基於主成分分析啟發之可擴展包立旋轉最小化演算法實現目標保真度之近似哈密頓量電路合成

Abstract

哈密頓量（Hamiltonian）電路合成對量子模擬至關重要。由於在容錯編譯中 T 閘成本高昂，於目標保真度下合成模擬電路自然成為近似合成（approximate synthesis）問題：經 Gridsynth 後，Pauli 旋轉數量大致決定 T 閘數量。本論文研究可擴展（scalable）的近似哈密頓量電路合成，在目標保真度 F_approx 約束下，最小化 non-Clifford Pauli 旋轉閘數量。本方法於 Qsyn 的 Continuous Phase Folding（CPF）流程中，在 Pauli-list 層級運作，結合主成分分析（PCA）啟發之縮減策略，並於化學與晶格基準進行系統性評估。

我們提出係數層級的保真度代理指標（fidelity surrogate），以及統一的 Pauli-list 近似架構，整合截斷（truncation）、前 k 項選取、預算約束（budget constraints）、結構合併（structure merge）、Clifford 吸收與混合縮減，並可在 O(#Pauli) 時間內完成。我們進一步提出 zero-sweep 貪婪啟發式方法，無需窮舉參數掃描（parameter sweeps），即可在目標保真度下獲得較低的 Pauli 旋轉數量。在 F_approx ≥ 0.99 下，以 LiH、H2O 與 N2 對照 Non-Clifford Fusion（NCF）單量子位元基準，本流程可將 Pauli 旋轉數量最多減少 99.7%（分子基準平均約 99%）。經後續 Qsyn 量子電路最佳化（QCO）後，T 閘數量與 Clifford 閘數量最多分別減少 90.7%（平均 82%）與 98.6%（平均 98%）。本流程實作於 Qsyn，並於 13 組 NCF 基準驗證，涵蓋分子與自旋晶格，規模達 60 量子位元。分子電路可大幅壓縮；相對地，均勻 Ising 與 Heisenberg 晶格需採較低目標保真度，始能達到相近的縮減效果。本研究使保真度感知（fidelity-aware）的哈密頓量合成得以在 Pauli-list 層級進行，並銜接量子模擬前端工作負載與高效的 Clifford+T 編譯後端。

Hamiltonian circuit synthesis is vital for quantum simulation. Because T gates are expensive in fault-tolerant compilation, synthesizing simulation circuits under a target fidelity naturally becomes an approximate synthesis problem: after Gridsynth, the number of Pauli rotations largely determines T-count. This thesis studies scalable approximate Hamiltonian circuit synthesis that minimizes non-Clifford Pauli rotations under a target compress fidelity F_approx. Our approach operates at the Pauli-list level within Qsyn's Continuous Phase Folding (CPF) flow, combining principal component analysis (PCA)-inspired reduction with systematic evaluation on chemistry and lattice benchmarks.

We develop a coefficient-level fidelity surrogate and a unified Pauli-list approximation framework that integrates truncation, top-$k$ selection, budget constraints, structure merge, Clifford absorption, and hybrid reduction in O(#Pauli) time. We further introduce a zero-sweep greedy heuristic that attains low rotation counts at target fidelities without exhaustive parameter sweeps. At F_approx ≥ 0.99, on LiH, H2O, and N2 against Non-Clifford Fusion (NCF) single-qubit baselines, our flow reduces Pauli rotations by up to 99.7% (avg. 99% over molecular benchmarks). After subsequent Qsyn quantum circuit optimization (QCO), T-count and Clifford count decrease by up to 90.7% (avg. 82%) and 98.6% (avg. 98%), respectively. Implemented in Qsyn, the flow is validated on 13 NCF benchmarks spanning molecules and spin lattices up to 60 qubits. Molecular circuits compress strongly, whereas uniform Ising and Heisenberg lattices require lower target fidelities to achieve comparable reduction. This work enables fidelity-aware Hamiltonian synthesis at the Pauli-list level, linking quantum simulation front-end workloads to efficient Clifford+T compilation back-ends.