The analysis of quantum state properties is of fundamental importance in quantum physics and serves as a primary motivation for the development of quantum computing. In particular, low-temperature and ground states are of significant interest, as they most prominently exhibit phenomena unique to quantum mechanics. However, extracting such properties typically requires costly numerical simulations or laboratory experiments, while simulation on quantum hardware remains limited by the capabilities of current devices. This thesis proposes two complementary approaches to address these challenges.The first contribution presents an algorithm for predicting ground-state properties across an entire family of quantum states within the same phase of matter, based on training data. Under slightly stronger assumptions, we improve upon the sample complexity of existing methods, prove the same rigorous guarantees for a deep neural network-based approach, and demonstrate its practical advantages through extensive numerical experiments.The second contribution addresses the challenge of preparing quantum states of interest using Variational Quantum Algorithms (VQAs), a framework based on the classical optimization of parametrized quantum circuits. We introduce multi-armed stochastic bandits into this setting, propose an instance-optimal algorithm for continuous single-parameter optimization, and and motivate further exploration of the framework we introduced with an array of experiments.
The analysis of quantum state properties is of fundamental importance in quantum physics and serves as a primary motivation for the development of quantum computing. In particular, low-temperature and ground states are of significant interest, as they most prominently exhibit phenomena unique to quantum mechanics. However, extracting such properties typically requires costly numerical simulations or laboratory experiments, while simulation on quantum hardware remains limited by the capabilities of current devices. This thesis proposes two complementary approaches to address these challenges.The first contribution presents an algorithm for predicting ground-state properties across an entire family of quantum states within the same phase of matter, based on training data. Under slightly stronger assumptions, we improve upon the sample complexity of existing methods, prove the same rigorous guarantees for a deep neural network-based approach, and demonstrate its practical advantages through extensive numerical experiments.The second contribution addresses the challenge of preparing quantum states of interest using Variational Quantum Algorithms (VQAs), a framework based on the classical optimization of parametrized quantum circuits. We introduce multi-armed stochastic bandits into this setting, propose an instance-optimal algorithm for continuous single-parameter optimization, and and motivate further exploration of the framework we introduced with an array of experiments.
Abstract Machine learning provides a powerful framework for predicting ground-state properties across families of quantum many-body problems, enabling amortized inference and reducing the cost of repeated simulations. Variational quantum algorithms (VQAs) are promising candidates for implementing such learnable solvers on quantum computers, yet rigorous guarantees for convergence and generalization remain scarce. Motivated by the fact that standard quantum phase estimation (QPE) provides provable performance when given a guiding state with non-trivial overlap with the ground state, we introduce a variational quantum algorithm with guiding states aiming towards predicting ground-state properties of quantum many-body systems. We then develop a proof technique—the linearization trick—that maps the training dynamics of the algorithm to those of a kernel model. This connection yields theoretical guarantees on both convergence and generalization for the VQA under the guiding state assumption. Our analysis shows that guiding states facilitate convergence, suppress finite-size error terms, and ensure stability across system dimensions. Finally, we validate our findings with numerical experiments on 2D random Heisenberg models.
