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When preparing a pure state with a quantum circuit, there is an inevitable
coherent error since each unitary gate suffers from the discretized coherent
error due to fault-tolerant implementation. A recently proposed approach called
probabilistic state synthesis, where the circuit is probabilistically sampled
to turn such coherent errors into incoherent ones, is able to reduce the order
of the approximation error compared to conventional deterministic synthesis. In
this paper, we demonstrate that the optimal probabilistic synthesis
quadratically reduces the approximation error with respect to the trace
distance. We also show that a deterministic synthesis algorithm can be
efficiently converted into a probabilistic one to achieve quadratic error
reduction. To estimate how the error reduction reduces the circuit size, we
show that probabilistic encoding asymptotically halves the length of the
classical bit string, which provides a general lower bound on the circuit size,
required to approximately encode a pure state. To derive these results, we
prove general theorems about the optimal convex approximation of a quantum
state by using a restricted subset of quantum states. As another application of
our theorem, we provide exact formulas for the minimum trace distance between
an entangled state and the set of separable states and alternate proof about a
recently identified coincidence between an entanglement measure and a coherence
measure.