arXiv subjects

Andrew Vlasic

Publications and source records attributed to Andrew Vlasic.

At least 19 recordsLinked to original sources

arXiv2026

Quantum encodings that preserve persistent homology

Given a data set with a notion of distance, such as a point cloud in Euclidean space, topological data analysis (TDA) uses techniques from algebraic topology and metric geometry to infer the topology of a hypothetical manifold from which the data are sampled. This inference is achieved by calculating topological invariants, some of which are difficult to compute classically. Meanwhile, quantum TDA utilizes quantum processes to extract the invariants used in making such inferences in an attempt to speed up the computations. Because applying transformations to the original classical dataset could alter the associated topological invariants, we investigate which quantum encodings would best preserve the invariants of the original dataset. This line of inquiry is distinct from standard approaches in quantum TDA, whose typical starting point is not from the classical dataset directly, but rather from the associated combinatorial objects, such as simplicial complexes, which typically demand a lot of resources to construct. We take the first step at a more direct approach by focusing on which quantum encodings acting directly on the data are admissible for applying quantum algorithms to extract topological features from classical datasets.

quant-ph↗

arXiv2025

Geodesics of Quantum Feature Maps on the Space of Quantum Operators

Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of data, under the manifold hypothesis, influences the subspace of quantum gates induced by quantum feature maps. While numerous encoding schemes have been proposed in quantum machine learning, little attention has been given to how data geometry is deformed when mapped into the Lie group of special unitary operators. Addressing this gap, we assume a point cloud forms a smooth Riemannian manifold and formally construct the induced Riemannian geometry of a broad class of Hamiltonian quantum feature maps, which encompasses the majority of derived schemes. Starting from first principles, we derive analytic and, consequently, computational formulas for fundamental geometric measurements, including curvature, volume forms, and harmonic maps, providing tools for systematic deformation analysis. Notably, the derivations of the formulas elucidates how changes along paths in the data manifold interplay to changes in the associated subspace of special unitary operators, thereby indicating a direct geometric effect of data on quantum circuits. This framework establishes the mathematical validity required for principled analysis beyond heuristic ansatz and enables future research into geometry-aware quantum algorithm design.

quant-ph↗

arXiv2024

Towards structure-preserving quantum encodings

Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such encodings may vary substantially from one task to another, and there exist only a few cases where structure has provided insight into their design and implementation, such as symmetry in geometric quantum learning. Here, we propose the perspective that category theory offers a natural mathematical framework for analyzing encodings that respect structure inherent in datasets and learning tasks. We illustrate this with pedagogical examples, which include geometric quantum machine learning, quantum metric learning, topological data analysis, and more. Moreover, our perspective provides a language in which to ask meaningful and mathematically precise questions for the design of quantum encodings and circuits for quantum machine learning tasks.

quant-ph↗

arXiv2024

Quantum Circuits, Feature Maps, and Expanded Pseudo-Entropy: Analysis of Encoding Real-World Data into a Quantum Computer

This manuscript introduces a computationally efficient method to calculate the nonlinearity of a quantum feature map, as well as a method for determining whether a quantum feature map will have a high concentration of quantum states. The technique analyzes quantum operators, through an extension of the functions of von Neumann entropy and state-transition pseudo-entropy, by deriving a method to extract the entropy of an operator. The technique is denoted as operator pseudo-entropy, is rigorously derived, and is generally complex valued; as with state-transition pseudo-entropy, complex values contain a lot of information about entanglement or nonlinearity. The characteristics of a class of quantum feature maps are rigorously shown. The operator pseudo-entropy is illuminated through experiments and compared with von Neumann entropy and state-transition pseudo-entropy. We end the manuscript with open questions and potential directions for further research.