Fault-tolerant quantum computers, when realized, guarantee significant speedups for particular computational tasks. Quantum hardware in the near term, however, is limited to low hundreds of noisy qubits that are not error-corrected. To leverage such devices as well as accommodate rapid developments in both hardware and algorithms, variational quantum algorithms (VQAs) were introduced. These algorithms have the potential to be useful in areas such as chemistry, optimization, machine learning, among others. Despite this potential, the largest experiment to date has employed just twelve qubits to implement the Variational Quantum Eigensolver (VQE), a VQA for simulating ground state energies of fermionic systems. In this dissertation, we present several methods for extending and further developing VQAs to step closer towards realizing applications in which quantum computers can provide meaningful advantages over classical computers. In the first part, we present methods for analyzing and optimizing parameterized quantum circuits (PQCs), a core component in VQAs for approximating target quantum states, e.g. ground state wavefunctions in VQE. To compare among PQCs and guide their design and selection, we introduce a set of algorithm-independent descriptors that can be estimated from simulations. Using these descriptors, we rule out circuits with limited capabilities in expressive and entangling powers. Next, we present a strategy for optimizing PQCs with large parameter counts and high redundancy by removing circuit operations that are less important for optimizing the algorithm objective, effectively reducing the search space. We demonstrate that this method enables larger VQE simulations by converging previously difficult optimizations while additionally reducing circuit resources. In the second part, we present methods for developing VQAs that are related to the quantum hardware. We introduce a new algorithm for simulating components of a quantum processor using an existing quantum computer. The goal of this application is to leverage near-term devices and algorithms to aid in the design of more powerful next-generation quantum hardware. Lastly, we present a guide for algorithm prototyping by deploying instances of VQAs on quantum computers via the cloud. We demonstrate instances of two algorithms, the quantum autoencoder and classifier, on an eight-qubit device. While further advancements in hardware and algorithms lie ahead, our work, along with developments from previous studies, represents significant progress towards realizing practical quantum advantage.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithms for learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learning properties of the ground state of an n-qubit gapped local Hamiltonian H from only nO(1) data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by [Lewis et al. Nature Communications 2024], to O(logn) samples when the geometry of the n-qubit system is known. In this work, we introduce two approaches that achieve a constant sample complexity, independent of system size n, for learning ground state properties. Our first algorithm consists of a simple modification of the ML model used by Lewis et al. and applies to a property of interest known beforehand. Our second algorithm, which applies even if a description of the property is not known, is a deep neural network model. While empirical results showing the performance of neural networks have been demonstrated, to our knowledge, this is the first rigorous sample complexity bound on a neural network model for predicting ground state properties. We also perform numerical experiments that confirm the improved scaling of our approach compared to earlier results.
In recent years, as datasets have grown larger, quantum computing has shown potential in addressing the rapidly increasing costs of data processing due to its superposition and entanglement capabilities. Limited by quantum hardware and the number of controllable qubits, variational quantum algorithms (VQAs) are algorithms suitable for NISQ devices, capable of computation on medium-scale noisy devices. VQAs are hybrid classical-quantum algorithms that use parameterized quantum circuits to prepare parameterized quantum states and employ classical optimizers to minimize objective functions, enabling efficient solutions for complex problems. Classical optimizers reduce the stringent requirements on quantum hardware and can achieve quantum advantage on current devices. This paper systematically elaborates on the core foundational theories of quantum computing and neural networks, with a focus on reviewing the research and application status of VQAs in three main scenarios: machine learning (supervised, unsupervised, and reinforcement learning), solving systems of equations (linear and nonlinear), and network attack detection. It analyzes the current technical bottlenecks in this field and provides prospects for future development, offering a reference for further research on variational quantum algorithms.
We introduce a novel approach to variational Quantum algorithms (VQA) via continuous bandits. VQA are a class of hybrid Quantum-classical algorithms where the parameters of Quantum circuits are optimized by classical algorithms. Previous work has used zero and first order gradient based methods, however such algorithms suffer from the barren plateau (BP) problem where gradients and loss differences are exponentially small. We introduce an approach using bandits methods which combine global exploration with local exploitation. We show how VQA can be formulated as a best arm identification problem in a continuous space of arms with Lipschitz smoothness. While regret minimization has been addressed in this setting, existing methods for pure exploration only cover discrete spaces. We give the first results for pure exploration in a continuous setting and derive a fixed-confidence, information-theoretic, instance specific lower bound. Under certain assumptions on the expected payoff, we derive a simple algorithm, which is near-optimal with respect to our lower bound. Finally, we apply our continuous bandit algorithm to two VQA schemes: a PQC and a QAOA quantum circuit, showing that we significantly outperform the previously known state of the art methods (which used gradient based methods).