量子机器学习

Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =\widetilde{O}\left(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},n^{3/2}t^{1/2})\right)$, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy–light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$-triangle cut sparsifiers of size $\tilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $\Omega(n/\varepsilon^2)$ on the size of any $\varepsilon$-triangle cut sparsifiers.

Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named \emph{quantum-inspired classical} algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.

We study the estimation of repeatedly nested expectations (RNEs) with a constant horizon (number of nestings) using quantum computing. We propose a quantum algorithm that achieves $\varepsilon$-error with cost $\tilde O(\varepsilon^{-1})$, up to logarithmic factors. Standard lower bounds show this scaling is essentially optimal, yielding an almost quadratic speedup over the best classical algorithm. Our results extend prior quantum speedups for single nested expectations to repeated nesting, and therefore cover a broader range of applications, including optimal stopping. This extension requires a new derandomized variant of the classical randomized Multilevel Monte Carlo (rMLMC) algorithm. Careful de-randomization is key to overcoming a variable-time issue that typically increases quantized versions of classical randomized algorithms.

Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks---including linear and multiple regression, Lasso, Ridge, Huber, $\ell_p$-, and $\delta_p$-type regressions---achieving up to a quadratic improvement in the number of samples $m$ over the best classical algorithms. This speedup is achieved by a non-trivial quantization of the recent classical breakthrough of Jambulapati et al. (2024), where we construct a full quantum pipeline that strategically employs quantum leverage score approximation to initialize and refine Multiscale Leverage Score Overestimates, enabling efficient importance sampling via the preparation of multiple state copies. For problems of dimension $n$, sparsity $r < n$, and error parameter $\epsilon$, our algorithm solves the problem in $\widetilde{O}(r\sqrt{mn}/\epsilon + \mathrm{poly}(n,1/\epsilon))$ quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.

Quantum machine learning (QML), as an interdisciplinary field bridging quantum computing and machine learning, has garnered significant attention in recent years. Currently, the field as a whole faces challenges due to incomplete theoretical foundations for the expressivity of quantum neural networks (QNNs). In this paper we propose a constructive QNN model and demonstrate that it possesses the universal approximation property (UAP), which means it can approximate any square-integrable function up to arbitrary accuracy. Furthermore, it supports switching function bases, thus adaptable to various scenarios in numerical approximation and machine learning. Our model has asymptotic advantages over the best classical feed-forward neural networks in terms of circuit size and achieves optimal parameter complexity when approximating Sobolev functions under $L_2$ norm.

Many successful families of generative models leverage a low-dimensional latent distribution that is mapped to a data distribution. Though simple latent distributions are often used, the choice of distribution has a strong impact on model performance. Recent experiments have suggested that the probability distributions produced by quantum processors, which are typically highly correlated and classically intractable, can lead to improved performance on some datasets. However, when and why latent distributions produced by quantum processors can improve performance, and whether these improvements are connected to quantum properties of these distributions, are open questions that we investigate in this work. We show in theory that, under certain conditions, these "quantum latent distributions" enable generative models to produce data distributions that classical latent distributions cannot efficiently produce. We provide intuition as to the underlying mechanisms that could explain a performance advantage on real datasets. Based on this, we perform extensive benchmarking on a synthetic quantum dataset and the QM9 molecular dataset, using both simulated and real photonic quantum processors. We find that the statistics arising from quantum interference lead to improved generative performance compared to classical baselines, suggesting that quantum processors can play a role in expanding the capabilities of deep generative models.

The technique of combining multiple votes to enhance the quality of a decision is the core of boosting algorithms in machine learning. In particular, boosting provably increases decision quality by combining multiple "weak learners"—hypotheses that are only slightly better than random guessing—into a single "strong learner" that classifies data well. There exist various versions of boosting algorithms, which we improve upon through the introduction of QuantumBoost. Inspired by classical work by Barak, Hardt and Kale, our QuantumBoost algorithm achieves the best known runtime over other boosting methods through two innovations. First, it uses a quantum algorithm to compute approximate Bregman projections faster. Second, it combines this with a lazy projection strategy, a technique from convex optimization where projections are performed infrequently rather than every iteration. To our knowledge, QuantumBoost is the first algorithm, classical or quantum, to successfully adopt a lazy projection strategy in the context of boosting.

Although transformers are most commonly optimized as statistical sequence models, it is unclear to what extent they can implement and learn exact algorithmic computations. Here, we specify a transformer implementation from first principles that executes a fundamental and widely used method for $k$-means clustering: Lloyd's algorithm. We theoretically prove and empirically demonstrate that this implementation of a transformer architecture, which we term the _$k$-means transformer_, exactly implements Lloyd's algorithm for $k$-means clustering using the standard circuit mechanisms of modern transformers: attention block, residual connections, and feed-forward block. In learning experiments, we find that training this base architecture on $k$-means clustering yields a generalizable clustering algorithm that surpasses Lloyd's algorithm in terms of clustering quality. Finally, we demonstrate that interpretable alterations (e.g., inclusion of layer normalizations) to this architecture yields diverse and novel variants of clustering algorithms, including soft $k$-means, spherical $k$-means, trimmed $k$-means. Overall, our results show that transformer circuit mechanisms can instantiate exact algorithmic routines for clustering, while simultaneously providing an effective learnable model.

Subsampled natural gradient descent (SNG) has been used to enable high-precision scientific machine learning, but standard analyses based on stochastic preconditioning fail to provide insight into realistic small-sample settings. We overcome this limitation by instead analyzing SNG as a sketch-and-project method. Motivated by this lens, we discard the usual theoretical proxy which decouples gradients and preconditioners using two independent mini-batches, and we replace it with a new proxy based on squared volume sampling. Under this new proxy the expectation of the SNG direction becomes equal to a preconditioned gradient descent step even in the presence of coupling, leading to (i) global convergence guarantees when using a single mini-batch of any size, and (ii) an explicit characterization of the convergence rate in terms of quantities related to the sketch-and-project structure. These findings in turn yield new insights into small-sample settings, for example by suggesting that the advantage of SNG over SGD is that it can more effectively exploit spectral decay in the model Jacobian. We also extend these ideas to explain a popular structured momentum scheme for SNG, known as SPRING, by showing that it arises naturally from accelerated sketch-and-project methods.

Automating the development of machine learning algorithms has the potential to unlock new breakthroughs. However, our ability to *improve* and *evaluate* algorithm discovery systems has thus far been limited by existing task suites. They suffer from many issues, such as: poor evaluation methodologies; data contamination; and containing saturated or very similar problems. Here, we introduce *DiscoGen*, a procedural generator of algorithm discovery tasks for machine learning, such as developing optimisers for reinforcement learning or loss functions for image classification. Motivated by the success of procedural generation in reinforcement learning, DiscoGen spans billions of tasks of varying difficulty and complexity from a range of machine learning fields. These tasks are specified by a small number of configuration parameters and can be used to optimise algorithm discovery agents (ADAs). We present *DiscoBench*, a fixed, small subset of DiscoGen tasks for principled evaluation of ADAs. Finally, we propose a number of ambitious, impactful research directions enabled by DiscoGen, and demonstrate its use for ADA optimisation through scaling experiments for automated prompt tuning. DiscoGen is released open-source.