Quantum Machine Learning

Graph sparsification serves as a foundation for many algorithms, such as approximation algorithms for graph cuts and Laplacian system solvers. As its natural generalization, hypergraph sparsification has recently gained increasing attention, with broad applications in graph machine learning and other areas. In this work, we propose the first quantum algorithm for hypergraph sparsification, addressing an open problem proposed by Apers and de Wolf (FOCS'20). For a weighted hypergraph with $n$ vertices, $m$ hyperedges, and rank $r$, our algorithm outputs a near-linear size $\varepsilon$-spectral sparsifier in time $\widetilde O(r\sqrt{mn}/\varepsilon)$. This algorithm matches the quantum lower bound for constant $r$ and demonstrates quantum speedup when compared with the state-of-the-art $\widetilde O(mr)$-time classical algorithm. As applications, our algorithm implies quantum speedups for computing hypergraph cut sparsifiers, approximating hypergraph mincuts and hypergraph $s$-$t$ mincuts.

With rapid advancements in machine learning, first-order algorithms have emerged as the backbone of modern optimization techniques, owing to their computational efficiency and low memory requirements. Recently, the connection between accelerated gradient methods and damped heavy-ball motion, particularly within the framework of Hamiltonian dynamics, has inspired the development of innovative quantum algorithms for continuous optimization. One such algorithm, Quantum Hamiltonian Descent (QHD), leverages quantum tunneling to escape saddle points and local minima, facilitating the discovery of global solutions in complex optimization landscapes. However, QHD faces several challenges, including slower convergence rates compared to classical gradient methods and limited robustness in highly non-convex problems due to the non-local nature of quantum states. Furthermore, the original QHD formulation primarily relies on function value information, which limits its effectiveness. Inspired by insights from high-resolution differential equations that have elucidated the acceleration mechanisms in classical methods, we propose an enhancement to QHD by incorporating gradient information, leading to what we call gradient-based QHD. This gradient-based QHD achieves faster convergence and significantly increases the likelihood of identifying global solutions. Numerical simulations on challenging problem instances demonstrate that this gradient-based QHD outperforms existing quantum and classical methods by at least an order of magnitude.

In this work, we design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs) in two distinct settings: (1) In the exact dynamics setting, where the agent has full knowledge of the environment's dynamics (i.e., transition probabilities), we prove that our **Quantum Value Iteration (QVI)** algorithm **QVI-1** achieves a quadratic speedup in the size of the action space $(A)$ compared with the classical value iteration algorithm for computing the optimal policy ($\pi^{\ast}$) and the optimal V-value function ($V_{0}^{\ast}$). Furthermore, our algorithm **QVI-2** provides an additional speedup in the size of the state space $(S)$ when obtaining near-optimal policies and V-value functions. Both **QVI-1** and **QVI-2** achieve quantum query complexities that provably improve upon classical lower bounds, particularly in their dependences on $S$ and $A$. (2) In the generative model setting, where samples from the environment are accessible in quantum superposition, we prove that our algorithms **QVI-3** and **QVI-4** achieve improvements in sample complexity over the state-of-the-art (SOTA) classical algorithm in terms of $A$, estimation error $(\epsilon)$, and time horizon $(H)$. More importantly, we prove quantum lower bounds to show that **QVI-3** and **QVI-4** are asymptotically optimal, up to logarithmic factors, assuming a constant time horizon.

Group equivariant neural networks have proven effective in modelling a wide range of tasks where the data lives in a classical geometric space and exhibits well-defined group symmetries. However, these networks are not suitable for learning from data that lives in a non-commutative geometry, described formally by non-commutative $\mathcal{C}^{\ast}$-algebras, since the $\mathcal{C}^{\ast}$-algebra of continuous functions on a compact matrix group is commutative. To address this limitation, we derive the existence of a new type of equivariant neural network, called compact matrix quantum group equivariant neural networks, which encode symmetries that are described by compact matrix quantum groups. We characterise the weight matrices that appear in these neural networks for the easy compact matrix quantum groups, which are defined by set partitions. As a result, we obtain new characterisations of equivariant weight matrices for some compact matrix groups that have not appeared previously in the machine learning literature.

