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The multiple scattering theory (MST) is a Green's function method that has been widely used in electronic structure calculations for crystalline disordered systems. The key property of the MST method is the scattering path matrix (SPM) that characterizes the Green's function within a local solution representation. This paper studies various approximations of the SPM, under the condition that an appropriate reference is used for perturbation. In particular, we justify the convergence of the SPM approximations with respect to the size of scattering region and the length of scattering path, which are the central numerical parameters to achieve a linear-scaling MST method. We present numerical experiments on several typical systems to support the theory.

Several quantum algorithms for linear algebra problems, and in particular quantum machine learning problems, have been "dequantized" in the past few years. These dequantization results typically hold when classical algorithms can access the data via length-squared sampling. In this work we investigate how robust these dequantization results are. We introduce the notion of approximate length-squared sampling, where classical algorithms are only able to sample from a distribution close to the ideal distribution in total variation distance. While quantum algorithms are natively robust against small perturbations, current techniques in dequantization are not. Our main technical contribution is showing how many techniques from randomized linear algebra can be adapted to work under this weaker assumption as well. We then use these techniques to show that the recent low-rank dequantization framework by Chia, Gily\'en, Li, Lin, Tang and Wang (JACM 2022) and the dequantization framework for sparse matrices by Gharibian and Le Gall (STOC 2022), which are both based on the Quantum Singular Value Transformation, can be generalized to the case of approximate length-squared sampling access to the input. We also apply these results to obtain a robust dequantization of many quantum machine learning algorithms, including quantum algorithms for recommendation systems, supervised clustering and low-rank matrix inversion.

Kernel methods map data into high-dimensional spaces, enabling linear algorithms to learn nonlinear functions without explicitly storing the feature vectors. Quantum kernel methods promise efficient learning by encoding feature maps into exponentially large Hilbert spaces inherent in quantum systems. In this work we implement quantum kernels on a 10-qubit star-topology register in a nuclear magnetic resonance (NMR) platform. We experimentally encode classical data in the evolution of multiple quantum coherence orders using data-dependent unitary transformations and then demonstrate one-dimensional regression and two-dimensional classification tasks. By extending the register to a double-layered star configuration, we propose an extended quantum kernel to handle non-parametrized operator inputs. By numerically simulating the extended quantum kernel, we show classification of entangling and nonentangling unitaries. These results confirm that quantum kernels exhibit strong capabilities in classical as well as quantum machine learning tasks.

We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + \eta_t$ where $\eta_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tilde{\Omega}(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tilde{\Omega}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

We discuss a nondeterministic variant of the recently introduced machine model of deterministic auxiliary depth-$k$ storage automata (or aux-$k$-sda's) by Yamakami. It was proven that all languages recognized by polynomial-time logarithmic-space aux-$k$-sda's are located between $\mathrm{LOGDCFL}$ and $\mathrm{SC}^k$ (the $k$th level of Steve's class SC). We further propose a new and simple computational model of semi-unbounded fan-in Boolean circuits composed partly of cascading blocks, in which the first few AND gates of unbounded fan-out (called AND$_{(\omega)}$ gates) at each layer from the left (where all gates at each layer are indexed from left to right) are linked in a "cascading" manner to their right neighbors though specific AND and OR gates. We use this new circuit model to characterize a nondeterministic variant of the aux-$2k$-sda's (called aux-$2k$-sna's) that run in polynomial time using logarithmic work space. By relaxing the requirement for cascading circuits, we also demonstrate how such cascading circuit families characterize the complexity class $\mathrm{P}$. This yields an upper bound on the computational complexity of $\mathrm{LOG}k\mathrm{SNA}$ by $\mathrm{P}$.

