Variational algorithms require architectures that naturally constrain the optimization space to run efficiently. Geometric quantum machine learning achieves this goal by encoding group structure into parameterized 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 $\unicode{x2013}$ circuits possessing spin-rotation symmetry. By changing to the basis that block diagonalizes the SU(2) group action, these networks provide a natural building block for constructing parameterized equivariant quantum circuits. We prove that our construction is mathematically equivalent to other known constructions, such as those based on twirling and generalized 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 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.

翻译：变分算法需要能够自然约束优化空间的结构以实现高效运行。几何量子机器学习通过将群结构编码到参数化量子电路中，从而将问题的对称性作为归纳偏置，实现了这一目标。然而，由于缺乏具体的指导原则，构建此类电路仍具有挑战性。本文提出利用自旋网络（一种在群变换下保持不变的定向张量网络）来设计SU(2)等变量子电路拟设——即具有自旋旋转对称性的电路。通过转换到块对角化SU(2)群作用的基底下，这些网络为构建参数化等变量子电路提供了天然的基础模块。我们证明，该构造在数学上等价于其他已知构造（如基于旋转和广义置换的构造），但在量子硬件上实现更为直接。通过求解一维三角晶格和Kagome晶格上SU(2)对称海森堡模型的基态问题，我们测试了所构造电路的有效性。结果表明，我们的等变电路提升了量子变分算法的性能，显示出对其他现实问题的广泛适用性。