Lower Bounds on Learning Pauli Channels with Individual Measurements

TL;DR

The paper establishes fundamental limits on learning n-qubit Pauli channels with ancilla-free, individual measurements, quantified in the diamond norm. It develops two main information-theoretic lower-bound frameworks: a non-adaptive bound N ≥ Ω(d^3/ε^2) and an adaptive bound N ≥ Ω(d^{2.5}/ε^2) in the high-precision regime, with a broader Ω(d^2/ε^2) baseline that holds generally. The results show that the popular non-adaptive Pauli-tomography procedure by Flammia and Wallman is essentially optimal up to logarithmic factors, and they reveal a substantial separation between adaptive and non-adaptive strategies in this constrained setting. Together, these bounds clarify the resource costs for ancilla-free Pauli-channel tomography and guide design of practical quantum-noise learning protocols.

Abstract

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.

Figures (4)

• Figure 1: Illustration of a strategy with individual measurements. The estimated channel $\widetilde{\mathcal{P}}$ is computed from $I_1, \dots, I_N$.
• Figure 2: Illustration of an adaptive strategy for learning Pauli channel.
• Figure 3: Illustration of a non-adaptive strategy for learning Pauli channel.
• Figure 4: Illustration of an adaptive strategy for learning Pauli channel using one channel per step.

Theorems & Definitions (37)

• Definition 3.1
• Theorem 3.2
• proof
• Lemma 3.3
• Lemma 3.4: fano1961transmission
• Lemma 3.5
• proof : Proof of Lemma \ref{['Upper bound on cond mutual info']}
• Lemma 3.6
• proof : Proof of Lemma \ref{['lem']}
• Lemma 3.7