An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations

Abstract

Many quantum computation algorithms, and processes like measurement based quantum computing, require the initial state of the quantum computer to be an eigenstate of a specific unitary operator. Here we study how quantum states that are eigenstates of finite dimensional irreducible representations of the special unitary (SU(d)) and the permutation (S_n) groups can be efficiently constructed in the computational basis formed by tensor products of the qudit states. The procedure is a unitary transform, which first uses Schur-Weyl duality to map every eigenstate to a unique Schur basis state, and then recursively uses the Clebsch - Gordan transform to rotate the Schur basis state to the computational basis. We explicitly provide an efficient quantum algorithm, and the corresponding quantum logic circuit, to generate any desired eigenstate of SU(2) and SU(3) irreducible representations in the computational basis.