In this thesis we have implemented three many-body methods in quantum mechanics on quantum computers, namely the Quantum Phase Estimation (QPE) algorithm, the Variational Quantum Eigensolver (VQE) algorithm and the Quantum Adiabatic Time Evolution (QATE) algorithm.We utilized these methods to approximate the ground state energy of the pairing model Hamiltonian.We provided a comparison between the ideal simulation of a quantum computer and a simulation including the noise model from some of IBM's actual quantum devices.For the VQE algortihm, we also run the method on the IBM Q London five-qubit device.As the time complexity (circuit depth) of a quantum algorithm (quantum circuit) is often the bottleneck on current quantum computers, we also introduced a circuit optimization scheme, called Recursive Circuit Optimization, which can be utilized to approximate the action of a quantum circuit by learning the parameters of a parametrized unitary operation and only evaluating small parts of the circuit at a time.We tested out the method by trying to optimize a random circuit, the QATE algorithm and the unitary coupled cluster doubles (UCCD) ansatz.Finally, we implemented a quantum circuit that allows us to encode a real data set into the amplitudes of a quantum state, and we utilized this circuit to implement several realizations of dense and recurrent neural networks on quantum computers.We proposed neural network layers that are consisting of hardware-and parameter-efficient parametrized quantum circuits, and we have made a Python framework to construct and train deep quantum neural networks.We then tested the neural networks by trying to learn a simple non-linear function as well as finding the smallest eigenvalue of the pairing Hamiltonian Full Configuration Interaction (FCI) matrix.We found that all the quantum many-body methods showed promise in providing an upper bound on the ground state energy for the ideal simulation of a quantum computer.When performing the simulation with noise and the execution on the real quantum device however, we found that only the VQE algorithm was able to yield reasonable results due to short circuit depth.Still, the success of the method was found to be dependent on the interaction strength g of the quantum system, as the approximation of the ground state energy started to deviate from the actual value as we increased g.For the recursive circuit optimization of a random circuit, we found that we were able to approximate the state given by the random circuit, and we learned that we should expect some error in the approximation due to finite measurements and the flexibility of the parametrized unitary.When utilizing the recursive optimization scheme to approximate the QATE algorithm, we found that although we were not able to exactly reproduce the action of the operator, we did provide an upper bound to the ground state energy close to the one given by the actual QATE algorithm.In the same time, the circuit depth of the algorithm was drastically decreased.When utilizing the scheme for the VQE algorithm, we got promising results while reducing the circuit depth of the UCCD ansatz.Over to the neural networks, we managed to approximate a non-linear function utilizing a hardware-efficient three layered network, with only three qubits in the intermediate layers.We showed that our proposed neural network are suitable for near-term quantum computers by successfully approximating the lowest eigenvalue of the FCI matrix when running a simulation with the noise model from the IBM Q London five-qubit computer.i I would first like to thank my supervisor, Morten Hjorth-Jensen, for being helpful and motivational throughout my time at the University of Oslo.Your door has always been open when help is needed, which has been a great relief in stressful times.I would also like to thank my friends and family for believing in me and providing emotional support.Your continuous interest in my studies and well-being has meant a lot to me.My next thanks goes to Heidi K. J.I highly appreciate your never-ending support.Finally I would like to thank the computational physics study group.You have been invaluable to me in both a social and educational manner.
