All you need is spin: SU(2) equivariant variational quantum circuits
  based on spin networks
Richard D. P. East Guillermo Alonso-Linaje Chae-Yeun Park
Variational algorithms require architectures that naturally constrain the
optimisation space to run efficiently. In geometric quantum machine learning,
one achieves this by encoding group structure into parameterised quantum
circuits to include the symmetries of a problem as an inductive bias. However,
constructing such circuits is challenging as a concrete guiding principle has
yet to emerge. In this paper, we propose the use of spin networks, a form of
directed tensor network invariant under a group transformation, to devise SU(2)
equivariant quantum circuit ans\"atze -- circuits possessing spin rotation
symmetry. By changing to the basis that block diagonalises SU(2) group action,
these networks provide a natural building block for constructing parameterised
equivariant quantum circuits. We prove that our construction is mathematically
equivalent to other known constructions, such as those based on twirling and
generalised permutations, but more direct to implement on quantum hardware. The
efficacy of our constructed circuits is tested by solving the ground state
problem of SU(2) symmetric Heisenberg models on the one-dimensional triangular
lattice and on the Kagome lattice. Our results highlight that our equivariant
circuits boost the performance of quantum variational algorithms, indicating
broader applicability to other real-world problems.