for the Trotter expansion, the error after the truncation is: After moving to the next bond, the state is, similarly: The error, after the second truncation, is: and so on, as we move from bond to bond. The second error source enfolded in the decomposition is more subtle and requires a little bit of calculation. As we calculated before, the normalization constant after making the truncation at bond is: Now let us go to the bond and calculate the norm of the right-hand Schmidt vectors ; taking into account the full Schmidt dimension, the norm is: , where . Taking into account the truncated space, the norm is: Taking the difference, , we get: Hence, when constructing the reduced density matrix, the trace of the matrix is multiplied by the factor: The total truncation error The total truncation error, considering both sources, is upper bounded by: When using the Trotter expansion, we do not move from bond to bond, but between bonds of same parity; moreover, for the ST2, we make a sweep of the even ones and two for the odd. But nevertheless, the calculation presented above still holds. The error is evaluated by successively multiplying with the normalization constant, each time we build the reduced density matrix and select its relevant eigenvalues. "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. References Quantum mechanics Computational physics