  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.
