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.

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