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arXiv:2108.08488 · Physical Review A

Quantum advantages for Pauli channel estimation

Senrui Chen, Sisi Zhou, Alireza Seif, Liang Jiang

Published Mar 22, 2022• 57 citations
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Abstract

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.

DOI
10.1103/physreva.105.032435

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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.
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 nnn-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∗ρ\otimesρ^*ρ⊗ρ∗. These results identify violation of the reduction criterion as a necessary condition for an exponential advantage in the learning tasks considered here.
Quantum resources enable us to achieve an exponential advantage in learning the properties of unknown physical systems by employing quantum memory. While entanglement with quantum memory is recognized as a necessary qualitative resource, its quantitative role remains less understood. In this work, we distinguish between two fundamental resources provided by quantum memory—entanglement and ancilla qubits—and analyze their separate contributions to the sampling complexity of quantum learning. Focusing on the task of Pauli channel learning, a prototypical example of quantum channel learning, remarkably, we prove that vanishingly small entanglement in the input state already suffices to accomplish the learning task with only a polynomial number of channel queries in the number of qubits. In contrast, we show that without a sufficient number of ancilla qubits, even learning partial information about the channel demands an exponentially large sample complexity. Thus, our findings reveal that while a large amount of entanglement is not necessary, the dimension of the quantum memory is a crucial resource. Hence, by identifying how the two resources contribute differently, our work offers deeper insight into the nature of the quantum learning advantage. While it is known that entanglement plays a role in learning the properties of unknown physical systems, the detailed quantitative dependence is unclear. Here, the authors investigate the fundamental resources behind quantum advantage in channel learning, revealing that while extensive entanglement is unnecessary, the number of ancilla qubits is crucial.
Quantum channels can be characterized by their action on an orthogonal operator basis, where these operators are related to observable properties of the quantum system. For qudit and multimode bosonic systems, this is encoded respectively in the Heisenberg--Weyl transfer matrix estimated from the Choi-state, and in the characteristic-function transfer map estimated from a two-mode squeezed vacuum based Choi-state. We derive sample-complexity bounds for estimating entries of the transfer matrix/map to additive accuracy εεε with success probability ≥1−δ\ge1-δ≥1−δ, under different resources: access to the complex-conjugate channel E∗\mathcal{E}^*E∗ and/or parallel access to ccc copies. In all settings, the learner uses parallel channel calls with adaptively chosen, ancilla-assisted input states and measurements. Absolute values of transfer-matrix entries can be learned efficiently with simultaneous access to E\mathcal{E}E and E∗\mathcal{E}^*E∗, with tight scaling ε−4ε^{-4}ε−4. Without conjugate access, any c<dc<dc<d copies are insufficient for efficient learning, requiring sample complexity exponential in the number of (ddd-level) qudits nnn (for prime ddd). Efficiency is recovered at c=dc=dc=d, with tight scaling ε−2dε^{-2d}ε−2d. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all c=O(1/ε)c=O(1/ε)c=O(1/ε). Although the task is learning a particular state, these bounds carry stronger implications than standard state-learning bounds since the learner controls the inputs and has ancillary assistance. This establishes a hierarchy of channel-learning resources: self-complex-conjugate channels require two-copy ancilla-assisted access for efficient learning, while for every square-free ddd, some channels require ddd-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.
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.