The proposed Variational Quantum Compilation (VQC) approach can be extended to larger system sizes and higher dimensions by adopting several strategies. One promising direction is to implement a local update scheme, where each layer of the quantum circuit is optimized sequentially. This method is akin to techniques used in density matrix renormalization group (DMRG) algorithms, which are effective for solving ground states in many-body systems. By optimizing each layer independently, the VQC can potentially navigate the parameter space more effectively, avoiding the pitfalls of barren plateaus that often arise in high-dimensional optimization landscapes.
Additionally, incorporating advanced tensor network techniques can facilitate the representation of quantum states in higher dimensions. For instance, using 2D isometric tensor networks can provide a more natural framework for simulating dynamics in two-dimensional systems, allowing for efficient computation of time-evolved states. This approach can leverage the inherent structure of the quantum dynamics, enabling the VQC to handle larger Hilbert spaces without a significant increase in computational cost.
Moreover, exploring the use of symmetries in the Hamiltonians can also enhance the scalability of the VQC. By recognizing and exploiting symmetries, such as translational invariance or conservation laws, the compilation process can be streamlined, reducing the complexity of the circuits needed for accurate simulations. This can lead to a more efficient training process, requiring fewer resources and less data to achieve high fidelity in the compiled circuits.