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We study the problem of finding $K$ collision pairs in a random function $f : [N] \rightarrow [N]$ by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using $S$ qubits of memory must perform a number $T$ of queries that satisfies the tradeoff $T^3 S \geq \Omega(K^3 N)$. Classically, the same question has only been settled recently by Dinur [Eurocrypt'20], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener achieves the optimal time-space tradeoff of $T^2 S = \Theta(K^2 N)$. Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry's recording query technique [Crypto'19] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time-space tradeoff $T^2 S \geq \Omega(N^3)$ for sorting $N$ numbers on a quantum computer, which was first obtained by Klauck, \v{S}palek and de Wolf [K\v{S}W07].

We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup over the number of classical i.i.d. samples needed to estimate the mean of a heavy-tailed distribution with a sub-Gaussian error rate. This result subsumes (up to logarithmic factors) earlier works on the mean estimation problem that were not optimal for heavy-tailed distributions [BHMT02,BDGT11], or that require prior information on the variance [Hein02,Mon15,HM19]. As an application, we obtain new quantum algorithms for the $(\epsilon,\delta)$-approximation problem with an optimal dependence on the coefficient of variation of the input random variable.

Nonsmooth optimization problems arising in practice tend to exhibit beneficial smooth substructure: their domains stratify into "active manifolds" of smooth variation, which common proximal algorithms "identify" in finite time. Identification then entails a transition to smooth dynamics, and accommodates second-order acceleration techniques. While identification is clearly useful algorithmically, empirical evidence suggests that even those algorithms that do not identify the active manifold in finite time -- notably the subgradient method -- are nonetheless affected by it. This work seeks to explain this phenomenon, asking: how do active manifolds impact the subgradient method in nonsmooth optimization? In this work, we answer this question by introducing two algorithmically useful properties -- aiming and subgradient approximation -- that fully expose the smooth substructure of the problem. We show that these properties imply that the shadow of the (stochastic) subgradient method along the active manifold is precisely an inexact Riemannian gradient method with an implicit retraction. We prove that these properties hold for a wide class of problems, including cone reducible/decomposable functions and generic semialgebraic problems. Moreover, we develop a thorough calculus, proving such properties are preserved under smooth deformations and spectral lifts. This viewpoint then leads to several algorithmic consequences that parallel results in smooth optimization, despite the nonsmoothness of the problem: local rates of convergence, asymptotic normality, and saddle point avoidance. The asymptotic normality results appear to be new even in the most classical setting of stochastic nonlinear programming. The results culminate in the following observation: the perturbed subgradient method on generic, Clarke regular semialgebraic problems, converges only to local minimizers.

A key problem in the field of quantum computing is understanding whether quantum machine learning (QML) models implemented on noisy intermediate-scale quantum (NISQ) machines can achieve quantum advantages. Recently, Huang et al. [Nat Commun 12, 2631] partially answered this question by the lens of quantum kernel learning. Namely, they exhibited that quantum kernels can learn specific datasets with lower generalization error over the optimal classical kernel methods. However, most of their results are established on the ideal setting and ignore the caveats of near-term quantum machines. To this end, a crucial open question is: does the power of quantum kernels still hold under the NISQ setting? In this study, we fill this knowledge gap by exploiting the power of quantum kernels when the quantum system noise and sample error are considered. Concretely, we first prove that the advantage of quantum kernels is vanished for large size of datasets, few number of measurements, and large system noise. With the aim of preserving the superiority of quantum kernels in the NISQ era, we further devise an effective method via indefinite kernel learning. Numerical simulations accord with our theoretical results. Our work provides theoretical guidance of exploring advanced quantum kernels to attain quantum advantages on NISQ devices.

Quantum computing is a new field that has recently attracted researchers from a broad range of fields due to its representation power, flexibility and promising results in both speed and scalability. Since 2020, laboratories around the globe have started to experiment with models that lie in the juxtaposition between machine learning and quantum computing. The availability of quantum processing units (QPUs) to the general scientific community through open APIs (e.g., Qiskit from IBM) have kindled the interest in developing and testing new approaches to old problems. In this paper, we present a hybrid quantum machine learning framework for health state diagnostics and prognostics. The framework is exemplified using a problem involving ball bearings dataset. To the best of our knowledge, this is the first attempt to harvest and leverage quantum computing to develop and apply a hybrid quantum-classical machine learning approach to a prognostics and health management (PHM) problem. We hope that this paper initiates the exploration and application of quantum machine learning algorithms in areas of risk and reliability.

We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter of multiple Gaussian distributions. We prove that the barycenter is Gaussian distribution when the entropic regularization parameter is small. We further derive closed-form expressions for the covariance matrix of the barycenter.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables stable single-player version of self-play training that helps the agent to escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high-quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.

Quantum machine learning is expected to be one of the first potential general-purpose applications of near-term quantum devices. A major recent breakthrough in classical machine learning is the notion of generative adversarial training, where the gradients of a discriminator model are used to train a separate generative model. In this work and a companion paper, we extend adversarial training to the quantum domain and show how to construct generative adversarial networks using quantum circuits. Furthermore, we also show how to compute gradients -- a key element in generative adversarial network training -- using another quantum circuit. We give an example of a simple practical circuit ansatz to parametrize quantum machine learning models and perform a simple numerical experiment to demonstrate that quantum generative adversarial networks can be trained successfully.

Singular value decomposition (SVD) is the mathematical basis of principal component analysis (PCA). Together, SVD and PCA are one of the most widely used mathematical formalism/decomposition in machine learning, data mining, pattern recognition, artificial intelligence, computer vision, signal processing, etc. In recent applications, regularization becomes an increasing trend. In this paper, we present a regularized SVD (RSVD), present an efficient computational algorithm, and provide several theoretical analysis. We show that although RSVD is non-convex, it has a closed-form global optimal solution. Finally, we apply RSVD to the application of recommender system and experimental result show that RSVD outperforms SVD significantly.