Abstract
We describe the hashing technique for obtaining a fast approximation of a target quantum gate in the unitary group SU(2) represented by a product of the elements of a universal basis. The hashing exploits the structure of the icosahedral group (or other finite subgroups of SU(2)) and its pseudogroup approximations to reduce the search within a small number of elements. One of the main advantages of the pseudogroup hashing is the possibility of iterating to obtain more accurate representations of the targets in the spirit of the renormalization group approach. We describe the iterative pseudogroup hashing algorithm using the universal basis given by the braidings of Fibonacci anyons. An analysis of the efficiency of the iterations based on the random matrix theory indicates that the runtime and braid length scale poly-logarithmically with the final error, comparing favorably to the Solovay–Kitaev algorithm.
Highlights
The possibility of physically implementing quantum computation opens new scenarios in the future technological development
We describe the hashing technique to obtain a fast approximation of a target quantum gate in the unitary group SU(2) represented by a product of the elements of a universal basis
For a generic universal topological quantum computer, that the iterative pseudogroup hashing algorithm allows an efficient search for a braid sequence that approximates an arbitrary given single-qubit gate
Summary
The possibility of physically implementing quantum computation opens new scenarios in the future technological development. This approach, consisting of a search algorithm over all the possible ordered product up to a certain length, has an extremely clear representation if one encodes qubits using non-abelian anyons In this case, the computational basis for the quantum gate is the set of the elementary braidings between every pair of anyons, and their products are represented by the braids describing the world lines of these anyonic quasiparticles. The computational basis for the quantum gate is the set of the elementary braidings between every pair of anyons, and their products are represented by the braids describing the world lines of these anyonic quasiparticles Starting from this kind of universal basis, the search among all the possible products of N elements gives, the optimal result, but the search time is exponential in N and it is impractical to reach sufficiently small error for arbitrary gates. In Appendix A we derive the distribution of the best approximation in a given set of braids, which can be used to estimate the performance of, e.g., the brute-force search
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