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arXiv:2511.13683

Quantum Advantage in Learning Mixed Unitary Channels

Yue Tu, Liang Jiang

quant-phmath-phSubmitted Nov 17, 2025• Announced Nov 18, 2025• 0 citations
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Abstract

We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as $\frac{r}{d\varepsilon^2}$, where $r$ is the rank of the channel (i.e.\ the number of different unitaries), $d$ is the dimension of the system, and $\varepsilon^2$ is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that $r$ is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.

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We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as rdε2\frac{r}{d\varepsilon^2}dε2r​, where rrr is the rank of the channel (i.e.\ the number of different unitaries), ddd is the dimension of the system, and ε2\varepsilon^2ε2 is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that rrr is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.
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.
We study the role of ancillary entanglement in quantum channel parameter estimation and investigate the minimal ancilla dimension required to achieve the maximum Fisher information. We introduce the kkk-ancilla Fisher information, which quantifies the optimal estimation precision achievable with input states of rank at most kkk, and derive a variational characterization in terms of a rank-constrained optimization problem. This formulation leads to a simple characterization of the minimum ancilla dimension k∗k^*k∗ required for optimal estimation, given by the minimum rank among the maximizers of the associated variational problem. We further identify sufficient conditions under which ancillary entanglement provides no advantage, including channel families admitting a fixed measure-and-prepare representation and channels satisfying a natural horizontality condition. In addition, we derive a bound on the incremental gain in Fisher information obtained by increasing the ancilla dimension under suitable structural assumptions on the optimal input states. The general results are illustrated through explicit examples, including unitary channels, qubit depolarizing and amplitude damping channels. These results provide a systematic framework for understanding and quantifying the entanglement resources required for optimal quantum channel estimation.
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.
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 nnn-qubit Pauli channel to $\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}$ precision. We give an estimation protocol with an nnn-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 kkk-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.
Quantum resources can dramatically reduce the data required to learn physical systems, with exponential improvements in sample complexity demonstrated in several quantum learning tasks. However, these advantages have typically relied on quantum resources that scale with problem complexity, most notably increasing system dimension or entanglement. This raises a fundamental question: can exponential quantum learning advantages arise within a fixed, unentangled quantum system? In this paper, we address this question by considering the learning of an unknown random-displacement distribution whose complexity is determined not by the dimensionality of the physical system, but by the Fourier resolution of its features. We show that the quantum-limited noise of vacuum probes progressively obscures high-frequency features, leading to an exponential growth in sample complexity. We establish an information-theoretic lower bound for arbitrary classical-state probes and show that squeezing overcomes this classical limit by extending the accessible Fourier bandwidth. Experimentally, we demonstrate an exponential reduction in sample complexity using squeezed vacuum probes for both binary hypothesis testing and characteristic-function reconstruction. Our results show that a single bosonic mode can exhibit exponential quantum learning advantages without entanglement or an increase in system size, identifying accessible Fourier bandwidth as a fundamentally distinct resource for quantum-enhanced learning.
We present a learning task for quantum channels that exhibits an exponential separation in query complexity (in terms of the number of qubits) between two access models: i) black-box channel access and ii) access to a Stinespring dilation unitary of the channel along with its inverse. We consider the task of learning how close the unknown channel is to the set of unitary channels. This is characterized by the \textit{optimal average gate fidelity (OAGF)}, defined as the average gate fidelity maximized over all unitary channels. For constant Kraus rank and a dilation environment of dimension polylog⁡(d)\operatorname{polylog}(d)polylog(d), we provide an algorithm that estimates the OAGF to fixed additive accuracy using polylog⁡(d)\operatorname{polylog}(d)polylog(d) queries to the Stinespring dilation unitary and its inverse. In contradistinction, under the black-box access model, this estimation requires at least Ω(d)Ω(\sqrt{d})Ω(d​) queries, even for channels of constant Kraus rank and constant additive accuracy. As a corollary, we rule out a universal simulator that uses only poly⁡(log⁡d,q)\operatorname{poly}(\log d,q)poly(logd,q) queries to the unknown channel to approximate the output of arbitrary qqq-query algorithms with access to a randomly-sampled Stinespring dilation unitary and its inverse, even to sufficiently small constant error. These results demonstrate that access to the system-environment joint evolution and its inverse can provide an exponential advantage in estimating an intrinsic property of a quantum channel.