Quantum Algorithms: Geometric Optimization in Variational Circuits and Methods for Simulating Quantum Systems

This thesis investigates methods to improve the scalability, robustness, and efficiency of quantum algorithms in the noisy intermediate-scale quantum (NISQ) era. It focuses on three main research directions: variational quantum algorithms (VQAs), quantum imaginary-time evolution (QITE), and classical simulation techniques based on Pauli Propagation. These approaches address key challenges of near-term quantum computing, including hardware noise, complex optimization landscapes, measurement overheads, and the computational cost of simulating many-body quantum systems.The first part of the thesis studies geometry-aware optimization methods for VQAs. In particular, it analyzesthe Quantum Natural Gradient (QNG), an optimization strategy that incorporates the geometry of the variational parameter space through the Quantum Fisher Information Matrix (QFIM). Benchmark studies based on the Quantum Approximate Optimization Algorithm applied to the transverse-field Ising model demonstrate that QNG achieves faster convergence and improved robustness compared with conventional gradient-descent methods, even under realistic noise conditions. To reduce the computational overhead associated with QNG, an efficient protocol for estimating the QFIM in commuting-block parameterized quantum circuits is introduced, substantially decreasing the number of required quantum circuit evaluations.The thesis also investigates variational approaches to quantum error mitigation. A variational protocol for reproducing the Quantum Fourier Transform under noisy conditions is proposed, combining trainable quantum circuits with training strategies based on Mutually Unbiased Bases. Numerical simulations show that the method can significantly improve the fidelity of the implemented transformation, particularly in the presence of coherent noise.A second major focus is quantum imaginary-time evolution for ground-state preparation. Two new methods are introduced. Adaptive-Time Compressed QITE combines adaptive time-stepping with circuit compression to reduce reconstruction costs and limit circuit-depth growth. Sparse-QITE incorporates L1-regularization into the QITE optimization procedure, producing sparse generators with significantly fewer Pauli operators while maintaining high accuracy. Beyond reducing resource requirements, this formulation provides a new perspective on QITE by combining the conventional locality parameter with a continuous relevance-based selection of operators.Finally, the thesis extends the Pauli Propagation framework to imaginary-time dynamics through the introduction of Imaginary-Time Pauli Propagation. The resulting method approximates imaginary-time evolution directly in the Pauli basis using truncated operator expansions. Numerical benchmarks demonstrate that it can accurately reproduce high-temperature properties while retaining only a relatively small subset of Pauli operators when sparse representations are available.Overall, this work develops complementary strategies for improving variational quantum algorithms and advancing both quantum and classical simulation methods for many-body quantum systems. The results contribute to the development of more scalable, robust, and resource-efficient techniques for near-term quantum computing.