IHGB

Optimal tomography of bosonic and fermionic Gaussian states

Senrui Chen, Marco Fanizza, FG, Ludovico Lami, Francesco Anna Mele, Michael Walter, Freek Witteveen
arXiv:2607.11847

The sample complexity is the minimum number of copies required to learn an accurate classical description of a quantum state. Bosonic and fermionic Gaussian quantum states are families of quantum states that play a key role in quantum science and technology, from quantum optics and many-body physics to quantum chemistry, quantum computing, and quantum information theory. Despite their importance, their sample complexity had not been fully determined. We settle this open problem and show that both bosonic and fermionic Gaussian states can be learned using a number of copies that scales quadratically in the number of modes, regardless of whether the state is pure or mixed, and independently of any energy bound on the state. We derive these results by using the representation theory of Gaussian unitaries and by putting forth a generalization of the random purification channel to this setting and beyond.

FOCS26

Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition

Kean Chen, FG, Aadil Oufkir, Nengkun Yu, Zhicheng Zhang
arXiv:2604.2604.17369

How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood.
In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, to within error $\varepsilon$. We identify the dilation rate $\tau = r d_2 / d_1$ (which always satisfies $\tau\geq 1$ due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of $\tau$. • In the boundary regime ($\tau = 1$): we show that the query complexity is $\Theta(r d_1 d_2/\varepsilon)$ for Choi trace norm error $\varepsilon$, and is upper bounded by $O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})$ and lower bounded by $\Omega(r d_1 d_2/\varepsilon)$ for diamond norm error $\varepsilon$. • In the away-from-boundary regime ($\tau \geq 1+\Omega(1)$): we show that the query complexity is $\Theta(r d_1 d_2/\varepsilon^2)$ for both Choi trace norm and diamond norm errors $\varepsilon$. Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at $\tau=1$, the optimal query complexity exhibits Heisenberg scaling $1/\varepsilon$, whereas for $\tau\geq 1+\Omega(1)$, it exhibits classical scaling $1/\varepsilon^2$. In addition, we show that in the near-boundary regime ($1< \tau < 1+o(1)$), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.

Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

Aadil Oufkir, FG
arXiv:2601.04180

We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\varepsilon$ requires $\Omega\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A= rd_B$, and $\Omega\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$. These lower bounds improve upon the best previous $\Omega(d_A d_B r)$ bound by introducing explicit $\varepsilon$-dependence, and they are optimal up to logarithmic factors. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

Random Stinespring superchannel: converting channel queries into dilation isometry queries

FG$^\ast$, Francesco Anna Mele$^\ast$, Haimeng Zhao$^\ast$, Marco Fanizza$^\dagger$, Ludovico Lami$^\dagger$
arXiv:2512.20599

The recently introduced random purification channel, which converts $n$ copies of an arbitrary mixed quantum state into $n$ copies of the same uniformly random purification, has emerged as a powerful tool in quantum information theory. Motivated by this development, we introduce a channel-level analogue, which we call the random Stinespring superchannel. This consists in a procedure to transform $n$ parallel queries of an arbitrary quantum channel into $n$ parallel queries of the same uniformly random Stinespring isometry, via universal encoding and decoding operations that are efficiently implementable. When the channel is promised to have Choi rank at most $r$, the procedure can be tailored to yield a Stinespring environment of dimension $r$. We present two applications of the random Stinespring superchannel, one in quantum Shannon theory and one in quantum learning theory. In quantum Shannon theory, we prove a channel-level analogue of Uhlmann's theorem for quantum divergences. In quantum learning theory, our construction shows that tomography of quantum channels reduces to tomography of isometries. This yields a simple channel learning algorithm, based on existing isometry learning protocols, that matches the performance of the two recently proposed channel tomography algorithms. Complementarily, whereas the optimality of these algorithms had previously been established only up to a logarithmic factor in the dimension, we close this gap by removing this logarithmic factor from the lower bound. Taken together, our results fully establish the optimality of these recently introduced channel learning algorithms, showing that the optimal query complexity of learning a quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ is $\Theta(d_A d_B r)$.

QIP26 QCTiP26

Is it Gaussian? Testing bosonic quantum states

FG$^\ast$, Freek Witteveen$^\ast$, Francesco Anna Mele, Lennart Bittel, Salvatore F. E. Oliviero, David Gross, Michael Walter
arXiv:2510.07305

Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

Quantitative convergence of trained quantum neural networks to a Gaussian process

Anderson Melchor Hernandez, FG, Davide Pastorello, Giacomo De Palma
Ann. Henri Poincaré (2025)

We study quantum neural networks where the generated function is the expectation value of the sum of single-qubit observables across all qubits. In [Girardi et al., arXiv:2402.08726], it is proven that the probability distributions of such generated functions converge in distribution to a Gaussian process in the limit of infinite width for both untrained networks with randomly initialized parameters and trained networks. In this paper, we provide a quantitative proof of this convergence in terms of the Wasserstein distance of order $1$. First, we establish an upper bound on the distance between the probability distribution of the function generated by any untrained network with finite width and the Gaussian process with the same covariance. This proof utilizes Stein's method to estimate the Wasserstein distance of order $1$. Next, we analyze the training dynamics of the network via gradient flow, proving an upper bound on the distance between the probability distribution of the function generated by the trained network and the corresponding Gaussian process. This proof is based on a quantitative upper bound on the maximum variation of a parameter during training. This bound implies that for sufficiently large widths, training occurs in the lazy regime, i.e., each parameter changes only by a small amount. While the convergence result of [Girardi et al., arXiv:2402.08726] holds at a fixed training time, our upper bounds are uniform in time and hold even as $t\to\infty$.

TQC24 ICMP24

Trained quantum neural networks are Gaussian processes

FG, Giacomo De Palma
Communications in Mathematical Physics 406 (4), 1-146 (2025)

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.