Patrick Rebentrost

Liming Zhao, Naixu Guo, Ming-Xing Luo et al. · berkeley

The complete learning of an $n$-qubit quantum state requires samples exponentially in $n$. Several works consider subclasses of quantum states that can be learned in polynomial sample complexity such as stabilizer states or high-temperature Gibbs states. Other works consider a weaker sense of learning, such as PAC learning and shadow tomography. In this work, we consider learning states that are close to neural network quantum states, which can efficiently be represented by a graphical model called restricted Boltzmann machines (RBMs). To this end, we exhibit robustness results for efficient provable two-hop neighborhood learning algorithms for ferromagnetic and locally consistent RBMs. We consider the $L_p$-norm as a measure of closeness, including both total variation distance and max-norm distance in the limit. Our results allow certain quantum states to be learned with a sample complexity \textit{exponentially} better than naive tomography. We hence provide new classes of efficiently learnable quantum states and apply new strategies to learn them.

Po-Wei Huang, Patrick Rebentrost

Hybrid quantum-classical computing in the noisy intermediate-scale quantum (NISQ) era with variational algorithms can exhibit barren plateau issues, causing difficult convergence of gradient-based optimization techniques. In this paper, we discuss "post-variational strategies", which shift tunable parameters from the quantum computer to the classical computer, opting for ensemble strategies when optimizing quantum models. We discuss various strategies and design principles for constructing individual quantum circuits, where the resulting ensembles can be optimized with convex programming. Further, we discuss architectural designs of post-variational quantum neural networks and analyze the propagation of estimation errors throughout such neural networks. Finally, we show that empirically, post-variational quantum neural networks using our architectural designs can potentially provide better results than variational algorithms and performance comparable to that of two-layer neural networks.

Beng Yee Gan, Po-Wei Huang, Elies Gil-Fuster et al.

Classical learning of the expectation values of observables for quantum states is a natural variant of learning quantum states or channels. While learning-theoretic frameworks establish the sample complexity and the number of measurement shots per sample required for learning such statistical quantities, the interplay between these two variables has not been adequately quantified before. In this work, we take the probabilistic nature of quantum measurements into account in classical modelling and discuss these quantities under a single unified learning framework. We provide provable guarantees for learning parameterized quantum models that also quantify the asymmetrical effects and interplay of the two variables on the performance of learning algorithms. These results show that while increasing the sample size enhances the learning performance of classical machines, even with single-shot estimates, the improvements from increasing measurements become asymptotically trivial beyond a constant factor. We further apply our framework and theoretical guarantees to study the impact of measurement noise on the classical surrogation of parameterized quantum circuit models. Our work provides new tools to analyse the operational influence of finite measurement noise in the classical learning of quantum systems.

Liming Zhao, Aman Agrawal, Patrick Rebentrost

Restricted Boltzmann Machines (RBMs) are widely used probabilistic undirected graphical models with visible and latent nodes, playing an important role in statistics and machine learning. The task of structure learning for RBMs involves inferring the underlying graph by using samples from the visible nodes. Specifically, learning the two-hop neighbors of each visible node allows for the inference of the graph structure. Prior research has addressed the structure learning problem for specific classes of RBMs, namely ferromagnetic and locally consistent RBMs. In this paper, we extend the scope to the quantum computing domain and propose corresponding quantum algorithms for this problem. Our study demonstrates that the proposed quantum algorithms yield a polynomial speedup compared to the classical algorithms for learning the structure of these two classes of RBMs.

Naixu Guo, Zhan Yu, Matthew Choi et al.

Powerful generative artificial intelligence from large language models (LLMs) harnesses extensive computational resources for inference. In this work, we investigate the transformer architecture, a key component of these models, under the lens of fault-tolerant quantum computing. We develop quantum subroutines to construct the building blocks in the transformer, including the self-attention, residual connection with layer normalization, and feed-forward network. As an important subroutine, we show how to efficiently implement the Hadamard product and element-wise functions of matrices on quantum computers. Our algorithm prepares an amplitude encoding of the transformer output, which can be measured for prediction or use in the next layer. We find that the matrix norm of the input sequence plays a dominant role in the quantum complexity. With numerical experiments on open-source LLMs, including for bio-informatics applications, we demonstrate the potential of a quantum speedup for transformer inference in practical regimes.

