure by Flammia and Wallman is essentially optimal. In the adaptive setting, we show a lower bound of $Ω(2^{2.5n}\varepsilon^{-2})$ for $\varepsilon=\mathcal{O}(2^{-n})$, and a lower bound of $Ω(2^{2n}\varepsilon^{-2} )$ for any $\varepsilon> 0$. This last lower bound holds even in a stronger model where in each step, before performing the measurement, the unknown channel may be used arbitrarily many times sequentially interspersed with unital operations.