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We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse $n$-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with $k$-th degree polynomial vector fields for arbitrary (finite) values of $k$. The steps involve: 1) mapping the $k$-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the $k$-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on $k$ and demonstrate the framework on an example.

Since the 1990's, many observed cognitive behaviors have been shown to violate rules based on classical probability and set theory. For example, the order in which questions are posed affects whether participants answer 'yes' or 'no', so the population that answers 'yes' to both questions cannot be modeled as the intersection of two fixed sets. It can however be modeled as a sequence of projections carried out in different orders. This and other examples have been described successfully using quantum probability, which relies on comparing angles between subspaces rather than volumes between subsets. Now in the early 2020's, quantum computers have reached the point where some of these quantum cognitive models can be implemented and investigated on quantum hardware, representing the mental states in qubit registers, and the cognitive operations and decisions using different gates and measurements. This paper develops such quantum circuit representations for quantum cognitive models, focusing particularly on modeling order effects and decision-making under uncertainty. The claim is not that the human brain uses qubits and quantum circuits explicitly (just like the use of Boolean set theory does not require the brain to be using classical bits), but that the mathematics shared between quantum cognition and quantum computing motivates the exploration of quantum computers for cognition modelling. Key quantum properties include superposition, entanglement, and collapse, as these mathematical elements provide a common language between cognitive models, quantum hardware, and circuit implementations.

Kernel methods are an important class of techniques in machine learning. To be effective, good feature maps are crucial for mapping non-linearly separable input data into a higher dimensional (feature) space, thus allowing the data to be linearly separable in feature space. Previous work has shown that quantum feature map design can be automated for a given dataset using NSGA-II, a genetic algorithm, while both minimizing circuit size and maximizing classification accuracy. However, the evaluation of the accuracy achieved by a candidate feature map is costly. In this work, we demonstrate the suitability of kernel-target alignment as a substitute for accuracy in genetic algorithm-based quantum feature map design. Kernel-target alignment is faster to evaluate than accuracy and doesn't require some data points to be reserved for its evaluation. To further accelerate the evaluation of genetic fitness, we provide a method to approximate kernel-target alignment. To improve kernel-target alignment and root mean squared error, the final trainable parameters of the generated circuits are further trained using COBYLA to determine whether a hybrid approach applying conventional circuit parameter training can easily complement the genetic structure optimization approach. A total of eight new approaches are compared to the original across nine varied binary classification problems from the UCI machine learning repository, showing that kernel-target alignment and its approximation produce feature map circuits enabling comparable accuracy to the previous work but with larger margins on training data (in excess of 20\% larger) that improve further with circuit parameter training.

Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. For example, edit distance of two strings of length $n$ can be solved in $O(n^2/w)$ time. In a reasonable classical model of computation, one can assume $w=\Theta(\log n)$. There are conditional lower bounds for such problems stating that speed-ups with factor $n^\epsilon$ for any $\epsilon>0$ would lead to breakthroughs in complexity theory. However, these conditional lower bounds do not cover quantum models of computing. Indeed, Boroujeni et al. (J. ACM, 2021) showed that edit distance can be approximated within a factor $3$ in sub-quadratic time $O(n^{1.81})$ using quantum computing. They also showed that, in their chosen model of quantum computing, the approximation factor cannot be improved using sub-quadractic time. To break through the aforementioned classical conditional lower bounds and this latest quantum lower bound, we enrich the model of computation with a quantum random access memory (QRAM), obtaining what we call the word QRAM model. Under this model, we show how to convert the bit-parallelism of quadratic time solvable problems into quantum algorithms that attain speed-ups with factor $n$. The technique we use is simple and general enough to apply to many bit-parallel algorithms that use Boolean logics and bit-shifts. To apply it to edit distance, we first show that the famous $O(n^2/w)$ time bit-parallel algorithm of Myers (J. ACM, 1999) can be adjusted to work without arithmetic + operations. As a direct consequence of applying our technique to this variant, we obtain linear time edit distance algorithm under the word QRAM model for constant alphabet. We give further results on a restricted variant of the word QRAM model to give more insights to the limits of the model.