Yuxuan Du, Xinbiao Wang, Naixu Guo et al.

This tutorial intends to introduce readers with a background in AI to quantum machine learning (QML) -- a rapidly evolving field that seeks to leverage the power of quantum computers to reshape the landscape of machine learning. For self-consistency, this tutorial covers foundational principles, representative QML algorithms, their potential applications, and critical aspects such as trainability, generalization, and computational complexity. In addition, practical code demonstrations are provided in https://qml-tutorial.github.io/ to illustrate real-world implementations and facilitate hands-on learning. Together, these elements offer readers a comprehensive overview of the latest advancements in QML. By bridging the gap between classical machine learning and quantum computing, this tutorial serves as a valuable resource for those looking to engage with QML and explore the forefront of AI in the quantum era.

Wei Zi, Siyi Wang, Hyunji Kim et al.

In recent years, Quantum Machine Learning (QML) has increasingly captured the interest of researchers. Among the components in this domain, activation functions hold a fundamental and indispensable role. Our research focuses on the development of activation functions quantum circuits for integration into fault-tolerant quantum computing architectures, with an emphasis on minimizing $T$-depth. Specifically, we present novel implementations of ReLU and leaky ReLU activation functions, achieving constant $T$-depths of 4 and 8, respectively. Leveraging quantum lookup tables, we extend our exploration to other activation functions such as the sigmoid. This approach enables us to customize precision and $T$-depth by adjusting the number of qubits, making our results more adaptable to various application scenarios. This study represents a significant advancement towards enhancing the practicality and application of quantum machine learning.

Yuxuan Du, Yan Zhu, Yuan-Hang Zhang et al.

Efficient characterization of large-scale quantum systems, especially those produced by quantum analog simulators and megaquop quantum computers, poses a central challenge in quantum science due to the exponential scaling of the Hilbert space with respect to system size. Recent advances in artificial intelligence (AI), with its aptitude for high-dimensional pattern recognition and function approximation, have emerged as a powerful tool to address this challenge. A growing body of research has leveraged AI to represent and characterize scalable quantum systems, spanning from theoretical foundations to experimental realizations. Depending on how prior knowledge and learning architectures are incorporated, the integration of AI into quantum system characterization can be categorized into three synergistic paradigms: machine learning, and, in particular, deep learning and language models. This review discusses how each of these AI paradigms contributes to two core tasks in quantum systems characterization: quantum property prediction and the construction of surrogates for quantum states. These tasks underlie diverse applications, from quantum certification and benchmarking to the enhancement of quantum algorithms and the understanding of strongly correlated phases of matter. Key challenges and open questions are also discussed, together with future prospects at the interface of AI and quantum science.

Joao F. Doriguello, Debbie Lim, Chi Seng Pun et al.

We present a novel quantum high-dimensional linear regression algorithm with an $\ell_1$-penalty based on the classical LARS (Least Angle Regression) pathwise algorithm. Similarly to available classical algorithms for Lasso, our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. A quadratic speedup on the number of features $d$ is possible by using the simple quantum minimum-finding subroutine from Dürr and Hoyer (arXiv'96) in order to obtain the joining time at each iteration. We then improve upon this simple quantum algorithm and obtain a quadratic speedup both in the number of features $d$ and the number of observations $n$ by using the approximate quantum minimum-finding subroutine from Chen and de Wolf (ICALP'23). In order to do so, we approximately compute the joining times to be searched over by the approximate quantum minimum-finding subroutine. As another main contribution, we prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm (and therefore our quantum algorithm) is robust to errors. This means that it still outputs a path that minimises the Lasso cost function up to a small error if the joining times are only approximately computed. Furthermore, we show that, when the observations are sampled from a Gaussian distribution, our quantum algorithm's complexity only depends polylogarithmically on $n$, exponentially better than the classical LARS algorithm, while keeping the quadratic improvement on $d$. Moreover, we propose a dequantised version of our quantum algorithm that also retains the polylogarithmic dependence on $n$, albeit presenting the linear scaling on $d$ from the standard LARS algorithm. Finally, we prove query lower bounds for classical and quantum Lasso algorithms.

