"Adaptive" Schmidt Dimension
One thing that can save a lot of computational time without loss of accuracy is to use a different Schmidt dimension for each bond instead of a fixed one for all bonds, keeping only the necessary amount of relevant coefficients, as usual. For example, taking the first bond, in the case of qubits, the Schmidt dimension is just two. Hence, at the first bond, instead of futilely diagonalizing, let us say, 10 by 10 or 20 by 20 matrices, we can just restrict ourselves to ordinary 2 by 2 ones, thus making the algorithm generally faster. What we can do instead is set a threshold for the eigenvalues of the SD, keeping only those that are above the threshold.
TEBD also offers the possibility of straightforward parallelization due to the factorization of the exponential time-evolution operator using the Suzuki-Trotter expansion. A parallel-TEBD has the same mathematics as its non-parallelized counterpart, the only difference is in the numerical implementation.
Read more about this topic: Time-evolving Block Decimation
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