Computing Kronecker products with cuBlasLt

One of the operations that I’d like to run on an RTX 2080-based GPU (for a quantum computer simulator), is Kronecker (tensor) products of square (specifically 2^n dimension unitary) matrices. One typical example would be to compute a 2^7 dimension matrix by taking the following tensor product of some Hadamard matrices and a Pauli-X matrix, where H and X are two-dimension square matrices.
X ⊗ H ⊗ H ⊗ H ⊗ H ⊗ H ⊗ H

I’d like to use single-precision complex numbers so that my 8GB GPU can do calculations that result in > 2^30 dimension matrices. Can something like this be done with the cuBlasLt library, or is there a better choice?

James Weaver