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $\delta = \delta(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{\delta \cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

Martin-L\"{o}f type theory $\mathbf{MLTT}$ was extended by Setzer with the so-called Mahlo universe types. The extension of $\mathbf{MLTT}$ with one Mahlo universe is called $\mathbf{MLM}$ and was introduced to develop a variant of $\mathbf{MLTT}$ equipped with an analogue of a large cardinal. Another instance of constructive systems extended with an analogue of a large set was formulated in the context of Aczel's constructive set theory: $\mathbf{CZF}$. Rathjen, Griffor and Palmgren extended $\mathbf{CZF}$ with inaccessible sets of all transfinite orders. While Rathjen proved that this extended system of $\mathbf{CZF}$ is interpretable in an extension of $\mathbf{MLM}$ with one usual universe type above the Mahlo universe, it is unknown whether it can be interpreted by the Mahlo universe without a universe type above it. We extend $\mathbf{MLM}$ not by a universe type but by the accessibility predicate, and show that $\mathbf{CZF}$ with inaccessible sets can be interpreted in $\mathbf{MLM}$ with the accessibility predicate. Our interpretation of this extension of $\mathbf{CZF}$ is the same as that of Rathjen, Griffor and Palmgren formulated by $\mathbf{MLTT}$ with second-order universe operators, except that we construct the inaccessible sets by using the Mahlo universe and the accessibility predicate. We formalised the main part of our interpretation in the proof assistant Agda.

We present MathDSL, a Domain-Specific Language (DSL) for mathematical equation solving, which, when deployed in program synthesis models, outperforms state-of-the-art reinforcement-learning-based methods. We also introduce a quantitative metric for measuring the conciseness of a mathematical solution and demonstrate the improvement in the quality of generated solutions compared to other methods. Our system demonstrates that a program synthesis system (DreamCoder) using MathDSL can generate programs that solve linear equations with greater accuracy and conciseness than using reinforcement learning systems. Additionally, we demonstrate that if we use the action spaces of previous reinforcement learning systems as DSLs, MathDSL outperforms the action-space-DSLs. We use DreamCoder to store equation-solving strategies as learned abstractions in its program library and demonstrate that by using MathDSL, these can be converted into human-interpretable solution strategies that could have applications in mathematical education.

The ideal estimand for comparing a new treatment $Rx$ with a control $C$ is the $\textit{counterfactual}$ efficacy $Rx:C$, the expected differential outcome between $Rx$ and $C$ if each patient were given $\textit{both}$. While counterfactual $\textit{point estimation}$ from $\textit{factual}$ Randomized Controlled Trials (RCTs) has been available, this article shows $\textit{counterfactual}$ uncertainty quantification (CUQ), quantifying uncertainty for factual point estimates but in a counterfactual setting, is surprisingly achievable. We achieve CUQ whose variability is typically smaller than factual UQ, by creating a new statistical modeling principle called ETZ which is applicable to RCTs with $\textit{Before-and-After}$ treatment Repeated Measures, common in many therapeutic areas. We urge caution when estimate of the unobservable true condition of a patient before treatment has measurement error, because that violation of standard regression assumption can cause attenuation in estimating treatment effects. Fortunately, we prove that, for traditional medicine in general, and for targeted therapy with efficacy defined as averaged over the population, counterfactual point estimation is unbiased. However, for targeted therapy, both Real Human and Digital Twins approaches should respect this limitation, lest predicted treatment effect in $\textit{subgroups}$ will have bias.

We introduce a generic framework that reduces the computational cost of object detection while retaining accuracy for scenarios where objects with varied sizes appear in high resolution images. Detection progresses in a coarse-to-fine manner, first on a down-sampled version of the image and then on a sequence of higher resolution regions identified as likely to improve the detection accuracy. Built upon reinforcement learning, our approach consists of a model (R-net) that uses coarse detection results to predict the potential accuracy gain for analyzing a region at a higher resolution and another model (Q-net) that sequentially selects regions to zoom in. Experiments on the Caltech Pedestrians dataset show that our approach reduces the number of processed pixels by over 50% without a drop in detection accuracy. The merits of our approach become more significant on a high resolution test set collected from YFCC100M dataset, where our approach maintains high detection performance while reducing the number of processed pixels by about 70% and the detection time by over 50%.