Maxime Meyer, Soumik Adhikary, Naixu Guo et al.

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the $ε$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.

Arthur G. Rattew, Po-Wei Huang, Naixu Guo et al.

Fault-tolerant Quantum Processing Units (QPUs) promise to deliver exponential speed-ups in select computational tasks, yet their integration into modern deep learning pipelines remains unclear. In this work, we take a step towards bridging this gap by presenting the first fully-coherent quantum implementation of a multilayer neural network with non-linear activation functions. Our constructions mirror widely used deep learning architectures based on ResNet, and consist of residual blocks with multi-filter 2D convolutions, sigmoid activations, skip-connections, and layer normalizations. We analyse the complexity of inference for networks under three quantum data access regimes. Without any assumptions, we establish a quadratic speedup over classical methods for shallow bilinear-style networks. With efficient quantum access to the weights, we obtain a quartic speedup over classical methods. With efficient quantum access to both the inputs and the network weights, we prove that a network with an $N$-dimensional vectorized input, $k$ residual block layers, and a final residual-linear-pooling layer can be implemented with an error of $ε$ with $O(\text{polylog}(N/ε)^k)$ inference cost.

Mathew Vanherreweghe, Lirandë Pira, Patrick Rebentrost

We introduce cumulative polynomial Kolmogorov-Arnold networks (CP-KAN), a neural architecture combining Chebyshev polynomial basis functions and quadratic unconstrained binary optimization (QUBO). Our primary contribution involves reformulating the degree selection problem as a QUBO task, reducing the complexity from $O(D^N)$ to a single optimization step per layer. This approach enables efficient degree selection across neurons while maintaining computational tractability. The architecture performs well in regression tasks with limited data, showing good robustness to input scales and natural regularization properties from its polynomial basis. Additionally, theoretical analysis establishes connections between CP-KAN's performance and properties of financial time series. Our empirical validation across multiple domains demonstrates competitive performance compared to several traditional architectures tested, especially in scenarios where data efficiency and numerical stability are important. Our implementation, including strategies for managing computational overhead in larger networks is available in Ref.~\citep{cpkan_implementation}.

Debbie Lim, Yixian Qiu, Patrick Rebentrost et al.

Logistic regression, the Support Vector Machine (SVM), and least squares are well-studied methods in the statistical and computer science community, with various practical applications. High-dimensional data arriving on a real-time basis makes the design of online learning algorithms that produce sparse solutions essential. The seminal work of \hyperlink{cite.langford2009sparse}{Langford, Li, and Zhang (2009)} developed a method to obtain sparsity via truncated gradient descent, showing a near-optimal online regret bound. Based on this method, we develop a quantum sparse online learning algorithm for logistic regression, the SVM, and least squares. Given efficient quantum access to the inputs, we show that a quadratic speedup in the time complexity with respect to the dimension of the problem is achievable, while maintaining a regret of $O(1/\sqrt{T})$, where $T$ is the number of iterations.

Yihui Quek, Clement Canonne, Patrick Rebentrost

We consider the problem of finding the minimum element in a list of length $N$ using a noisy comparator. The noise is modelled as follows: given two elements to compare, if the values of the elements differ by at least $α$ by some metric defined on the elements, then the comparison will be made correctly; if the values of the elements are closer than $α$, the outcome of the comparison is not subject to any guarantees. We demonstrate a quantum algorithm for noisy quantum minimum-finding that preserves the quadratic speedup of the noiseless case: our algorithm runs in time $\tilde O(\sqrt{N (1+Δ)})$, where $Δ$ is an upper-bound on the number of elements within the interval $α$, and outputs a good approximation of the true minimum with high probability. Our noisy comparator model is motivated by the problem of hypothesis selection, where given a set of $N$ known candidate probability distributions and samples from an unknown target distribution, one seeks to output some candidate distribution $O(\varepsilon)$-close to the unknown target. Much work on the classical front has been devoted to speeding up the run time of classical hypothesis selection from $O(N^2)$ to $O(N)$, in part by using statistical primitives such as the Scheffé test. Assuming a quantum oracle generalization of the classical data access and applying our noisy quantum minimum-finding algorithm, we take this run time into the sublinear regime. The final expected run time is $\tilde O( \sqrt{N(1+Δ)})$, with the same $O(\log N)$ sample complexity from the unknown distribution as the classical algorithm. We expect robust quantum minimum-finding to be a useful building block for algorithms in situations where the comparator (which may be another quantum or classical algorithm) is resolution-limited or subject to some uncertainty.

