  In usual (non-stochastic) tensor network calculations, the truncated singular
value decomposition (SVD) is often used for approximating a tensor, and it
causes systematic errors. By introducing stochastic noise in the approximation,
however, one can avoid such systematic errors at the expense of statistical
errors which can be straightforwardly controlled. Therefore in principle, exact
results can be obtained even at finite bond dimension up to the statistical
errors. A previous study of the unbiased method implemented in tensor
renormalization group (TRG) algorithm, however, showed that the statistical
errors for physical quantity are not negligible, and furthermore the
computational cost is linearly proportional to a system volume. In this paper,
we introduce a new way of stochastic noise such that the statistical error is
suppressed, and moreover, in order to reduce the computational cost we propose
common noise method whose cost is proportional to the logarithm of volume. We
find that the method provides better accuracy for the free energy compared with
the truncated SVD when applying to TRG for Ising model on square lattice.
Although the common noise method introduces systematic error originated from a
correlation of noises, we show that the error can be described by a simple
functional form in terms of the number of noises, thus the error can be
straightforwardly controlled in an actual analysis. We also apply the method to
the graph independent local truncation algorithm and show that the accuracy is
further improved.
