This work investigates the relationships between quantum state synthesis complexity classes (a recent concept in computational complexity that focuses on the complexity of preparing quantum states) and traditional decision complexity classes. We especially investigate the role of the synthesis error parameter, which characterizes the quality of the synthesis in quantum state synthesis complexity classes. We first show that in the high synthesis error regime, collapse of synthesis classes implies collapse of the equivalent decision classes. For more reasonable synthesis error, we then show a similar relationships for BQP and QCMA. Finally, we show that for quantum state synthesis classes it is in general impossible to improve the quality of the synthesis: unlike the completeness and soundness parameters (which can be improved via repetition), the synthesis error cannot be reduced, even with arbitrary computational power.

翻译：本研究探讨了量子态合成复杂性类（计算复杂性理论中关注量子态制备复杂性的新兴概念）与传统判定复杂性类之间的关系。我们特别研究了合成误差参数在量子态合成复杂性类中的作用，该参数表征了量子态合成的质量。我们首先证明在高合成误差区域，合成复杂性类的坍缩会引发等价判定复杂性类的坍缩。针对更合理的合成误差范围，我们进一步展示了BQP与QCMA复杂性类之间的类似关联关系。最后，我们证明对于量子态合成复杂性类而言，合成质量在本质上无法提升：与可重复性优化的完备性和稳健性参数不同，即使拥有任意计算资源，合成误差仍具有不可约简性。