A myriad of approaches have been proposed to characterise the mesoscale structure of networks - most often as a partition based on patterns variously called communities, blocks, or clusters. Clearly, distinct methods designed to detect different types of patterns may provide a variety of answers to the network's mesoscale structure. Yet, even multiple runs of a given method can sometimes yield diverse and conflicting results, yielding entire landscapes of partitions which potentially include multiple (locally optimal) mesoscale explanations of the network. Such ambiguity motivates a closer look at the ability of these methods to find multiple qualitatively different 'ground truth' partitions in a network. Here, we propose a generative model which allows for two distinct partitions to be built into the mesoscale structure of a single benchmark network. We demonstrate a use case of the benchmark model by exploring the power of stochastic block models (SBMs) to detect coexisting bi-community and core-periphery structures of different strengths. We find that the ability to detect the two partitions individually varies considerably by SBM variant and that coexistence of both partitions is recovered only in a very limited number of cases. Our findings suggest that in most instances only one - in some way dominating - structure can be detected, even in the presence of other partitions in the generated network. They underline the need for considering entire landscapes of partitions when different competing explanations exist and motivate future research to advance partition coexistence detection methods. Our model also contributes to the field of benchmark networks more generally by enabling further exploration of the ability of new and existing methods to detect ambiguity in mesoscale structure of networks.

We study the combinatorial assignment domain, which includes combinatorial auctions and course allocation. The main challenge in this domain is that the bundle space grows exponentially in the number of items. To address this, several papers have recently proposed machine learning-based preference elicitation algorithms that aim to elicit only the most important information from agents. However, the main shortcoming of this prior work is that it does not model a mechanism's uncertainty over values for not yet elicited bundles. In this paper, we address this shortcoming by presenting a Bayesian optimization-based combinatorial assignment (BOCA) mechanism. Our key technical contribution is to integrate a method for capturing model uncertainty into an iterative combinatorial auction mechanism. Concretely, we design a new method for estimating an upper uncertainty bound that can be used to define an acquisition function to determine the next query to the agents. This enables the mechanism to properly explore (and not just exploit) the bundle space during its preference elicitation phase. We run computational experiments in several spectrum auction domains to evaluate BOCA's performance. Our results show that BOCA achieves higher allocative efficiency than state-of-the-art approaches.

The problem of computing minimally sparse solutions of under-determined linear systems is $NP$ hard in general. Subsets with extra properties, may allow efficient algorithms, most notably problems with the restricted isometry property (RIP) can be solved by convex $\ell_1$-minimization. While these classes have been very successful, they leave out many practical applications. In this paper, we consider adaptable classes that are tractable after training on a curriculum of increasingly difficult samples. The setup is intended as a candidate model for a human mathematician, who may not be able to tackle an arbitrary proof right away, but may be successful in relatively flexible subclasses, or areas of expertise, after training on a suitable curriculum.

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharp lower bound for the Quantum Coupon Collector problem, with the exact leading order term, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.

This work studies the pure-exploration setting for the convex hull feasibility (CHF) problem where one aims to efficiently and accurately determine if a given point lies in the convex hull of means of a finite set of distributions. We give a complete characterization of the sample complexity of the CHF problem in the one-dimensional setting. We present the first asymptotically optimal algorithm called Thompson-CHF, whose modular design consists of a stopping rule and a sampling rule. In addition, we provide an extension of the algorithm that generalizes several important problems in the multi-armed bandit literature. Finally, we further investigate the Gaussian bandit case with unknown variances and address how the Thompson-CHF algorithm can be adjusted to be asymptotically optimal in this setting.

The ability to interpret machine learning models has become increasingly important as their usage in data science continues to rise. Most current interpretability methods are optimized to work on either (\textit{i}) a global scale, where the goal is to rank features based on their contributions to overall variation in an observed population, or (\textit{ii}) the local level, which aims to detail on how important a feature is to a particular individual in the dataset. In this work, we present the ``GlObal And Local Score'' (GOALS) operator: a simple \textit{post hoc} approach to simultaneously assess local and global feature variable importance in nonlinear models. Motivated by problems in statistical genetics, we demonstrate our approach using Gaussian process regression where understanding how genetic markers affect trait architecture both among individuals and across populations is of high interest. With detailed simulations and real data analyses, we illustrate the flexible and efficient utility of GOALS over state-of-the-art variable importance strategies.