quant-ph↗

arXiv2024

Empirical Power of Quantum Encoding Methods for Binary Classification

Quantum machine learning is one of the many potential applications of quantum computing, each of which is hoped to provide some novel computational advantage. However, quantum machine learning applications often fail to outperform classical approaches on real-world classical data. The ability of these models to generalize well from few training data points is typically considered one of the few definitive advantages of this approach. In this work, we will instead focus on encoding schemes and their effects on various machine learning metrics. Specifically, we focus on real-world data encoding to demonstrate differences between quantum encoding strategies for several real-world datasets and the classification model standard, LightGBM. In particular, we apply the following encoding strategies, including three standard approaches and two modified approaches: Angle, Amplitude, IQP, Entangled Angle, and Alternative IQP. As these approaches require either a significant number of qubits or gates to encode larger datasets, we perform feature selection to support the limited computing power of quantum simulators. This feature selection is performed through a quantum annealing enhanced approach that builds on a QUBO formulation of the problem. In this work, we provide a preliminary demonstration that quantum machine learning with the IQP encoding and LightGBM produce statistically equivalent results for a large majority of the assigned learning tasks.

quant-ph↗

arXiv2024

QuOp: A Quantum Operator Representation for Nodes

We derive an intuitive and novel method to represent nodes in a graph with special unitary operators, or quantum operators, which does not require parameter training and is competitive with classical methods on scoring similarity between nodes. This method opens up future possibilities to apply quantum algorithms for NLP or other applications that need to detect anomalies within a network structure. Specifically, this technique leverages the advantage of quantum computation, representing nodes in higher dimensional Hilbert spaces. To create the representations, the local topology around each node with a predetermined number of hops is calculated and the respective adjacency matrix is used to derive the Hamiltonian. While using the local topology of a node to derive a Hamiltonian is a natural extension of a graph into a quantum circuit, our method differs by not assuming the quantum operators in the representation a priori, but letting the adjacency matrix dictate the representation. As a consequence of this simplicity, the set of adjacency matrices of size $2^n \times 2^n$ generates a sub-vector space of the Lie algebra of the special unitary operators, $\mathfrak{su}(2^n)$. This sub-vector space in turn generates a subgroup of the Lie group of special unitary operators, $\mathrm{SU}(2^n)$. Applications of our quantum embedding method, in comparison with the classical algorithms GloVe (a natural language processing embedding method) and FastRP (a general graph embedding method, display superior performance in measuring similarity between nodes in graph structures.

quant-ph↗

arXiv2024

Predicting Ising Model Performance on Quantum Annealers

By analyzing the characteristics of hardware-native Ising Models and their performance on current and next generation quantum annealers, we provide a framework for determining the prospect of advantage utilizing adiabatic evolution compared to classical heuristics like simulated annealing. We conduct Ising Model experiments with coefficients drawn from a variety of different distributions and provide a range for the necessary moments of the distributions that lead to frustration in classical heuristics. By identifying the relationships between the linear and quadratic terms of the models, analysis can be done a priori to determine problem instance suitability on annealers. We then extend these experiments to a prototype of D-Wave's next generation device, showing further performance improvements compared to the current Advantage annealers.

quant-ph↗

arXiv2024

Investigate the Performance of Distribution Loading with Conditional Quantum Generative Adversarial Network Algorithm on Quantum Hardware with Error Suppression

The study examines the efficacy of the Fire Opal error suppression and AI circuit optimization system integrated with IBM's quantum computing platform for a multi-modal distribution loading algorithm. Using Kullback-Leibler (KL) divergence as a quantitative error analysis, the results indicate that Fire Opal can improve on the time-dependent distributions generated by our Conditional Quantum Generative Adversarial algorithm by 30-40\% in comparison with the results on the simulator. In addition, Fire Opal's performance remains consistent for complex circuits despite the needs to run more trials. The research concludes that Fire Opal's error suppression and circuit optimization significantly enhanced quantum computing processes, highlighting its potential for practical applications. In addition, the study also reviews leading error mitigation strategies, including zero noise extrapolation (ZNE), probabilistic error cancellation (PEC), Pauli twirling, measurement error mitigation, and machine learning methods, assessing their advantages and disadvantages in terms of technical implementation, quantum resources, and scalability.

