量子机器学习

$k$-Clustering in $\mathbb{R}^d$ (e.g., $k$-median and $k$-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality $n$, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for $k$-clustering in $\mathbb{R}^d$ with $\tilde{O}(\sqrt{nk}d^{3/2})$ query complexity. Our coreset reduces the input size from $n$ to $\mathrm{poly}(k\epsilon^{-1}d)$, so that existing $\alpha$-approximation algorithms for clustering can run on top of it and yield $(1 + \epsilon)\alpha$-approximation. This eventually yields a quadratic speedup for various $k$-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make $\Omega(\sqrt{nk})$ queries in order to achieve even $O(1)$-approximation for $k$-clustering.

A significant challenge in the field of quantum machine learning (QML) is to establish applications of quantum computation to accelerate common tasks in machine learning such as those for neural networks. Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks, but the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime $\exp(O(D))$ as data dimension $D$ increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime $O(D)$ of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for QML to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.

Quantum computing is an emerging technology that has been rapidly advancing in the past decades. In this paper, we conduct a systematic study of quantum lower bounds on finding $\epsilon$-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to $p$-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds are $\Omega\big(\epsilon^{-\frac{1+p}{p}}\big)$ regarding the first setting and $\Omega(\epsilon^{-4})$ regarding the second setting (or $\Omega(\epsilon^{-3})$ if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding $\epsilon$-stationary points of nonconvex functions with $p$-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, we prove our quantum lower bounds by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.

With the arrival of the Noisy Intermediate-Scale Quantum (NISQ) era and the fast development of machine learning, variational quantum algorithms (VQA) including Variational Quantum Eigensolver (VQE) and quantum neural network (QNN) have received increasing attention with wide potential applications in foreseeable near future. We study the problem of quantum architecture search (QAS) for VQA to automatically design parameterized quantum circuits (PQC). We devise a differentiable searching algorithm based on Gumbel-Softmax in contrast to peer methods that often require numerous circuit sampling and evaluation. Two versions of our algorithm are provided, namely macro search and micro search, where macro search directly searches for the whole circuit like other literature while the innovative micro search is able to infer the sub-circuit structure from a small-scale and then transfer that to a large-scale problem. We conduct intensive experiments on unweighted Max-Cut, ground state energy estimation, and image classification. The superior performance shows the efficiency and capability of macro search, which requires little prior knowledge. Moreover, the experiments on micro search show the potential of our algorithm for large-scale QAS problems.

In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time $T$, where the system is governed by a time-dependent Schrödinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schrödinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.

Shadow tomography for quantum states provides a sample efficient approach for predicting the measurement outcomes of quantum systems. However, these shadow tomography procedures yield poor bounds if there are more than two outcomes per measurement. In this paper, we consider a general problem of learning properties from quantum states: given an unknown $d$-dimensional quantum state $\rho$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we have to learn unknown distributions instead of values. Here, we propose an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound showing that at least $\Omega(\min\{d^2,K+\log M\}/\epsilon^2)$ copies of $\rho$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal concerning the dependence on $K$.

In this paper, we propose a novel Hadamard Transform (HT)-based neural network layer for hybrid quantum-classical computing. It implements the regular convolutional layers in the Hadamard transform domain. The idea is based on the HT convolution theorem which states that the dyadic convolution between two vectors is equivalent to the element-wise multiplication of their HT representation. Computing the HT is simply the application of a Hadamard gate to each qubit individually, so the HT computations of our proposed layer can be implemented on a quantum computer. Compared to the regular Conv2D layer, the proposed HT-perceptron layer is computationally more efficient. Compared to a CNN with the same number of trainable parameters and 99.26% test accuracy, our HT network reaches 99.31% test accuracy with 57.1% MACs reduced in the MNIST dataset; and in our ImageNet-1K experiments, our HT-based ResNet-50 exceeds the accuracy of the baseline ResNet-50 by 0.59% center-crop top-1 accuracy using 11.5% fewer parameters with 12.6% fewer MACs.

When two different parties use the same learning rule on their own data, how can we test whether the distributions of the two outcomes are similar? In this paper, we study the similarity of outcomes of learning rules through the lens of the Total Variation (TV) distance of distributions. We say that a learning rule is TV indistinguishable if the expected TV distance between the posterior distributions of its outputs, executed on two training data sets drawn independently from the same distribution, is small. We first investigate the learnability of hypothesis classes using TV indistinguishable learners. Our main results are information-theoretic equivalences between TV indistinguishability and existing algorithmic stability notions such as replicability and approximate differential privacy. Then, we provide statistical amplification and boosting algorithms for TV indistinguishable learners.