In this thesis we have implemented three many-body methods in quantum mechanics on quantum computers, namely the Quantum Phase Estimation (QPE) algorithm, the Variational Quantum Eigensolver (VQE) algorithm and the Quantum Adiabatic Time Evolution (QATE) algorithm.We utilized these methods to approximate the ground state energy of the pairing model Hamiltonian.We provided a comparison between the ideal simulation of a quantum computer and a simulation including the noise model from some of IBM's actual quantum devices.For the VQE algortihm, we also run the method on the IBM Q London five-qubit device.As the time complexity (circuit depth) of a quantum algorithm (quantum circuit) is often the bottleneck on current quantum computers, we also introduced a circuit optimization scheme, called Recursive Circuit Optimization, which can be utilized to approximate the action of a quantum circuit by learning the parameters of a parametrized unitary operation and only evaluating small parts of the circuit at a time.We tested out the method by trying to optimize a random circuit, the QATE algorithm and the unitary coupled cluster doubles (UCCD) ansatz.Finally, we implemented a quantum circuit that allows us to encode a real data set into the amplitudes of a quantum state, and we utilized this circuit to implement several realizations of dense and recurrent neural networks on quantum computers.We proposed neural network layers that are consisting of hardware-and parameter-efficient parametrized quantum circuits, and we have made a Python framework to construct and train deep quantum neural networks.We then tested the neural networks by trying to learn a simple non-linear function as well as finding the smallest eigenvalue of the pairing Hamiltonian Full Configuration Interaction (FCI) matrix.We found that all the quantum many-body methods showed promise in providing an upper bound on the ground state energy for the ideal simulation of a quantum computer.When performing the simulation with noise and the execution on the real quantum device however, we found that only the VQE algorithm was able to yield reasonable results due to short circuit depth.Still, the success of the method was found to be dependent on the interaction strength g of the quantum system, as the approximation of the ground state energy started to deviate from the actual value as we increased g.For the recursive circuit optimization of a random circuit, we found that we were able to approximate the state given by the random circuit, and we learned that we should expect some error in the approximation due to finite measurements and the flexibility of the parametrized unitary.When utilizing the recursive optimization scheme to approximate the QATE algorithm, we found that although we were not able to exactly reproduce the action of the operator, we did provide an upper bound to the ground state energy close to the one given by the actual QATE algorithm.In the same time, the circuit depth of the algorithm was drastically decreased.When utilizing the scheme for the VQE algorithm, we got promising results while reducing the circuit depth of the UCCD ansatz.Over to the neural networks, we managed to approximate a non-linear function utilizing a hardware-efficient three layered network, with only three qubits in the intermediate layers.We showed that our proposed neural network are suitable for near-term quantum computers by successfully approximating the lowest eigenvalue of the FCI matrix when running a simulation with the noise model from the IBM Q London five-qubit computer.i I would first like to thank my supervisor, Morten Hjorth-Jensen, for being helpful and motivational throughout my time at the University of Oslo.Your door has always been open when help is needed, which has been a great relief in stressful times.I would also like to thank my friends and family for believing in me and providing emotional support.Your continuous interest in my studies and well-being has meant a lot to me.My next thanks goes to Heidi K. J.I highly appreciate your never-ending support.Finally I would like to thank the computational physics study group.You have been invaluable to me in both a social and educational manner.
One of the first practical applications of quantum computers is expected to be molecular modelling. Performing this task would profoundly affect areas such as chemistry, materials science and drug synthesis. Modelling of molecules, which are classically intractable, can be achieved with just over 30 qubits, whereas state of the art quantum computers already have more than 50 qubits. The Variational Quantum Eigensolver (VQE) algorithm and VQE based protocols, are promising candidates to enable this task on emerging Noisy Intermediate-Scale Quantum (NISQ) computers. These protocols require short quantum circuits and short coherence times, and are particularly resilient to quantum errors. Nevertheless, there is still a significant gap between the accuracy and the coherence times of current NISQ computers, and the hardware requirements of VQE protocols to simulate practically interesting molecules. In this thesis, I present my contribution to narrowing this gap by developing VQE protocols for molecular modelling that are less demanding on quantum hardware. The VQE relies on the Rayleigh-Ritz variational principle to estimate the eigenvalues of a Hamiltonian operator, by minimizing its expectation value with respect to a trial quantum state, prepared by an ansatz. A major challenge for the practical realisation of VQE protocols on NISQ computers is to construct an ansatz that: (1) can accurately approximate the eigenstates of the Hamiltonian; (2) is easy to optimize; and (3) can be implemented by a shallow circuit, within the capabilities of a NISQ computer. The most widely used, unitary coupled cluster (UCC), type of ans\"atze mathematically correspond to a product of unitary evolutions of fermionic excitation operators. Owing to their fermionic structure, UCC ans\"atze preserve the symmetries of electronic wavefunctions, and thus are accurate and easy to optimize. Nevertheless, UCC ans\"atze are implemented by high depth circuits, which severely limit the size of the molecules that can be reliably simulated on NISQ computers. In this thesis, I begin by constructing efficient quantum circuits to perform evolutions of fermionic excitation operators. The circuits are optimized in the number of two-qubit entangling gates, which are the current bottleneck of NISQ computers. Compared to the standard circuits used to implement evolutions of fermionic excitation operators, the circuits derived in this thesis reduce the number of two-qubit entangling gates by more than 70% on average. As an intermediate result, I also derive efficient circuits to perform evolutions of qubit excitation operators (excitation operators that account for qubit, rather than fermionic commutation relations). Even with the fermionic-excitation-evolution circuits derived here, UCC ans\"atze still require very long circuits, with a particularly large number of two-qubit entangling gates. In this thesis, I consider the use of alternative VQE ans\"atze, based on evolutions of qubit excitation operators. Due to not accounting for fermionic anticommutation, evolutions of qubit excitation operators can be performed by circuits that require asymptotically fewer two-qubit entangling gates. Furthermore, qubit excitation operators preserve many of the physical properties of fermionic excitation operators. Performing a number of classical numerical VQE simulations for small molecules, I show that qubit-excitation-based ans\"atze can approximate molecular electronic wavefunctions almost as accurately as fermionic-excitation-based ans\"atze. Hence, I argue that evolutions of qubit excitation operators are more suitable to construct molecular ans\"atze than evolutions of fermionic excitation operators, especially in the era of NISQ computers. Motivated by the advantage of qubit-excitation-based ans\"atze, I introduce the qubit-excitation-based adaptive variational quantum eigensolver (QEB-ADAPT-VQE). The QEB-ADAPT-VQE belongs to a family of ADAPT-VQE protocols for molecular modelling that grow a problem-tailored ansatz by iteratively appending unitary operators sampled from a predefined finite-size pool of operators. The operator at each iteration is sampled based on an ansatz-growing strategy, which aims to achieve the lowest estimate for the Hamiltonian expectation value at each iteration. In this way, ADAPT-VQE protocols construct shallow-circuit, few-parameter ans\"atze tailored specifically to the molecular systems of interest. In the case of the QEB-ADAPT-VQE, the operator pool is defined by a set of evolutions of single and double qubit excitation operators. I benchmark the performance of the QEB-ADAPT-VQE, by performing classical numerical simulations. I demonstrate that it can construct ans\"atze that are several orders of magnitude more accurate, and require significantly shallower circuits, than standard UCC ans\"atze. I also compare the QEB-ADAPT-VQE against the original fermionic-ADAPT-VQE, which utilizes a pool of fermionic excitation evolutions, and the qubit-ADAPT-VQE, which utilizes a pool of Pauli-string evolutions. I demonstrate that, in terms of circuit efficiency and convergence speed, the QEB-ADAPT-VQE systematically outperforms the qubit-ADAPT-VQE, which to my knowledge was the previous most circuit-efficient, scalable VQE protocol for molecular modeling. The QEB-ADAPT-VQE protocol, therefore represents a significant improvement in the field of VQE protocols for molecular modelling and brings us closer to achieving practical quantum advantage. Lastly, I outline a modified version of the QEB-ADAPT-VQE, the excited-QEB-ADAPT-VQE, designed to estimate energies of excited molecular states. The excited-QEB-ADAPT-VQE is more robust to initial simulation conditions, at the expense of increased computational complexity.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H2) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the RY-RZ-CZ ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers (p=1 to 5) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced RY-CZ ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at p∈{8,64} characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.
This thesis explores variational quantum algorithms in order to solve optimization combinatorial problems using a meta-learning approach. This variational algorithm sare promising due to its capability to be used in the near future in the so-called NoisyIntermediate-Scale Quantum (NISQ) era, in which algorithms with high tolerance to noise could perform better than its classical counterparts. In this approach, a Recurrent Neural Network (RNN) optimizer tries to obtain the best set of parameters ofa quantum circuit for a given problem using the Qauntum Approximation Optimiza-tion Algorithm (QAOA), with the minimal amount of queries possible to a quantumcomputer. Thanks to the advances of parameter shift methods to compute the gradi-ents of parametrized quantum circuits, these algorithms could be even trained in realquantum computers without the need of highly demanding simulations. The resultspresented show that this approach is able to generalize to other problem instanceswhich the model has not seen before during training.