quant-ph↗

arXiv2024

Hybrid Quantum Graph Neural Network for Molecular Property Prediction

To accelerate the process of materials design, materials science has increasingly used data driven techniques to extract information from collected data. Specially, machine learning (ML) algorithms, which span the ML discipline, have demonstrated ability to predict various properties of materials with the level of accuracy similar to explicit calculation of quantum mechanical theories, but with significantly reduced run time and computational resources. Within ML, graph neural networks have emerged as an important algorithm within the field of machine learning, since they are capable of predicting accurately a wide range of important physical, chemical and electronic properties due to their higher learning ability based on the graph representation of material and molecular descriptors through the aggregation of information embedded within the graph. In parallel with the development of state of the art classical machine learning applications, the fusion of quantum computing and machine learning have created a new paradigm where classical machine learning model can be augmented with quantum layers which are able to encode high dimensional data more efficiently. Leveraging the structure of existing algorithms, we developed a unique and novel gradient free hybrid quantum classical convoluted graph neural network (HyQCGNN) to predict formation energies of perovskite materials. The performance of our hybrid statistical model is competitive with the results obtained purely from a classical convoluted graph neural network, and other classical machine learning algorithms, such as XGBoost. Consequently, our study suggests a new pathway to explore how quantum feature encoding and parametric quantum circuits can yield drastic improvements of complex ML algorithm like graph neural network.

quant-ph↗

arXiv2023

Scoring Anomalous Vertices Through Quantum Walks

With the constant flow of data from vast sources over the past decades, a plethora of advanced analytical techniques have been developed to extract relevant information from different data types ranging from labeled data, quasi-labeled data, and data with no labels known a priori. For data with at best quasi-labels, graphs are a natural representation of these data types and have important applications in many industries and scientific disciplines. Specifically, for unlabeled data, anomaly detection on graphs is a method to determine which data points do not posses the latent characteristics that is present in most other data. There have been a variety of classical methods to compute an anomalous score for the individual vertices of a respected graph, such as checking the local topology of a node,random walks, and complex neural networks. Leveraging the structure of the graph, we propose a first quantum algorithm to calculate the anomaly score of each node by continuously traversing the graph with a uniform starting position of all nodes. The proposed algorithm incorporates well-known characteristics of quantum random walks, and, taking into consideration the NISQ era and subsequent ISQ era, an adjustment to the algorithm is given to mitigate the increasing depth of the circuit. This algorithm is rigorously shown to converge to the expected probability, with respect to the initial condition.

quant-ph↗

arXiv2023

Conditional Generative Models for Learning Stochastic Processes

A framework to learn a multi-modal distribution is proposed, denoted as the Conditional Quantum Generative Adversarial Network (C-qGAN). The neural network structure is strictly within a quantum circuit and, as a consequence, is shown to represent a more efficient state preparation procedure than current methods. This methodology has the potential to speed-up algorithms, such as Monte Carlo analysis. In particular, after demonstrating the effectiveness of the network in the learning task, the technique is applied to price Asian option derivatives, providing the foundation for further research on other path-dependent options.

quant-ph↗

arXiv2023

Complement Grover's Search Algorithm: An Amplitude Suppression Implementation

Grover's search algorithm was a groundbreaking advancement in quantum algorithms, displaying a quadratic speed-up of querying for items. Since the creation of this algorithm it has been utilized in various ways, including in preparing specific states for the general circuit. However, as the number of desired items increases so does the gate complexity of the sub-process that conducts the query. To counter this complexity, an extension of Grover's search algorithm is derived where the focus of the query is on the undesirable items in order to suppress the amplitude of the queried items. To display the efficacy the algorithm is implemented as a sub-process into QAOA and applied to a traveling salesman problem. For a basis of comparison, the results are compared against QAOA.

quant-ph↗

arXiv2023

Understanding the Mapping of Encode Data Through An Implementation of Quantum Topological Analysis