In recent times, there have been major developments in two distinct yet connected domains of quantum information. On the one hand, substantial progress has been made in so-called randomized measurement protocols. Here, a number of properties of unknown quantum states can be deduced from surprisingly few measurement outcomes, using schemes such as classical shadows. On the other hand, significant progress has been made in quantum machine learning. For example, exponential advantages have been proven when the data consists of quantum states and quantum algorithms can coherently measure multiple copies of input states. In this work, we aim to understand the implications and limitations of combining randomized measurement protocols with quantum machine learning, although the implications are broader. Specifically, we investigate quantum machine learning algorithms that, when dealing with quantum data, can either process it entirely using quantum methods or measure the input data through a fixed measurement scheme and utilize the resulting classical information. We prove limitations for the general class of quantum machine learning algorithms that use fixed measurement schemes on the input quantum states. Our results have several implications. From the perspective of randomized measurement procedures, we show limitations of measure-first protocols in the average case, improving on the state-of-the-art which only focuses on worst-case scenarios. Additionally, previous lower bounds were only known for physically unrealizable states. We improve upon this by employing quantum pseudorandom functions to prove that a learning separation also exists when dealing with physically realizable states, which may be encountered in experiments. From a machine learning perspective, our results are crucial for defining a physically meaningful task that shows fully quantum machine learning processing is not only more efficient but also necessary for solving certain problems. The tasks at hand are also realistic, as the algorithms and proven separations hold when working with efficiently preparable states and remain robust in the presence of measurement and preparation errors.

The Quantum Alternating Operator Ansatz (QAOA) is a hybrid quantum-classical variational algorithm for approximately solving combinatorial optimization problems on Noisy Intermediate-Scale Quantum (NISQ) devices. Although it has been successfully applied to a variety of problems, there is only limited work on correlation clustering due to the difficulty of modelling the problem constraints with the ansatz. Motivated by this, we present a generalization of QAOA that is more suitable for this problem. In particular, we modify QAOA in two ways: Firstly, we use nucleus sampling for the computation of the expected cost. Secondly, we split the problem into sub-problems, solving each individually with QAOA. We call this generalization the Sub-Problem Quantum Alternating Operator Ansatz (SQAOA) and show theoretically that optimal solutions for correlation clustering instances can be obtained with certainty when the depth of the ansatz tends to infinity. Further, we show experimentally that SQAOA achieves better approximation ratios than QAOA for correlation clustering, while using only one qubit per node of the respective problem instance and reducing the runtime (of simulations).

Benchmarking and establishing proper statistical validation metrics for reinforcement learning (RL) remain ongoing challenges, where no consensus has been established yet. The emergence of quantum computing and its potential applications in quantum reinforcement learning (QRL) further complicate benchmarking efforts. To enable valid performance comparisons and to streamline current research in this area, we propose a novel benchmarking methodology, which is based on a statistical estimator for sample complexity and a definition of statistical outperformance. Furthermore, considering QRL, our methodology casts doubt on some previous claims regarding its superiority. We conducted experiments on a novel benchmarking environment with flexible levels of complexity. While we still identify possible advantages, our findings are more nuanced overall. We discuss the potential limitations of these results and explore their implications for empirical research on quantum advantage in QRL.

Quantum machine learning models incorporating data re-uploading circuits have garnered significant attention due to their exceptional expressivity and trainability. However, their ability to generate accurate predictions on unseen data, referred to as the predictive performance, remains insufficiently investigated. This study reveals a fundamental limitation in predictive performance when deep encoding layers are employed within the data re-uploading model. Concretely, we theoretically demonstrate that when processing high-dimensional data with limited-qubit data re-uploading models, their predictive performance progressively degenerates to near random-guessing levels as the number of encoding layers increases. In this context, the repeated data uploading cannot mitigate the performance degradation. These findings are validated through experiments on both synthetic linearly separable datasets and real-world datasets. Our results demonstrate that when processing high-dimensional data, the quantum data re-uploading models should be designed with wider circuit architectures rather than deeper and narrower ones.