All you need is spin: SU equivariant variational quantum circuits based on spin networks

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.

On the geometry of learning neural quantum states

Combining insights from machine learning and quantum Monte Carlo, the stochastic reconfiguration method with neural network Ansatz states is a promising new direction for high precision ground state estimation of quantum many body problems. At present, the method is heuristic, lacking a proper theoretical foundation. We initiate a thorough analysis of the learning landscape, and show that it reveals universal behavior reflecting a combination of the underlying physics and of the learning dynamics. In particular, the spectrum of the quantum Fisher matrix of complex restricted Boltzmann machine states can dramatically change across a phase transition. In contrast to the spectral properties of the quantum Fisher matrix, the actual weights of the network at convergence do not reveal much information about the system or the dynamics. Furthermore, we identify a new measure of correlation in the state by analyzing entanglement the eigenvectors. We show that, generically, the learning landscape modes with least entanglement have largest eigenvalue, suggesting that correlations are encoded in large flat valleys of the learning landscape, favoring stable representations of the ground state.