When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the T-count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure.

当用量子电路准备纯态时，由于容错实现中的编译错误，不可避免地会出现近似误差。最近提出的一种称为概率状态合成的方法，其中对电路进行概率采样，与传统的确定性合成相比，能够减少近似误差。在本文中，我们证明了最佳概率综合可以二次减少近似误差。此外，我们表明确定性综合算法可以有效地转换为概率性算法，从而实现二次误差减少。我们还以数值方式演示了这种转换如何减少T计数，并通过分析证明这种转换将电路尺寸的信息论下限减半。为了得出这些结果，我们证明了关于量子态的最佳凸近似的一般定理。此外，我们证明该定理可用于分析纠缠度量。