We formalize the problem of machine unlearning as design of efficient unlearning algorithms corresponding to learning algorithms which perform a selection of adaptive queries from structured query classes. We give efficient unlearning algorithms for linear and prefix-sum query classes. As applications, we show that unlearning in many problems, in particular, stochastic convex optimization (SCO), can be reduced to the above, yielding improved guarantees for the problem. In particular, for smooth Lipschitz losses and any $\rho>0$, our results yield an unlearning algorithm with excess population risk of $\tilde O\big(\frac{1}{\sqrt{n}}+\frac{\sqrt{d}}{n\rho}\big)$ with unlearning query (gradient) complexity $\tilde O(\rho \cdot \text{Retraining Complexity})$, where $d$ is the model dimensionality and $n$ is the initial number of samples. For non-smooth Lipschitz losses, we give an unlearning algorithm with excess population risk $\tilde O\big(\frac{1}{\sqrt{n}}+\big(\frac{\sqrt{d}}{n\rho}\big)^{1/2}\big)$ with the same unlearning query (gradient) complexity. Furthermore, in the special case of Generalized Linear Models (GLMs), such as those in linear and logistic regression, we get dimension-independent rates of $\tilde O\big(\frac{1}{\sqrt{n}} +\frac{1}{(n\rho)^{2/3}}\big)$ and $\tilde O\big(\frac{1}{\sqrt{n}} +\frac{1}{(n\rho)^{1/3}}\big)$ for smooth Lipschitz and non-smooth Lipschitz losses respectively. Finally, we give generalizations of the above from one unlearning request to *dynamic* streams consisting of insertions and deletions.

Machine learning force fields (MLFF) have been proposed to accelerate molecular dynamics (MD) simulation, which finds widespread applications in chemistry and biomedical research. Even for the most data-efficient MLFFs, reaching chemical accuracy can require hundreds of frames of force and energy labels generated by expensive quantum mechanical algorithms, which may scale as $O(n^3)$ to $O(n^7)$, with $n$ proportional to the number of basis functions. To address this issue, we propose a multi-stage computational framework -- ASTEROID, which lowers the data cost of MLFFs by leveraging a combination of cheap inaccurate data and expensive accurate data. The motivation behind ASTEROID is that inaccurate data, though incurring large bias, can help capture the sophisticated structures of the underlying force field. Therefore, we first train a MLFF model on a large amount of inaccurate training data, employing a bias-aware loss function to prevent the model from overfitting the potential bias of this data. We then fine-tune the obtained model using a small amount of accurate training data, which preserves the knowledge learned from the inaccurate training data while significantly improving the model's accuracy. Moreover, we propose a variant of ASTEROID based on score matching for the setting where the inaccurate training data are unlabeled. Extensive experiments on MD datasets and downstream tasks validate the efficacy of ASTEROID. Our code and data are available at https://github.com/abukharin3/asteroid.

Automatic quantum architecture search (QAS) has been widely studied across disciplines with different implications. In this paper, beyond a particular domain, we formulate the QAS problem into two basic (and relatively even ideal) tasks: i) arbitrary quantum circuit (QC) regeneration given a target QC; ii) approximating an arbitrary unitary (oracle). The latter can be connected to the setting of various quantum machine learning tasks and other QAS applications. Based on these two tasks, we generate a public QAS benchmark including 900 random QCs and 400 random unitary matrices which is still missing in the literature. We evaluate six baseline algorithms including brute force search, simulated annealing, genetic algorithm, reinforcement learning, hybrid algorithm, and differentiable algorithm as part of our benchmark. One characteristic of our proposed evaluation protocol on the basic tasks is that it deprives the domain-specific designs and techniques as used in existing QAS literature, making a unified evaluation possible and focusing on the vanilla search methods themselves without coupling with domain prior. In fact, the unitary approximation task could be algorithmically more difficult than the specific problems as it needs to explore the whole matrix space to fit the unitary. While specific tasks often only need to fit a partial observation of the unitary as the objective for search. Data and code are available at https://github.com/Lucky-Lance/QAS-Bench.

Differential privacy (DP) is by far the most widely accepted framework for mitigating privacy risks in machine learning. However, exactly how small the privacy parameter $\epsilon$ needs to be to protect against certain privacy risks in practice is still not well-understood. In this work, we study data reconstruction attacks for discrete data and analyze it under the framework of multiple hypothesis testing. For a learning algorithm satisfying $(\alpha, \epsilon)$-Renyi DP, we utilize different variants of the celebrated Fano's inequality to upper bound the attack advantage of a data reconstruction adversary. Our bound can be numerically computed to relate the parameter $\epsilon$ to the desired level of privacy protection in practice, and complements the empirical evidence for the effectiveness of DP against data reconstruction attacks even at relatively large values of $\epsilon$.

