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State distillation is the process of taking a number of imperfect copies of aparticular quantum state and producing fewer better copies. Until recently, thelowest 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 codestate distillation methods can produce k improved |A> states given 3k+8 inputcopies, potentially significantly reducing the overhead associated with statedistillation. We construct an explicit surface code implementation of blockcode state distillation and quantitatively compare the overhead of thisapproach to the old. We find that, using the best available techniques, forparameters of practical interest, block code state distillation does not alwayslead to lower overhead, and, when it does, the overhead reduction is typicallyless than a factor of three.

Surface code implementation of block code state distillation