All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks
This work addresses a bottleneck in geometric quantum machine learning for researchers developing efficient quantum variational algorithms, though it is incremental as it builds on existing equivariance methods.
The paper tackled the challenge of constructing group-equivariant variational quantum circuits by proposing spin networks as a direct building block for SU(2) equivariant circuits, and demonstrated that these circuits improve performance in solving ground state problems for SU(2) symmetric Heisenberg models on specific lattices.
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 ansätze -- 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.