Encoding quantum circuits in optimal tensor network structures with gradient descent methods

Tensor networks are a well-known structure for the simulation of quantum systems and a useful alternative to current quantum devices, also being flexible enough to have applications in other computing fields such as machine learning. The expressibility available to a specific tensor network depends on how big the bond dimensions between tensors are and how much they are connected, at the cost of computational difficulty. Structures that are adapted to a specific problem offer a way to achieve higher faithfulness while keeping the computational cost down but are in general hard to find. Here we propose an algorithm to find optimal structures for an arbitrary problem and showcase some examples where a tree structure offers an advantage over a chain while not being much more costly. To achieve so, we use different flavours of gradient descent to iterate over the connections between tensors. The tools used in this algorithm have been developed specifically with high-performance computing in mind so that these structures can be used in large simulations to explore quantum systems that are not yet within reach of quantum computing. In addition, it allows us to identify correlation patterns in complex condensed matter systems, for example, so that multipartite entanglement can be better understood and characterized.