  Spin network states are a powerful tool for constructing the $SU(2)$ gauge
theories on a graph. In loop quantum gravity (LQG), they have yielded many
promising predictions, although progress has been limited by the computational
challenge of dealing with high-dimensional Hilbert spaces. To explore more
general configurations, quantum computing methods can be applied by
representing spin network states as quantum circuits. In this article, we
introduce an improved method for constructing quantum circuits for 4-valent
Ising spin networks, which utilizes a smaller number of qubits than previous
approaches. This has practical implications for the implementation of quantum
circuits. We also demonstrate the procedure with various examples, including
the construction of a 10-node Ising spin network state. The key ingredient of
the method is the variational transfer of partial states, which we illustrate
through numerous examples. Our improved construction provides a promising
avenue for further exploring the potential of quantum computing methods in
quantum gravity research.
