Preparing quantum samples (QSAMPLES), coherent encodings of stationary distributions of reversible Markov chains, is a fundamental primitive in quantum sampling, particularly for quantum simulated annealing. A central limitation of existing phase-estimation-based frameworks is the ancilla qubit overhead. In this work, we present a new end-to-end framework requiring only one ancilla qubit in the working register. The key technical ingredient is a selective phase compiler circuit using one ancilla qubit, built from a generalized quantum signal processing (GQSP)-based projector onto the 1-eigenspace of the qubitized Szegedy walk. Embedding these selective phase compilers into the fixed-point amplitude amplification (FPAA) procedure and iterating yields a quantum algorithm that, given an initial state, oracle access, lower bounds on the overlaps between adjacent states, and lower bounds on the phase gaps, outputs a QSAMPLE within any desired trace distance and thus total variation distance. The query complexity scales inversely with the square roots of both the minimum overlap and the minimum spectral gap of the Markov chains across the cooling schedule, up to polylogarithmic factors. We also perform simulations to verify how our qubit and query complexity evolve with the trace distance, and how this work compares to the previous framework. These results establish two improvements over the previous framework by Wocjan and Abeyesinghe. First, the working-register ancilla cost is reduced to one. Second, by inserting our GQSP-based selective phase compiler into the FPAA procedure, we improve the QSAMPLE transport overlap dependence from inverse minimum overlap to inverse square-root minimum overlap, relative to their Grover pi-over-three fixed-point method. Finally, as a direct application, we apply the quantum algorithm to prepare a Gibbs QSAMPLE and obtain a rigorous complexity analysis.

翻译：制备量子样本（QSAMPLE），即可逆马尔可夫链平稳分布的相干编码，是量子采样中的基本原语，尤其在量子模拟退火中。现有基于相位估计的框架的一个核心限制是辅助量子比特的开销。在本工作中，我们提出一个新的端到端框架，仅需在工作寄存器中使用一个辅助量子比特。关键技术要素是一个使用一个辅助量子比特的选择性相位编译器电路，该电路基于广义量子信号处理（GQSP）构建的投影算符，投影到量子化 Szegedy 游走的 1-特征空间。将这些选择性相位编译器嵌入到定点振幅放大（FPAA）过程中并迭代，得到一个量子算法，该算法给定初始态、预言机访问、相邻态之间重叠的下界以及相位间隙的下界，能在任意所需的迹距离（进而全变差距离）内输出一个 QSAMPLE。查询复杂度与冷却时间表上马尔可夫链的最小重叠和最小谱间隙的平方根的倒数成正比，直到 polylog 因子。我们还进行仿真，以验证我们的量子比特和查询复杂度如何随迹距离变化，以及本工作与先前框架的比较。这些结果确立了相对于先前 Wocjan 和 Abeyesinghe 框架的两项改进。第一，工作寄存器的辅助量子比特成本降低到一个。第二，通过将我们的基于 GQSP 的选择性相位编译器插入 FPAA 过程，我们改进了 QSAMPLE 传输重叠依赖性，从逆最小重叠改进为逆平方根最小重叠，相对于他们的 Grover π/3 定点方法。最后，作为直接应用，我们应用该量子算法制备 Gibbs QSAMPLE，并获得了严格的复杂度分析。