AI 中文总结
本论文针对量子信号与图像处理的数据加载步骤，提出两种基于DsiHT的量子态合成方法，经多IBM量子处理器基准测试，其CNOT代价低于多数从零实现的方法，实合成的经典构建代价、复合成的采样误差表现优异。
AI 中文摘要
本论文研究任意稠密n量子比特态的精确确定性制备，这是量子信号与图像处理中的数据加载步骤。它推导了两种基于数字信号诱导堆变换（Digital Signal-induced Heap Transform, DsiHT）的合成方法：QsiHT快速路径实合成（QsiHT Fast Path Real Synthesis）和QsiHT快速路径复合成（QsiHT Fast Path Complex Synthesis）。将这两种方法与覆盖UCR、等距、多路复用器、施密特/奇异值分解（SVD）、量子状态分解（QSD）及堆变换家族的10种配置进行基准测试，其中部分配置通过Qiskit构建器或编译器优化实现，而非从零开始重新实现。n=3的含噪对比在ibm_fez、ibm_kingston和ibm_marrakesh这3款156量子比特的IBM Heron r2处理器上开展，单次提交的9种方法作业允许会话内家族范围的本贾米尼-霍奇伯格校正对比。前沿方法与QSD的分离在每台设备上的一次校准内即可复现，但前沿方法内部的顺序无法跨设备或校准日复现。在ibm_fez上开展的n=8深电路运行显示，在复目标上，执行计数排序在运行间波动之外重新显现，且深度未与映射、门组合、校准或会话效应分离。所有方法均精确到机器精度，仅在代价上存在差异。在噪声下，粗误差层级跟踪已执行（路由）的两量子比特计数，将Θ(2ⁿ)前沿方法与QSD的Θ(4ⁿ)及更精细的内容区分开。两种合成方法均实现了已部署Qiskit StatePreparation的2ⁿ-n-1个受控非门（CNOT）下限，在实际构建的CNOT轴上低于所有其他从零开始的方法。实合成适用于带符号的实目标，通过一次快速沃尔什-哈达玛变换，在大寄存器尺寸的精确加载器中具有最低的经典构建代价；复合成适用于任意复目标，在模拟采样误差上与Qiskit的StatePreparation持平。
英文摘要
This thesis studies exact, deterministic preparation of arbitrary dense n-qubit states, the data-loading step in quantum signal and image processing. It derives two syntheses built on the Digital Signal-induced Heap Transform (DsiHT): the QsiHT Fast Path Real Synthesis and the QsiHT Fast Path Complex Synthesis. Both are benchmarked against ten configurations spanning the UCR, isometry, multiplexor, Schmidt/SVD, QSD, and heap-transform families. Several of those are realizations through Qiskit builders or compiler optimization, not from-scratch reimplementations. The n=3 noisy comparison spans ibm_fez, ibm_kingston, and ibm_marrakesh, three 156-qubit IBM Heron r2 processors, where single-submission nine-method jobs permit within-session family-wide Benjamini-Hochberg-corrected comparisons. The frontier-versus-QSD separation reproduces within one calibration on every device, but the within-frontier order does not reproduce across devices or calibration days. A deep-circuit n=8 run on ibm_fez shows the executed-count ordering re-emerge outside the run-to-run spread on the complex target. Depth was not isolated from mapping, gate composition, calibration, or session effects. All methods are exact to machine precision and differ only in cost. Under noise the coarse error tier tracks the executed (routed) two-qubit count, separating the Theta(2^n) frontier from QSD's Theta(4^n) and nothing finer. Both syntheses realize the deployed Qiskit StatePreparation floor of 2^n-n-1 CNOTs, undercutting every other from-scratch method on the as-built CNOT axis. The Real Synthesis serves real (sign-bearing) targets and holds the lowest classical build cost among the exact loaders at large register sizes, through one fast Walsh-Hadamard pass. The Complex Synthesis serves arbitrary complex targets and ties Qiskit's StatePreparation for the lowest simulated sampled error.
