Scaling of symmetry-restricted quantum circuits

The intrinsic symmetries of physical systems have been employed to reduce the number of degrees of freedom of systems, thereby simplifying computations. In this work, we investigate the properties of MSU (2N), M-invariant subspaces of the special unitary Lie group SU (2N) acting on N qubits, for some M M₂₍ (C). We demonstrate that for certain choices of M, the subset MSU (2N) inherits many topological and group properties from SU (2N). We then present a combinatorial method for computing the dimension of such subspaces when M is a representation of a permutation group acting on qubits (GSU (2N) ), or a Hamiltonian (H^ (N) SU (2N) ). The Kronecker product of su (2) matrices is employed to construct the Lie algebras associated with different permutation-invariant groups GSU (2N). Numerical results on the number of dimensions support the the developed theory.