Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this Letter, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that Θ(2^{n}ϵ^{-2}) rounds of measurements are required to estimate each eigenvalue of an n-qubit Pauli channel to ϵ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs Θ(ϵ^{-2}) copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.
Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this Letter, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that Θ(2^{n}ϵ^{-2}) rounds of measurements are required to estimate each eigenvalue of an n-qubit Pauli channel to ϵ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs Θ(ϵ^{-2}) copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.
We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an n-qubit Pauli channel to $\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}$ precision. We give an estimation protocol with an n-qubit ancilla that succeeds with high probability using only $O(n/{\ensuremath{\varepsilon}}^{2})$ copies of the Pauli channel, while proving that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least $\mathrm{\ensuremath{\Omega}}({2}^{n/3})$ rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a k-qubit ancilla ($k\ensuremath{\le}n$) is available, we obtain a sample complexity lower bound of $\mathrm{\ensuremath{\Omega}}({2}^{(n\ensuremath{-}k)/3})$ for any nonconcatenating protocol, and a stronger lower bound of $\mathrm{\ensuremath{\Omega}}(n{2}^{n\ensuremath{-}k})$ for any nonadaptive, nonconcatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically interesting example for quantum advantages in learning and also bring insights for quantum benchmarking.
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(23nε−2) to learn an n-qubit Pauli channel. In particular, this shows that the recently introduced learning procedure by [1] is essentially optimal. In the adaptive setting, we show a lower bound of Ω(22.5nε−2) for ε = O(2−n), and a lower bound of Ω(22nε−2) for any ε > 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.
We prove that adaptive strategies offer no advantage over non-adaptive ones for learning and testing Pauli channels using entangled inputs. This key observation allows us to characterize the query complexity for several fundamental tasks by translating optimal classical estimation algorithms into the quantum setting. First, we determine the tight query complexity for learning a Pauli channel under the general ℓp norm, providing results that improve upon or match the best-known bounds for the ℓ1,ℓ2, and ℓ∞ distances. Second, we resolve the complexity of testing whether a Pauli channel is a white noise source. Finally, we show that the optimal query complexities for estimating the Shannon entropy and support size of the channel's error distribution, and for estimating the diamond distance between two Pauli channels, are all Θ(4nnϵ2) .
While entanglement is known to enable exponential improvements in the sample complexity of quantum learning, it remains unclear which properties of entangled resources are responsible for such improvements. We address this question through the reduction criterion, a condition obeyed by all bound-entangled states. In n-qubit Pauli-channel learning, we show that restricting either the input states or the measurement effects to satisfy this criterion rules out an exponential advantage for incoherent adaptive protocols. An exponential lower bound persists for the one-sided coherent adaptive protocols considered here, even when the unrestricted side retains quantum correlations across channel uses. Using conditional min-entropy, we further quantify how the sample-complexity lower bounds weaken as larger violations of the reduction criterion are allowed. Finally, we show that the same obstruction appears in conjugate-state learning: restricted joint measurements cannot reproduce the logarithmic-sample advantage of unrestricted joint measurements on ρ⊗ρ∗. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here, we prove several results on efficiently learning Pauli channels and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits with high probability to a relative precision ϵ using O (ϵ -2 n2 n ) measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors, which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next, we show that the error rates for an arbitrary set of s Pauli errors can be estimated to a relative precision ϵ using O (ϵ -4 log s log s/ϵ) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k -local correlations, we can learn an entire n -qubit Pauli channel to relative precision ϵ with only O k (ϵ -2 n 2 log n ) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.
In this work, we consider one of the prototypical tasks for characterizing the structure of noise in quantum devices: estimating eigenvalues of an n -qubit Pauli noise channel. Prior work [Chen , Phys. Rev. A 105, 032435 (2022)] has proved no-go theorems for this task in the practical regime in which one has a limited amount of quantum memory; i.e., any protocol with ≤ 0.99 n ancilla qubits of quantum memory must make exponentially many measurements, provided that it is . Such protocols can only interact with the channel by repeatedly preparing a state, passing it through the channel, and measuring immediately afterward. Surprisingly, in this work we show that protocols with an extremely small amount of quantum memory achieve an exponential advantage for this task. We give a protocol that can estimate any prescribed set of eigenvalues A of a Pauli channel to error ε using only O ( log log ( | A | ) / ε 2 ) ancilla qubits, O ~ ( n 2 / ε 2 ) measurements, and O ( n 2 | A | / ε 2 ) queries to the Pauli channel. In contrast, we show that any protocol with zero ancilla—even a concatenating one—must make Ω ( 2 n / ε 2 ) measurements. We also prove that the number of queries cannot be improved significantly: with k ancilla qubits, Ω ( 2 ( n − k ) / 3 ) queries to the channel are necessary to learn all eigenvalues, even for concatenating protocols.