We introduce hybrid classical-quantum algorithms for problems involving a large classical data set X and a space of models Y such that a quantum computer has superposition access to Y but not X. These algorithms use data reduction techniques to construct a weighted subset of X called a coreset that yields approximately the same loss for each model. The coreset can be constructed by the classical computer alone, or via an interactive protocol in which the outputs of the quantum computer are used to help decide which elements of X to use. By using the quantum computer to perform Grover search or rejection sampling, this yields quantum speedups for maximum likelihood estimation, Bayesian inference and saddle-point optimization. Concrete applications include k-means clustering, logistical regression, zero-sum games and boosting.
We introduce hybrid classical-quantum algorithms for problems involving a large classical data set X and a space of models Y such that a quantum computer has superposition access to Y but not X. These algorithms use data reduction techniques to construct a weighted subset of X called a coreset that yields approximately the same loss for each model. The coreset can be constructed by the classical computer alone, or via an interactive protocol in which the outputs of the quantum computer are used to help decide which elements of X to use. By using the quantum computer to perform Grover search or rejection sampling, this yields quantum speedups for maximum likelihood estimation, Bayesian inference and saddle-point optimization. Concrete applications include k-means clustering, logistical regression, zero-sum games and boosting.
Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigated using this paradigm to perform k-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We used numerical simulations to compare the performance of this approach to classical k-means clustering. We were able to find data sets with which coresets work well relative to random sampling and where QAOA could potentially outperform standard k-means on a coreset. However, finding data sets where both coresets and QAOA work well—which is necessary for a quantum advantage over k-means on the entire data set—appears to be challenging.
Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigated using this paradigm to perform k-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We used numerical simulations to compare the performance of this approach to classical k-means clustering. We were able to find data sets with which coresets work well relative to random sampling and where QAOA could potentially outperform standard k-means on a coreset. However, finding data sets where both coresets and QAOA work well—which is necessary for a quantum advantage over k-means on the entire data set—appears to be challenging.
Current quantum hardware prohibits any direct use of large classical datasets. Coresets allow for a succinct description of these large datasets and their solution in a computational task is competitive with the solution on the original dataset. The method of combining coresets with small quantum computers to solve a given task that requires a large number of data points was first introduced by Harrow [arXiv:2004.00026]. In this paper, we apply the coreset method in three different well-studied classical machine learning problems, namely Divisive Clustering, 3-means Clustering, and Gaussian Mixture Model Clustering. We provide a Hamiltonian formulation of the aforementioned problems for which the number of qubits scales linearly with the size of the coreset. Then, we evaluate how the variational quantum eigensolver (VQE) performs on these problems and demonstrate the practical efficiency of coresets when used along with a small quantum computer. We perform noiseless simulations on instances of sizes up to 25 qubits on CUDA Quantum and show that our approach provides comparable performance to classical solvers.
Quantum computers acting as ML accelerators have attracted considerable interest as a promising avenue for addressing the limitations of classical computing confronted with solving complex, high-dimensional problems and handling large data sets. The most exciting aspect of quantum computing lies in its quantum advantage via techniques such as quantum parallelism, superposition, and entanglement which allows not only for rapid data processing, but also a means to resolve complex optimization problems beyond the capacity of classical algorithms. Nevertheless, the usage of quantum computing in machine learning is still a fledgling domain, with challenges concerning quantum noise, hardware restrictions, and algorithmic development. In particular, this research is concerned with the development of and study on hybrid quantum-classical algorithms that target to improve ML models in terms of computational capabilities by making use of the best of both worlds. To address this, researchers are developing hybrid algorithms that leverage the strengths of quantum computing for specific tasks, like optimization and pattern recognition, while still relying on classical computing for more flexible and larger-scale processing, ultimately achieving more efficient machine learning systems that can handle large datasets and complex problem-solving more effectively. It includes potential hybrid approaches like quantum-enhanced optimization algorithms, quantum neural networks, and quantum support vector machines, and examines how such algorithms can be used for practical ML problems including data classification, regression and clustering. Furthermore, the paper proposes that hybrid quantum algorithms offer several advantages over their classical counterparts, such as faster convergence rates, higher accuracy in high-dimensional spaces, and improved handling of non-linear and complex problem landscapes. Key advancements in the area of data processing speed and efficiency in problem-solving with and without quantum in the machine learning workflow are also highlighted in the research through comparative analysis and simulation studies. Lastly, the paper concludes by outlining the obstacles yet to be surmounted and the opportunities that lie ahead for hybrid quantum-classical algorithms, such as further improvement of quantum hardware and error mitigation methods, as well as the design of new algorithmic paradigms that seamlessly integrates the classical and quantum computing spaces.
k-Clustering in Rd (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 Rd with O~(nkd3/2) query complexity. Our coreset reduces the input size from n to poly(kε−1d), so that existing α-approximation algorithms for clustering can run on top of it and yield (1+ε)α-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 Ω(nk) queries in order to achieve even O(1)-approximation for k-clustering.
This thesis presents two main projects focused on hybrid quantum search algorithms. The first, Particle-Guided Grover's Search, applies Bayesian inference to quantum search when the number of solutions is a priori unknown. It begins with a prior distribution that encodes our initial "beliefs" about the true number of solutions in the search space, and recursively updates these beliefs using Bayesian inference. Measurement outcomes of the quantum search that do not yield a solution are interpreted as "evidence", guiding the Bayesian update to determine the optimal number of Grover iterations for subsequent rounds. This approach leverages Particle Filtering -- a Bayesian filtering technique -- to avoid explicitly storing the entire prior distribution at each step. Instead of requiring O(N) memory, the method draws a set of "particles" from the prior and updates their associated weights as quantum search failures accumulate. Particle filtering also offers the advantage of incorporating information from all past failures, rather than only the most recent one. The algorithm's performance is benchmarked against a prior work. The second project, Runtime--Coherence Trade-offs for Hybrid SAT Solvers, is focused on a class of hybrid k-SAT solvers. These solvers are based on a well-known random walk technique called Schoening's algorithm, which uses two sources of randomness to perform a local search over the space of all assignments for a k-SAT problem. In this project, we assume that these sources of randomness are encoded into a long bitstring, allowing the random walk to be interpreted as a deterministic function mapping each bitstring to either 0 or 1, depending on whether the walk successfully finds a satisfying assignment. This functional perspective enables us to pose the question: Given a bound on the coherence time of a quantum computer, which portion of the bitstring should we search coherently, and which portion should be left to classical search? We address this question by introducing a family of hybrid algorithms called "partial Groverizations", which strategically combine quantum and classical resources. We analyze the runtime performance of this family as a function of the available quantum coherence time. The results of this work have been published in IEEE Transactions on Quantum Engineering.