When applying differential privacy to sensitive data, we can often improve performance using external information such as other sensitive data, public data, or human priors. We propose to use the learning-augmented algorithms (or algorithms with predictions) framework---previously applied largely to improve time complexity or competitive ratios---as a powerful way of designing and analyzing privacy-preserving methods that can take advantage of such external information to improve utility. This idea is instantiated on the important task of multiple quantile release, for which we derive error guarantees that scale with a natural measure of prediction quality while (almost) recovering state-of-the-art prediction-independent guarantees. Our analysis enjoys several advantages, including minimal assumptions about the data, a natural way of adding robustness, and the provision of useful surrogate losses for two novel ''meta'' algorithms that learn predictions from other (potentially sensitive) data. We conclude with experiments on challenging tasks demonstrating that learning predictions across one or more instances can lead to large error reductions while preserving privacy.

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.

The relational data model was designed to facilitate large-scale data management and analytics. We consider the problem of how to differentiate computations expressed relationally. We show experimentally that a relational engine running an auto-differentiated relational algorithm can easily scale to very large datasets, and is competitive with state-of-the-art, special-purpose systems for large-scale distributed machine learning.

Combinatorial optimization (CO) on the graph is a crucial but challenging research topic. Recent quantum algorithms provide a new perspective for solving CO problems and have the potential to demonstrate quantum advantage. Quantum Approximate Optimization Algorithm (QAOA) is a well-known quantum heuristic for CO constructed by a parametric quantum circuit. However, QAOA is originally designed for unconstrained problems and the circuit parameters and solutions are jointly solved with time-consuming iterations. In this paper, we propose a novel quantum neural network (QNN) for learning CO problems in a supervised manner to achieve better and faster results. We focus on the Quadratic Assignment Problem (QAP) with matching constraints and the node permutation invariance property. To this end, a quantum neural network called QAP-QNN is devised to translate the QAP into a constrained vertex classification task. Moreover, we study two QAP tasks: Graph Matching and Traveling Salesman Problem on TorchQauntum simulators, and empirically show the effectiveness of our approach.

We study the problem of adaptively identifying patient subpopulations that benefit from a given treatment during a confirmatory clinical trial. This type of adaptive clinical trial has been thoroughly studied in biostatistics, but has been allowed only limited adaptivity so far. Here, we aim to relax classical restrictions on such designs and investigate how to incorporate ideas from the recent machine learning literature on adaptive and online experimentation to make trials more flexible and efficient. We find that the unique characteristics of the subpopulation selection problem -- most importantly that (i) one is usually interested in finding subpopulations with any treatment benefit (and not necessarily the single subgroup with largest effect) given a limited budget and that (ii) effectiveness only has to be demonstrated across the subpopulation on average -- give rise to interesting challenges and new desiderata when designing algorithmic solutions. Building on these findings, we propose AdaGGI and AdaGCPI, two meta-algorithms for subpopulation construction. We empirically investigate their performance across a range of simulation scenarios and derive insights into their (dis)advantages across different settings.

This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a variable-duplication technique that copies each variable $K$ times and reduces the optimality gap and constraint violation by a factor of $\sqrt{K}$. Furthermore, we show how online algorithms can be effectively integrated into sifting, a column generation scheme for large-scale LPs. Numerical experiments demonstrate that our methods can serve as either an approximate direct solver, or an initialization subroutine for exact LP solving.

Data is pooled across entities (individuals or enterprises) to create machine learning models, and sometimes, the entities that contribute the data also benefit from the models. Consider for instance a recommender system (e.g. Spotify, Instagram or YouTube), a health care app that predicts the risk for some disease, or a service built by pooling data across enterprises. In this work we propose a framework to study this value exchange, i.e., we model and measure contributions (outflows), benefits (inflows) and the balance between contributions and benefits (the degree of reciprocity). We show theoretically, and via experiments that under certain distributional assumptions, some classes of models are approximately reciprocal. These results only scratch the surface; we conclude with several open directions.

The concept class of low-degree polynomial threshold functions (PTFs) plays a fundamental role in machine learning. In this paper, we study PAC learning of $K$-sparse degree-$d$ PTFs on $\mathbb{R}^n$, where any such concept depends only on $K$ out of $n$ attributes of the input. Our main contribution is a new algorithm that runs in time $({nd}/{\epsilon})^{O(d)}$ and under the Gaussian marginal distribution, PAC learns the class up to error rate $\epsilon$ with $O(\frac{K^{4d}}{\epsilon^{2d}} \cdot \log^{5d} n)$ samples even when an $\eta \leq O(\epsilon^d)$ fraction of them are corrupted by the nasty noise of Bshouty et al. (2002), possibly the strongest corruption model. Prior to this work, attribute-efficient robust algorithms are established only for the special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a structural result that translates the attribute sparsity to a sparsity pattern of the Chow vector under the basis of Hermite polynomials, and 2) a novel attribute-efficient robust Chow vector estimation algorithm which uses exclusively a restricted Frobenius norm to either certify a good approximation or to validate a sparsity-induced degree-$2d$ polynomial as a filter to detect corrupted samples.