Quantum algorithms offer the hope to potentially solve computationally expensive problems that are too difficult for classical computers. Superposition and entanglement allow these algorithms to use quantum states to better approximate heuristic solutions. However, current hardware limitations face significant challenges from gate noise and short coherence times, forcing optimized circuits to be created. The Fermi-Hubbard model is a system used to determine the ground state energy of molecules by capturing the behavior of interacting particles in lattice systems. Many classically complex optimization problems can be run on this model, allowing them to be studied using similar computational approaches. However, quantum simulation methods depend on carefully designed variational circuits, known as ansatz structures, within algorithms such as the Variational Quantum Eigensolver. Ansatz creation requires meticulously following minimal gradients, which oftentimes results in the occurrence of a barren plateau. This work aims to investigate the effectiveness of a machine learning-based approach to generating stable, effective quantum circuits to simulate mapped problems on the Fermi-Hubbard model using a deep neural network. By deriving circuit parameters from the energy of the system, this study evaluates the performance of the generated circuits based on their ability to reach ground state convergence with strong gradients.
Quantum algorithms offer the hope to potentially solve computationally expensive problems that are too difficult for classical computers. Superposition and entanglement allow these algorithms to use quantum states to better approximate heuristic solutions. However, current hardware limitations face significant challenges from gate noise and short coherence times, forcing optimized circuits to be created. The Fermi-Hubbard model is a system used to determine the ground state energy of molecules by capturing the behavior of interacting particles in lattice systems. Many classically complex optimization problems can be run on this model, allowing them to be studied using similar computational approaches. However, quantum simulation methods depend on carefully designed variational circuits, known as ansatz structures, within algorithms such as the Variational Quantum Eigensolver. Ansatz creation requires meticulously following minimal gradients, which oftentimes results in the occurrence of a barren plateau. This work aims to investigate the effectiveness of a machine learning-based approach to generating stable, effective quantum circuits to simulate mapped problems on the Fermi-Hubbard model using a deep neural network. By deriving circuit parameters from the energy of the system, this study evaluates the performance of the generated circuits based on their ability to reach ground state convergence with strong gradients.
Quantum simulation and near-term quantum algorithms have gained significant momentum over the past few years, with researchers exploring quantum-based approaches to solve classically intractable problems. Out of many research problems, certain problems are well suited for quantum algorithms. Fermi Hubbard model based on lattice structures is one of the quantum-mechanical models used in many-body and condensed matter physics to capture the essence of electron-correlated systems. In this work, we show one of the first kinds of experiments on simulating the ground state properties of 1D and 2D Fermi Hubbard-based lattice models compared with the reference ground state energy values using Qiskit Nature. We utilize the Variational Quantum Eigensolver (VQE) and different gradient-based optimizers to achieve results that closely converge towards the exact values, showcasing the potential of variational quantum algorithms.
The problem of the neural network optimization is local minima and flat areas. According to quantum-inspired methods, quantum mechanics, superposition, entanglement and interference are inspirational in order to possibly surpass these constraints in a more efficient manner. Variational quantum algorithms use parameterized quantum circuits parameterized quantum circuits (ansatz) to optimize weights. Quantum kernel methods also take advantage of quantum embeddings in order to transform features better. Hybrid quantum-classical systems incorporate quantum-inspired processing with already known reliability of optimization. There are, however, serious weaknesses that limit implementation. There are exponentially decreasing gradients due to the barren plateau phenomenon that makes it hard to train at scale. The modern quantum machines have a 0.1-1% error and a microsecond coherence time. The overheads in state preparation, the cost of measurement, circuit depth, and poor qubit connectivity are still impediments. The quantum-inspired classical algorithms, which include the tensor networks and optimization heuristics and run-on standard computers, are the most effective in the near future. These come with small advances in certain areas without involving quantum hardware. Although both quantum kernel methods and variational approaches show theoretical potential, the advantages are still small (relative to the advantages of optimized classical methods). Recent evaluations show that quantum benefits are more constrained than originally claimed them to be. Real quantum applications benefits in neural network optimization may only happen 10-20 years ahead, notwithstanding failure resistant quantum hardware and algorithm developments.
