  Dual unitary gates are highly non-local two-qudit unitary gates that have
been studied extensively in quantum many-body physics and quantum information
in the recent past. A special subset of dual unitary gates consists of
rank-four perfect tensors, which are equivalent to highly entangled
multipartite pure states called absolutely maximally entangled (AME) states. In
this work, numerical and analytical constructions of dual unitary gates and
perfect tensors that are diagonal in a special maximally entangled basis are
presented. The main ingredient in our construction is a phase-valued
(unimodular) two-dimensional array whose discrete Fourier transform is also
unimodular. We obtain perfect tensors for several local Hilbert space
dimensions, particularly, in dimension six. A perfect tensor in local dimension
six is equivalent to an AME state of four qudits, denoted as AME(4,6), and such
a state cannot be constructed from existing constructions of AME states based
on error-correcting codes and graph states. The existence of AME(4,6) states
featured in well-known open problem lists in quantum information, and was
settled positively in Phys. Rev. Lett. 128 080507 (2022). We provide an
explicit construction of perfect tensors in local dimension six that can be
written in terms of controlled unitary gates in the computational basis, making
them amenable for quantum circuit implementations.
