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A symmetric ideal is an ideal in a polynomial ring which is stable under all permutations of the variables. In this paper we initiate a global study of zero-dimensional symmetric ideals. By this we mean a geometric study of the invariant Hilbert schemes $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$ parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $\rho$. In the case that $\rho = M^\lambda$ is a permutation module corresponding to certain special types of partitions $\lambda$ of $n$, we prove that $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim \rho \leq 2n$ and the invariant Hilbert scheme is non-empty. In this same range, we classify all homogeneous symmetric ideals and decide which of these define singular points of $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial theory of higher Specht polynomials., Comment: Comments welcome!

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $\rho$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $\rho$ at least as well as any state in $\mathcal{C}$ (up to some small error $\varepsilon$). This task generalizes ordinary quantum tomography of states in $\mathcal{C}$ and is more challenging because the learning algorithm must be robust to perturbations of $\rho$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $\rho$ has fidelity at least $\tau$ with a stabilizer product state, the algorithm runs in time $n^{O(1 + \log(1/\tau))} / \varepsilon^2$. This runtime is quasipolynomial in all parameters, and polynomial if $\tau$ is a constant., Comment: 20 pages

We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $\Omega(t)$ non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $\frac{t}{2}$ bits of the true entanglement entropy., Comment: 15 pages; v2: slight edits to concurrent work section

Weather regimes are recurrent and persistent large-scale atmospheric circulation patterns that modulate the occurrence of local impact variables such as extreme precipitation. In their capacity as mediators between long-range teleconnections and these local extremes, they have shown potential for improving sub-seasonal forecasting as well as long-term climate projections. However, existing methods for identifying weather regimes are not designed to capture the physical processes relevant to the impact variable in question while still representing the full atmospheric phase space. This paper introduces a novel probabilistic machine learning method, RMM-VAE, for identifying weather regimes targeted to a local-scale impact variable. Based on a variational autoencoder architecture, the method combines non-linear dimensionality reduction with a prediction task and probabilistic clustering in a coherent architecture. The new method is applied to identify circulation patterns over the Mediterranean region targeted to precipitation over Morocco and compared to three existing approaches, two established linear methods and another machine learning approach. The RMM-VAE method identifies regimes that are more predictive of the target variable compared to the two linear methods, and more robust and persistent compared to the alternative machine learning method, while also improving the reconstruction of the input space. The results demonstrate the potential benefit of the new method for use in various climate applications such as sub-seasonal forecasting, while also highlighting the trade-offs involved in targeted clustering.

Scientific and technological advances in numerical modelling have improved the quality of climate predictions over recent decades, but predictive skill remains limited in many aspects. Extreme events such as heat and cold waves, droughts, heavy rain and storms are particularly challenging to predict accurately due to their rarity and non-linear chaotic nature, and because of model limitations. However, recent studies have shown that predictive skill of extremes can be increased when using more sophisticated approaches, indicating that there might be systemic predictability that is not being leveraged. Recently, numerous studies have been devoted to the exploitation of Artificial Intelligence (AI) to study the predictability and make predictions of weather and climate. AI techniques have shown great potential to improve the prediction of extreme events and uncover their links to large-scale and local drivers. Machine and deep learning, causal discovery, explainable AI, are only some of the approaches that have been tested to both improve our understanding of the processes underlying predictability and enhance prediction skill of extreme events. Results are promising, especially for hybrid predictions that combine the AI, which can reveal and exploit unknown spatio-temporal connections from data, and climate models, that provide the theoretical foundation and interpretability of the physical world. On the other hand, challenges are multiple in many aspects, from data curation to model uncertainty and generalizability, to the reproducibility of methods and workflows. A few best practices are identified to increase trust in these novel techniques, and future perspectives are envisaged for further scientific development., Comment: 20 pages, 4 figures, 2 tables

Recent work has shown that $n$-qubit quantum states output by circuits with at most $t$ single-qubit non-Clifford gates can be learned to trace distance $\epsilon$ using $\mathsf{poly}(n,2^t,1/\epsilon)$ time and samples. All prior algorithms achieving this runtime use entangled measurements across two copies of the input state. In this work, we give a similarly efficient algorithm that learns the same class of states using only single-copy measurements., Comment: This work has been merged into arXiv:2305.13409

We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|\psi\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|\psi\rangle$ to learn $|\psi\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|\psi\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest., Comment: 54 pages. Merged v3 with arXiv:2308.07175. This version now subsumes arXiv:2308.07175

We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $\Omega(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, we show that a modification of the above algorithm runs in polynomial time. - We exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory., Comment: 41 pages, 2 figures. v3: changed presentation of tolerant testing algorithm and other minor edits

The paper explores the optical properties of an exfoliated MoSe$_2$ monolayer implanted with Cr$^+$ ions, accelerated to 25 eV. Photoluminescence of the implanted MoSe$_2$ reveals an emission line from Cr-related defects that is present only under weak electron doping. Unlike band-to-band transition, the Cr-introduced emission is characterised by non-zero activation energy, long lifetimes, and weak response to the magnetic field. To rationalise the experimental results and get insights into the atomic structure of the defects, we modelled the Cr-ion irradiation process using ab-initio molecular dynamics simulations followed by the electronic structure calculations of the system with defects. The experimental and theoretical results suggest that the recombination of electrons on the acceptors, which could be introduced by the Cr implantation-induced defects, with the valence band holes is the most likely origin of the low energy emission. Our results demonstrate the potential of low-energy ion implantation as a tool to tailor the properties of 2D materials by doping.

Explainable artificial intelligence (XAI) methods shed light on the predictions of machine learning algorithms. Several different approaches exist and have already been applied in climate science. However, usually missing ground truth explanations complicate their evaluation and comparison, subsequently impeding the choice of the XAI method. Therefore, in this work, we introduce XAI evaluation in the climate context and discuss different desired explanation properties, namely robustness, faithfulness, randomization, complexity, and localization. To this end, we chose previous work as a case study where the decade of annual-mean temperature maps is predicted. After training both a multi-layer perceptron (MLP) and a convolutional neural network (CNN), multiple XAI methods are applied and their skill scores in reference to a random uniform explanation are calculated for each property. Independent of the network, we find that XAI methods Integrated Gradients, layer-wise relevance propagation, and input times gradients exhibit considerable robustness, faithfulness, and complexity while sacrificing randomization performance. Sensitivity methods -- gradient, SmoothGrad, NoiseGrad, and FusionGrad, match the robustness skill but sacrifice faithfulness and complexity for randomization skill. We find architecture-dependent performance differences regarding robustness, complexity and localization skills of different XAI methods, highlighting the necessity for research task-specific evaluation. Overall, our work offers an overview of different evaluation properties in the climate science context and shows how to compare and benchmark different explanation methods, assessing their suitability based on strengths and weaknesses, for the specific research problem at hand. By that, we aim to support climate researchers in the selection of a suitable XAI method., Comment: 19 pages, 10 figure, accepted at AIES journal by AMS