Floating point representations in quantum circuit synthesis
Nathan Wiebe, Vadym Kliuchnikov · New Journal of Physics · 2013
We provide a non-deterministic quantum protocol that approximates the single qubit rotations R x (2 ϕ 2 1 ϕ 2 2 ) using R x (2 ϕ 1 ) and R x (2 ϕ 2 ) and a constant number of Clifford and T operations. We then use this method to construct a 'floating point' implementation of a small rotation wherein we use the aforementioned method to construct the exponent part of the rotation and also to combine it with a mantissa. This causes the cost of the synthesis to depend more strongly on the relative (rather than absolute) precision required. We analyze the mean and variance of the T -count required to use our techniques and provide new lower bounds for the T -count for ancilla free synthesis of small single-qubit axial rotations. We further show that our techniques can use ancillas to beat these lower bounds with high probability. We also discuss the T -depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths.