Theoretical Physics has made progress in the research of shrinking arbitrary tensor networks

[ Instrument network instrument research and development ] tensor network has a wide range of applications in physics. In quantum physics, a tensor network can be used as an efficient variational wave function. In statistical physics, a partition function can be converted into a reduction of the tensor network, and then renormalized groups and low-rank approximation methods can be used for effective Calculation; In addition, in quantum computing, quantum circuits can be regarded as a special tensor network with unity, and the calculation of its single amplitude can also be converted into a tensor network shrinkage problem.

However, the classic tensor network shrinking methods, such as the tensor renormalization group method, usually assume that the system is defined on the grid. However, there is no mature method for shrinking tensor networks with irregular connections. The difficulty lies in the fact that the intermediate tensors with huge dimensions will be encountered during the shrinking process, but there is no effective method to approximate them in low dimensions. This limits the application of the tensor network method to a wider range of physical problems.

Zhang Pan, a researcher at the Institute of Theoretical Physics, Chinese Academy of Sciences, and PhD students Pan Feng, Zhou Pengfei, and Li Sujie proposed a new method of tensor network shrinkage, which uses matrix product states to express the intermediate tensors generated in the tensor network shrinkage. The density matrix renormalization group performs an effective approximation, compresses the dimensionality of the intermediate tensor, and then can shrink any tensor network. Each fourth-order tensor in (1) is represented as a matrix product state (MPS) in (2), and then different MPSs are merged into longer MPSs, and their dimensionality is reduced by a low-rank approximation, and finally (9 ) In the condensed result. The new method has shown strong ability in calculation of free energy in statistical physics. Compared with traditional methods, it has higher calculation accuracy and faster calculation speed in the calculation of free energy of spin glass with multiple topologically connected structures. In addition, the new method can perform classical simulations of shallow quantum circuits with smaller calculation errors, and has the potential to simulate and verify noise-containing medium-sized (NISQ) quantum computers.

This research work was supported by the National Natural Science Foundation of China's general project and the Chinese Academy of Sciences' Frontier Science Key Research Project.