This thesis focuses on the development, application, and high-performance implementation of numerical methods for simulating quantum many-body systems. In the first chapter, I provide an overview of several most successful conventional approaches for simulating quantum many-body systems, along with their current limitations. Then, the basic properties of the quantum spin model, which will be the subject of study in the following chapters, and its ground state are briefly summarized. Additionally, I discuss the background and the motivations of the research projects carried out during my PhD research. Chapter 2 introduces two numerical algorithms that are extensively used in this thesis, which are exact diagonalization (ED) and variational Monte Carlo. I outline the process of constructing a Hamiltonian matrix and employing iterative methods for exact diagonalization. Subsequently, I introduce the mechanism of the four important components in the ``standard" variational Monte Carlo that are widely used in the most recent research. In Chapter 3, we propose a symmetry-projected variational ansatz in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J1−J2 model on the square lattice, a paradigmatic problem under debate in condensed matter physics,and achieve state-of-the-art accuracy in the representation of the ground state. In Chapter 4, we present a variational approach based on a Chebyshev expansion of the spectral function, and each Chebyshev polynomial is represented by a neural network quantum state. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the "Chebyshev wave functions." We present results for the one-dimensional and two-dimensional Heisenberg models on the square lattice and compare them with those obtained by other methods in the literature. In Chapter 5, we describe a method to carry out thermodynamic simulations of quantum many-body models using typicality and variational representations of quantum states. Minimally entangled typical thermal states (METTS) are a construction that allows one to solve for the imaginary time evolution of quantum many-body systems. By using wave functions that are weakly entangled, one can take advantage of efficient representations in the form of matrixproduct states. We generalize these ideas to arbitrary variational wave functions and we focus, as an illustration, on the particular case of restricted Boltzmann machines. We demonstrate the application of the method to quantum spin systems in one and two spatial dimensions. In Chapter 6, we study the properties of the impurity band in heavily doped nonmagnetic semiconductors using the Jacobi-Davidson algorithm and the supervised deep learning method. The disorder averaged inverse participation ratio and Thouless number calculation show us the rich structure inside the impurity band. A convolutional neural network model, which is trained to distinguish the extended/localized phase of the Anderson model with high accuracy, shows us the results in good agreement with the conventional approach. Together, we find that there are three mobility edges in the impurity band for a specific on-site impurity potential, which means the presence of the extended states while filling the impurity band. In the final chapter, Chapter 7, we discuss the implementation of architecture-aware optimizations for spin dynamics simulations on x86 CPUs and Nvidia GPUs. The time cost of evaluating the spin-spin correlation functions in real space increases quadratically as the number of spins N, leading to significant computational effort, making the simulation of large spin systems very challenging. We propose to use a highly optimized general matrix multiply (GEMM) subroutine to calculate the dynamical spin-spin correlation function that can achieve near-optimal hardware utilization. Furthermore, we fuse the element-wise operations in the calculation of S(q,t) into the in-house GEMM kernel to reduce memory latency. On Nvidia GPU, we perform the on-the-fly calculation in the epilogue of the GEMM subroutine to avoid saving intermediate results on global memory, which significantly reduces memory usage and achieves better performance than cuBLAS.--Author's abstract
In recent years, we have observed a new revolution in computer science, driven by the increasing availability of computational power and data. This historic event has seen the capillary diffusion of what is commonly known as artificial intelligence (AI). In particular, we have seen a rapid improvement of large language models that now can interact with people on a level that appeared impossible just a few years ago. Despite this newly found focus on AI and computer science, the technical difficulties that limit the increase in computational power are increasing, and the prediction described by Moore's law no longer holds. For this reason physicists, computer scientists, mathematicians, and engineers are studying another computational paradigm called quantum computing. Similar to the rise of large language models, quantum computing has gained significant attention in the last ten years due to the development of initial quantum hardware and its rapid advancements. The quantum computing paradigm offers a fundamentally different set of tools with respect to what is called classical computation. Quantum properties and effects like superposition, entanglement, and interference give us the ability to solve problems that we could not approach classically. In this thesis, we explore the intersection between artificial intelligence and quantum computing from an experimental point of view. We researched quantum AI models, utilizing both the classical and quantum paradigms using variational architectures. This thesis is organized as follows: part 1 contains an introduction to quantum computation, starting from the postulate of quantum mechanics, through an introduction to QML and finally the state of the art. Part 2 contains the description of the research we performed. It's divided into two main chapters containing fundamentally different approaches to quantum computing. Each chapter contains section illustrating our research on that approach.