本論文探討了量子計算 (quantum computation) 中的兩個關鍵領域：高效量子電路的構建以及分子基態模擬 (molecular ground state simulation)。
第一部分處理「么正分解 (Unitary Decomposition)」問題，即如何用一組物理上可實現的基本量子閘來建構一個特定的么正操作。傳統電腦面臨「指數牆 (exponential wall)」的障礙，難以模擬複雜的量子系統。量子電腦或可克服此問題，但任意操作的實現並非易事。
本文詳述一種在特殊么正群流形 SU(2n) 上，透過尋找最短路徑 (geodesic) 來分解目標么正算符的方法。此方法利用廣義泡利矩陣 (generalized Pauli matrices) 定義了一個右不變度量，並推導了測地線方程式以確定最佳演化路徑，從而建立了一個數學框架。 其目標是使用一系列單量子位元 (qubit) 和雙量子位元閘來近似目標操作，並基於 Nielsen 等人的基礎性工作，對所需閘數與合成誤差的關係進行了公式化分析。
第二部分著重於減少「分子基態模擬」所需的量子位元數量，這是一個前景廣闊但資源密集的量子計算應用。本文引入了一個名為「子空間限制 (subspace restriction)」的數學框架，用以分析並擴展如「量子位元高效編碼 (Qubit Efficient Encoding, QEE)」等方法。 該框架的核心是在將分子哈密頓量編碼至量子位元前，先將其限制在一個更小且具物理意義的子空間中。本文提出利用「廣義洪德定則 (generalized Hund's rule)」，並結合粒子數守恆 (particle conservation) 和分子多重度 (molecular multiplicity) 來選擇這些子空間。 論文測試了四種子空間：粒子數守恆、多重度、洪德，以及多重度洪德。 針對 15 種分子的數值模擬和 5 種分子的位能面計算顯示，多重度洪德與洪德子空間提供了最顯著的量子位元數量縮減。 儘管這種縮減會與精確的全組態交互作用 (Full Configuration Interaction, FCI) 結果產生微小且可預測的偏差，但該方法仍能以高空間效率提供可靠的預測。

This thesis explores two key areas in quantum computation: efficient quantum circuit construction and molecular ground state simulation.
The first part addresses the "Unitary Decomposition" problem, which is how to construct a desired unitary operation from a set of basic, physically realizable quantum gates. Classical computers face an "exponential wall," making it difficult to simulate complex quantum systems. Quantum computers may overcome this, but implementing arbitrary operations is not straightforward. This work details a method for decomposing a target unitary operator by finding the shortest path (geodesic) on the special unitary group manifold SU(2n). It establishes a mathematical framework by defining a right-invariant metric using generalized Pauli matrices and derives the geodesic equation to determine the optimal evolution path. The goal is to approximate the desired operation with a sequence of one- and two-qubit gates, with a formal analysis connecting the number of gates to the synthesis error, building upon foundational work by Nielsen and collaborators.
The second part focuses on reducing the qubit requirements for "Molecular Ground State Simulation," a promising but resource-intensive application for quantum computers. The author introduces a mathematical framework called "subspace restriction" to analyze and extend methods like Qubit Efficient Encoding (QEE). This involves restricting the molecular Hamiltonian to a smaller, physically relevant subspace before encoding it into qubits. The thesis proposes using a "generalized Hund's rule" alongside particle conservation and molecular multiplicity to select these subspaces. Four subspaces are tested: Particle Conservation, Multiplicity, Hund, and Multiplicity Hund. Numerical simulations on 15 molecules and potential energy surfaces of 5 molecules demonstrate that the Multiplicity, Hund and Hund subspaces offer the most significant qubit reduction. While this reduction introduces a small, predictable deviation from exact Full Configuration Interaction (FCI) results, the method provides reliable predictions with high spatial efficiency.

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