There is increasing interest in the area of quantum computing, but using and developing new quantum algorithms, often represented as quantum circuits, is arduous and requires designers with a wide area of experience. The exploration of techniques for the automatic design of quantum algorithms, i.e., quantum architecture search, is therefore of high importance. This work explores the use of neural networks for quantum architecture search, since neural networks have proven quite successful in various complex search domains, and current approaches in quantum architecture search make very restricted use of them. There are many different domains in which quantum architecture search can be applied. They can be used to compile unitaries into quantum circuits, develop ansätze for variational quantum circuits, or develop new quantum algorithms by applying it to input output pairs of one task one wishes to solve. The focus in this thesis is on the task of quantum state preparation, a very important task in the field of quantum computing, as it is a subroutine in a large number of quantum algorithms like quantum machine learning and Hamiltonian simulation. This thesis approaches the task of quantum architecture search first by establishing the importance of the structure of quantum circuits. This is done with a large-scale study of the influence different circuit structure parameters, such as qubits and depth, have on the performance on variational quantum machine learning. During this research, four guidelines for designing circuits for variational quantum machine learning were established. DeepQPrep, a neural network-augmented deductive search for the automatic design of quantum circuits for quantum state preparation, was proposed. It is able to create non-parameterized quantum circuits to solve 76.9% of a restrictive set of quantum states with quantum circuits that were much shallower than the ones found in the ground-truth data. The model was trained on a dataset created in this thesis as none were available at the time. During the exploration of extensions of DeepQPrep to enable it to solve more arbitrary quantum state preparation problems without the need for labeled data, different challenges became apparent. A study on the difficulty of training neural networks to create parameterized quantum circuits using reinforcement learning found that training is hampered by the barren plateau problem of quantum computing. It shows that the variance of the gradient of the quantum circuit output with respect to its parameters goes exponentially to zero. A study showed that this problem makes the application of neural networks to encode quantum circuits highly challenging when no intermediate quantum state information is used. This culminated in DeepQPrepGrad, which combines parameterized circuits, the inductive search of DeepQPrep, and the reinforcement learning-based approach with more compact representations. By combining these parts with the gradients of quantum circuits, an approach that is able to create parameterized quantum circuits for arbitrary quantum states was obtained. The approach was evaluated on 20000 arbitrary 3-qubit quantum states without the need for any labeled data, resulting in a mean minimal similarity of 0.97 alleviated the found challenges to a large degree. This thesis explored key challenges faced when training neural networks to encode quantum circuits. These challenges are of paramount importance as they potentially affect a wide range of tasks in the field of quantum computing to which neural networks could be applied. Lastly, DeepQPrepGrad provides a significant step forward for the creation of quantum circuits using neural networks. DeepQPrepGrad makes great progress in solving the challenges highlighted above and has the potential to be applicable in a wide range of quantum architecture search tasks, such as variational eigensolver and variational quantum machine learning.
Variational quantum eigensolver ansätze hold considerable promise for ground-state energy calculations on near-term quantum hardware, yet most promising ansatz designs currently strongly depend on how well the molecular orbital basis captures the electronic correlation of the system. Computing optimized orbital coefficients via classical routines is computationally expensive and must be performed independently for each molecular geometry -- a bottleneck that limits scalability across chemical space. We present a graph neural network framework that predicts optimized orbital coefficients directly from molecular geometry and pair-wise bonding structure. Trained on hydrogenic systems of modest size (H4 and H6) across tens of thousands of geometries, our model transfers to larger, unseen systems (H8, H10 and H12) without retraining -- demonstrating strong out-of-distribution generalization with respect to system size. When evaluating on structured and random configurations, and comparing against energies obtained with full classical optimization, our model reaches mean absolute energy errors O(102) and O(10) milli-Hartrees, respectively. Beyond energy estimation, the predicted orbitals serve as high-quality warm-start initializations that substantially reduce optimizer iterations to ground-state energy convergence. These results establish graph neural networks as an effective and scalable strategy for accelerating orbital optimization in hybrid quantum-classical workflows, directly reducing the classical pre-processing overhead that currently limits the practical deployment of variational quantum eigensolver on near-term quantum hardware.
Abstract Variational hybrid quantum–classical algorithms are powerful tools to maximize the use of noisy intermediate-scale quantum devices. While past studies have developed powerful and expressive ansatze, their near-term applications have been limited by the difficulty of optimizing in the vast parameter space. In this work, we propose a heuristic optimization strategy for such ansatze used in variational quantum algorithms, which we call ‘parameter-efficient circuit training (PECT)’. Instead of optimizing all of the ansatz parameters at once, PECT launches a sequence of variational algorithms, in which each iteration of the algorithm activates and optimizes a subset of the total parameter set. To update the parameter subset between iterations, we adapt the Dynamic Sparse Reparameterization scheme which was originally proposed for training deep convolutional neural networks. We demonstrate PECT for the Variational Quantum Eigensolver, in which we benchmark unitary coupled-cluster ansatze including UCCSD and k -UpCCGSD, as well as the Low-Depth Circuit Ansatz (LDCA), to estimate ground state energies of molecular systems. We additionally use a layerwise variant of PECT to optimize a hardware-efficient circuit for the Sycamore processor to estimate the ground state energy densities of the one-dimensional Fermi-Hubbard model. From our numerical data, we find that PECT can enable optimizations of certain ansatze that were previously difficult to converge and more generally can improve the performance of variational algorithms by reducing the optimization runtime and/or the depth of circuits that encode the solution candidate(s).
Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an ε-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in ε−1, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.