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.

我们提出旋转网络的使用，它是一种在群变换下保持不变的有向张量网络，用于设计具有自旋旋转对称性的参数化等变量量子电路。我们证明了该构建与其他已知构建的数学等价性，并且在量子硬件上更直接实现。我们通过求解一维三角晶格和 Kagome 晶格上具有 SU (2) 对称性的海森堡模型的基态问题来测试我们构建的电路的有效性，并且结果表明我们的等变电路提升了量子变分算法的性能，暗示了对其他实际问题的更广泛适用性。