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Zeitentwicklungsoperatoren für Pauli-String-basierte Quanten-Hamiltonoperator

Huang, Qunsheng

Published Nov 20, 2025• 0 citations
OpenAlex

Abstract

Quantum computing promises a paradigm shift in computational capabilities, but its realization demands problem representations fundamentally different from classical methods. To harness quantum advantages, problems must be encoded in forms native to quantum systems—commonly via the Pauli basis, which maps operations to rotations on the Bloch sphere. This dissertation investigates the bottlenecks limiting the implementation of such encodings across quantum, hybrid, and classical contexts.

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Quantum computing promises a paradigm shift in computational capabilities, but its realization demands problem representations fundamentally different from classical methods. To harness quantum advantages, problems must be encoded in forms native to quantum systems—commonly via the Pauli basis, which maps operations to rotations on the Bloch sphere. This dissertation investigates the bottlenecks limiting the implementation of such encodings across quantum, hybrid, and classical contexts.
Pauli matrices and Pauli strings are widely used in quantum computing. These mathematical objects are useful to describe or manipulate the quantum state of qubits. They offer a convenient basis to express operators and observables used in different problem instances such as molecular simulation and combinatorial optimization. Therefore, it is important to have a well-rounded, versatile and efficient tool to handle a large number of Pauli strings and operators expressed in this basis. This is the objective behind the development of the PauliArray library presented in this work. This library introduces data structures to represent arrays of Pauli strings and operators as well as various methods to modify and combine them. Built using NumPy, PauliArray offers fast operations and the ability to use broadcasting to easily carry out otherwise cumbersome manipulations. Applications to the fermion-to-qubit mapping, to the estimation of expectation values and to the computation of commutators are considered to illustrate how PauliArray can simplify some relevant tasks and accomplish them faster than current libraries.
The complexity of simulating the out-of-equilibrium evolution of local operators in the Heisenberg picture is governed by the operator entanglement, which grows linearly in time for generic nonintegrable systems, leading to an exponential increase in computational resources. A promising approach to simplify this challenge involves discarding parts of the operator and focusing on a subspace formed by ``light'' Pauli strings---strings with few Pauli matrices---as proposed by Rakovszki et al. [Phys. Rev. B 105, 075131 (2022)]. In this work, we investigate whether this strategy can be applied to quenches starting from homogeneous product states. For ergodic dynamics, these initial states grant access to a wide range of equilibration temperatures. By concentrating on the desired matrix elements and retaining only the portion of the operator that contains Pauli strings parallel to the initial state, we uncover a complex scenario. In some cases, the light Pauli strings suffice to describe the dynamics, enabling efficient simulation with current algorithms. However, in other cases, heavier strings become necessary, pushing computational demands beyond our current capabilities. We analyze this behavior by introducing a measure of complexity, the operator weight entropy, which we compute for different operators across most points on the Bloch sphere.
While quantum computing holds immense potential for tackling previously intractable problems, its current practicality remains limited. A critical aspect of realizing quantum utility is the ability to efficiently interface with data from the classical world. This research focuses on the crucial phase of quantum encoding, which enables the transformation of classical information into quantum states for processing within quantum systems. We focus on three prominent encoding models: Phase Encoding, Qubit Lattice, and Flexible Representation of Quantum Images (FRQI) for cost and efficiency analysis. The aim of quantifying their different characteristics is to analyze their impact on quantum processing workflows. This comparative analysis offers valuable insights into their limitations and potential to accelerate the development of practical quantum computing solutions.
Quantum computing represents a fundamentally new paradigm in information processing, in which information is encoded and manipulated using two-level quantum systems known as qubits, whose evolution is governed by the principles of quantum mechanics. This chapter presents a structured introduction to the theoretical foundations of quantum computation, beginning with the limitations of classical computation and the physical constraints that motivate the need for a quantum model of computation. We develop the mathematical framework required to describe quantum information, introducing qubits as state vectors in Hilbert space and examining the roles of superposition, measurement, and unitary evolution. Single-qubit operations are presented through the Pauli and Hadamard gates, along with their geometric interpretation on the Bloch sphere. Building on this formalism, the discussion is extended to multi-qubit systems using tensor products, and quantum entanglement is introduced as a non-classical correlation fundamental to quantum information processing. The chapter then explores quantum algorithms, focusing on Grover’s unstructured search algorithm, to demonstrate how superposition, interference, and entanglement can be harnessed to achieve computational advantages over classical approaches. Finally, the chapter discusses applications of quantum computation, including quantum simulation, cryptography, search and optimization problems, and near-term hybrid quantum algorithms such as the Variational Quantum Eigensolver (VQE)and the Quantum Approximate Optimization (QAO) Algorithm, highlighting their relevance in the noisy intermediate-scale quantum (NISQ) era.
While quantum computing holds immense potential for tackling previously intractable problems, its current practicality remains limited. A critical aspect of realizing quantum utility is the ability to efficiently interface with data from the classical world. This research focuses on the crucial phase of quantum encoding, which enables the transformation of classical information into quantum states for processing within quantum systems. We focus on three prominent encoding models: Phase Encoding, Qubit Lattice, and Flexible Representation of Quantum Images (FRQI) for cost and efficiency analysis. The aim of quantifying their different characteristics is to analyze their impact on quantum processing workflows. This comparative analysis offers valuable insights into their limitations and potential to accelerate the development of practical quantum computing solutions.