  State distillation is the process of taking a number of imperfect copies of a
particular quantum state and producing fewer better copies. Until recently, the
lowest overhead method of distilling states |A>=(|0>+e^{i\pi/4}|1>)/\sqrt{2}
produced a single improved |A> state given 15 input copies. New block code
state distillation methods can produce k improved |A> states given 3k+8 input
copies, potentially significantly reducing the overhead associated with state
distillation. We construct an explicit surface code implementation of block
code state distillation and quantitatively compare the overhead of this
approach to the old. We find that, using the best available techniques, for
parameters of practical interest, block code state distillation does not always
lead to lower overhead, and, when it does, the overhead reduction is typically
less than a factor of three.