Yassine Hamoudi, Patrick Rebentrost, Ansis Rosmanis et al.

Submodular functions are set functions mapping every subset of some ground set of size $n$ into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization [LSW15] runs in time $\widetilde{O}(n^3 \cdot \mathrm{EO} + n^4)$ where $\mathrm{EO}$ denotes the cost to evaluate the function on any set. For functions with range $[-1,1]$, the best $ε$-additive approximation algorithm [CLSW17] runs in time $\widetilde{O}(n^{5/3}/ε^{2} \cdot \mathrm{EO})$. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of [CLSW17] and runs in time $\widetilde{O}(n^{3/2}/ε^2 \cdot \mathrm{EO})$. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time $\widetilde{O}(n^{5/4}/ε^{5/2} \cdot \log(1/ε) \cdot \mathrm{EO})$. The main ingredient of the quantum result is a new method for sampling with high probability $T$ independent elements from any discrete probability distribution of support size $n$ in time $O(\sqrt{Tn})$. Previous quantum algorithms for this problem were of complexity $O(T\sqrt{n})$.

Zhikuan Zhao, Alejandro Pozas-Kerstjens, Patrick Rebentrost et al.

Bayesian methods in machine learning, such as Gaussian processes, have great advantages com-pared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected deep feedforward neural networks with Gaussian processes, allowing training without backpropagation. This connection enables us to leverage a quantum algorithm designed for Gaussian processes and develop a new algorithm for Bayesian deep learning on quantum computers. The properties of the kernel matrix in the Gaussian process ensure the efficient execution of the core component of the protocol, quantum matrix inversion, providing an at least polynomial speedup over classical algorithms. Furthermore, we demonstrate the execution of the algorithm on contemporary quantum computers and analyze its robustness with respect to realistic noise models.

Zhikuan Zhao, Jack K. Fitzsimons, Patrick Rebentrost et al.

Machine learning has recently emerged as a fruitful area for finding potential quantum computational advantage. Many of the quantum enhanced machine learning algorithms critically hinge upon the ability to efficiently produce states proportional to high-dimensional data points stored in a quantum accessible memory. Even given query access to exponentially many entries stored in a database, the construction of which is considered a one-off overhead, it has been argued that the cost of preparing such amplitude-encoded states may offset any exponential quantum advantage. Here we prove using smoothed analysis, that if the data-analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries. This criterion is typically satisfied in realistic machine learning applications, where input data is subjective to moderate noise. Our results are equally applicable to the recent seminal progress in quantum-inspired algorithms, where specially constructed databases suffice for polylogarithmic classical algorithm in low-rank cases. The consequence of our finding is that for the purpose of practical machine learning, polylogarithmic processing time is possible under a general and flexible input model with quantum algorithms or quantum-inspired classical algorithms in the low-rank cases.

Jacob Biamonte, Peter Wittek, Nicola Pancotti et al.

Fuelled by increasing computer power and algorithmic advances, machine learning techniques have become powerful tools for finding patterns in data. Since quantum systems produce counter-intuitive patterns believed not to be efficiently produced by classical systems, it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. The field of quantum machine learning explores how to devise and implement concrete quantum software that offers such advantages. Recent work has made clear that the hardware and software challenges are still considerable but has also opened paths towards solutions.

Patrick Rebentrost, Masoud Mohseni, Seth Lloyd

Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized binary classifier, can be implemented on a quantum computer, with complexity logarithmic in the size of the vectors and the number of training examples. In cases when classical sampling algorithms require polynomial time, an exponential speed-up is obtained. At the core of this quantum big data algorithm is a non-sparse matrix exponentiation technique for efficiently performing a matrix inversion of the training data inner-product (kernel) matrix.