A potential advantage of quantum machine learning stems from the ability of encoding classical data into high dimensional complex Hilbert space using quantum circuits. Recent studies exhibit that not all encoding methods are the same when representing classical data since certain parameterized circuit structures are more expressive than the others. In this study, we show the difference in encoding techniques can be visualized by investigating the topology of the data embedded in complex Hilbert space. The technique for visualization is a hybrid quantum based topological analysis which uses simple diagonalization of the boundary operators to compute the persistent Betti numbers and the persistent homology graph. To augment the computation of Betti numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through a illuminating example of a synthetic data set and the methods of angle encoding, amplitude encoding, and IQP encoding, we reveal topological differences with the encoding methods, as well as the original data. Consequently, our results suggest the encoding method needs to be considered carefully within different quantum machine learning models since it can strongly affect downstream analysis like clustering or classification.

quant-ph↗

arXiv2022

An Advantage Using Feature Selection with a Quantum Annealer

Feature selection is a technique in statistical prediction modeling that identifies features in a record with a strong statistical connection to the target variable. Excluding features with a weak statistical connection to the target variable in training not only drops the dimension of the data, which decreases the time complexity of the algorithm, it also decreases noise within the data which assists in avoiding overfitting. In all, feature selection assists in training a robust statistical model that performs well and is stable. Given the lack of scalability in classical computation, current techniques only consider the predictive power of the feature and not redundancy between the features themselves. Recent advancements in feature selection that leverages quantum annealing (QA) gives a scalable technique that aims to maximize the predictive power of the features while minimizing redundancy. As a consequence, it is expected that this algorithm would assist in the bias/variance trade-off yielding better features for training a statistical model. This paper tests this intuition against classical methods by utilizing open-source data sets and evaluate the efficacy of each trained statistical model well-known prediction algorithms. The numerical results display an advantage utilizing the features selected from the algorithm that leveraged QA.

quant-ph↗

arXiv2022

$α$QBoost: An Iteratively Weighted Adiabatic Trained Classifier

A new implementation of an adiabatically-trained ensemble model is derived that shows significant improvements over classical methods. In particular, empirical results of this new algorithm show that it offers not just higher performance, but also more stability with less classifiers, an attribute that is critically important in areas like explainability and speed-of-inference. In all, the empirical analysis displays that the algorithm can provide an increase in performance on unseen data by strengthening stability of the statistical model through further minimizing and balancing variance and bias, while decreasing the time to convergence over its predecessors.

cs.LG↗

arXiv2020

Markovian Switching of Mutation Rates in Evolutionary Network Dynamics

The replicator-mutator dynamic was originally derived to model the evolution of language, and since the model was derived in such a general manner, it has been applied to the dynamics of social behavior and decision making in multi-agent networks. For the two type population, a bifurcation point of the mutation rate is derived, displaying different long-run behaviors above and below this point. The long-run behavior would naturally be subjected to noise from the environment, however, to date there does not exist a model that dynamically accounts for the effects of the environment. To account for the environmental impacts on the evolution of the populace, mutation rates above and below this bifurcation point are switched according to a continuous-time Markov chain. The long-run behaviors of this model are derived, showing a counterintuitive result that the majority of initial conditions will favor the dominated type.

math.PR↗

arXiv2014

Stochastic Replicator Dynamics Subject to Markovian Switching

Population dynamics are often subject to random independent changes in the environment. For the two strategy stochastic replicator dynamic, we assume that stochastic changes in the environment replace the payoffs and variance. This is modeled by a continuous time Markov chain in a finite atom space. We establish conditions for this dynamic to have an analogous characterization of the long-run behavior to that of the deterministic dynamic. To create intuition, we first consider the case when the Markov chain has two states. A very natural extension to the general finite state space of the Markov chain will be given.

q-bio.PE↗

arXiv2014

Long-Run Analysis of the Stochastic Replicator Dynamics in the Presence of Random Jumps

A further generalization of the stochastic replicator dynamic derived by Fudenberg and Harris \cite{FH92} is considered. In particular, a Poissonian integral is introduced to the fitness to simulate the affects of anomalous events. For the two strategy population, an estimation of the long run behavior of the dynamic is derived. For the population with many strategies, conditions for stability to pure strict Nash equilibria, extinction of dominated pure strategies, and recurrence in a neighborhood of an internal evolutionary stable strategy are derived. This extends the results given by Imhof \cite{I05}.

q-bio.PE↗