This work proves convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels and proposes an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure.
Abstract
Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.
Quantum state ensembles are important in quantum information processing. For example, quantum $t$-designs model highly entangled states in complex systems, while projected ensembles appear in generative quantum machine learning and studies of thermalization. With their sample state accompanied by a classical label, these ensembles contain operational information beyond their average density operators. Yet an ensemble differs from a classical-quantum state because it is invariant under permutations of labels. We formulate binary hypothesis testing between finite quantum ensembles and derive fundamental limits on error probability. Given an observed label pattern, we show that the joint sampled state can be described by power-weighted ensemble moments. This yields the Bayes-optimal measurement and exact finite-sample error, revealing that discrimination is governed by the full moment hierarchy up to the number of samples. In the many-sample limit, we derive Chernoff bounds and obtain exact error exponents for finite uniform pure-state ensembles. We apply these results to optical communication and $t$-designs. For finite uniform pure-state $t$-designs with large $t$, the maximal discrimination exponent scales sharply as $\sim t^{-2}$, while equal-prior fixed-error testing requires $\sim t^2$ samples.
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 ancillary assistance, and 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 $d$, some channels require $d-copy access.
M. Subramanian, Hyukgun Kwon, Liang Jiang· 0 citations
This work establishes canonical quantization as a principled framework for constructing quantum machine learning primitives and provides a foundation for developing neural architectures tailored to quantum data.
A protocol for approximating the measurement distributions of quantum states, extending beyond standard observable estimation is introduced, and tightened gate complexity bounds for practically relevant systems, including those with k-local interactions, long-tailed matrix ensembles, and conserved quantities are provided.
A. Mazumder, James D. Watson, Samson Wang· 0 citations
Phenomena that slip beyond the grasp of our classical intuition reveal uniquely quantum effects that deepen our understanding of the physical world and enable advances in information processing, particularly in quantum communication and computation. One such phenomenon is quantum steering, whereby measurements performed by one party influence the conditional states of another when the two share an entangled quantum system. If the observed correlations cannot be explained by a local hidden state model, the state is said to be steerable. In this work, we investigate the typicality of this behavior: given a generic two-qubit state and $m$ Haar-random projective measurements, what is the probability of observing steering? We derive analytical expressions for the steering probability $\mathcal{P}_S$ of Werner states in two- and three-setting scenarios, the latter restricted to coplanar projective measurements on the Bloch sphere. For larger numbers of settings and various random states ensembles, we perform numerical analyses showing that $\mathcal{P}_S$ increases systematically with the number of measurements and substantially exceeds the corresponding probabilities associated with Bell nonlocality. Our results demonstrate that random states with minimal environmental coupling exhibit a high probability of steering for finite $m$ and approach genuine typicality, $\mathcal{P}_S=100\%$, as the number of settings increases. We provide a detailed characterization of $\mathcal{P}_S$ across different state ensembles and specific families, including Bell-diagonal and Werner states, identifying those with the greatest non-classical potential and highlighting their relevance for protocols in which steering serves as a key resource.
Gerard Anglès Munné, Paweł Cieśliński, Tamás Vértesi et al.· 0 citations