We have already used the fact that FSMs and MPOs are not unique, and for the arithmetics we introduced the maximally branched representation. On the other hand, MPOs with a small bond dimensionw are desirable in order to decrease the necessary matrix-product state (MPS) truncation after the application of the MPO. Note that the bond dimension of an MPO is equal to the number of nodes of an FSM by which it is created. We are therefore interested in a lossless8 way to obtain a representation with as few nodes as possible, especially for FSMs that result of FSM arithmetics.
The topic of obtaining the optimal MPO representation of a general operator is also discussed in [HMS17] in terms of MPO arithmetics. Here, we present similar compression schemes based on graph manipulations and show the connection to the schemes in [HMS17].
Note that not all compressions presented here are used in our current implementation, because sums of different operators on a single edge are not supported, yet. We present them, nevertheless, in this general way in order to show possible enhancements and as a guideline to the manual construction of FSMs.
Deparallelization
The idea behind the deparallelization [McC07] is to combine columns in the MPO site tensor, or analogously in the state-transition table, which only differ by a constant factor in every element and store these factors in a transfer matrixT, which is afterwards multiplied onto the next site.
The numerical deparallelization acts purely on the MPO site tensors and is therefore naturally capable of handling site-dependent parameters. In the case of the symbolic deparallelization, which is applied on FSMs, the site-dependent parameters are only handled correctly if they are not moved away from their corresponding operator or the distance is unique and stored within the parameters.
8In this case, it is important to obtain a lossless compression, because it is possible that the resulting MPO will be applied very often, which can lead to an accumulating error.
62 Chapter 7. Finite-State Machines
The procedure and the differences between the numeric and the symbolic realization can best be described by comparing graphs and state-transition tables of a concrete example. Note that we consider the operator-valued state-transition table as direct representation of the MPO site tensor and assume site-independent parameters for this example in order to be able to directly apply the transfer matrix from the left, leading always to square matrices. Consider the uncompressed FSM in its maximally branched form with its graph representation and its state-transition table given in Fig. 7.9.
Figure 7.9: (Adopted from Fig. 7 in [PKM17]) Initial maximally-branched example FSM for the subsequent compression via deparallelization. (Left) the graph representation, (right) the state-transition table.
The columns A1 and A2 are equal and hence can be merged. Simultaneously, the edges from node I to node A1 and to node A2 are identical and therefore the nodes A1 and A2 can be merged. The resulting FSM is given in Fig. 7.10.
At this point, the columns A3 and A4 differ only by a factor β/α and can be compressed by the numerical deparallelization method. The transfer matrix includes the two parameters αand β on the same row, resulting in a sum in the MPO site tensor. In order to obtain the same compression within the FSM, we first need to move the parameters to their corresponding next edge (αˆk2⊗kˆ3→kˆ2⊗α[−1]kˆ3 and βˆk2⊗kˆ4→kˆ2⊗β[−1]ˆk4)9. Afterwards, the twoˆk2 edges can be joined, leading to a sum on the last edge as depicted in Fig. 7.11.
Note that the symbolic deparalellization is fully automatized and lossless. A row-wise deparal-lelization corresponds with merging equal edges starting from the final node and can be applied subsequently.
Delinearization
The delinearization exploits linear dependencies between columns in the reshaped MPO site tensors. As this method heavily depends on sums of operators, we present it here only for completeness and as an impulse for further development.
The idea is, again, best illustrated by an example. Therefore, consider the MPO represented by the FSM in Fig. 7.12.
9Here we introduce an offset in superscripted squared brackets that indicates to which site the parameter belongs. This becomes important when the parameters are site-dependent.
Section 7.3. Compressing MPOs in Terms of FSMs 63
Figure 7.10: (Adopted from Fig. 7 in [PKM17]) Intermediate state of the compression of the example FSM. Again, (left) the graph representation, (right) the state-transition table. The identical edges from node I to A1 and to A2 are merged into one edge to the new nodeB. This graphical compression is identical to a first step of the deparallelization, which also would return the compressed MPO site tensor after applying the transfer matrix.
Figure 7.11: Fully compressed FSM. Again, (left) the graph representation, (right) the state-transition table.
It is not obvious how to compress this FSM from the graph representation, but it can be read off the state-transition table that column C is the sum of columns A and B with prefactors α and β. In order to utilize this insight, it is necessary to find corresponding prefactors for the transition from A to F and from B to F, so that discarding the row and the column denoted withCis compensated. In [HMS17] this is done numerically via an algorithm that involves a QR decomposition of a matrix that includes all columns that can be used to construct the column at hand. In this context, a column can be used to construct another column if it does not have a finite element in a row in which the column at hand has a zero. This condition is employed in order to prevent errors due to numerical precision that could arise by subtracting nearly equal values. An example for the resulting compressed FSM is shown in Fig. 7.13.
64 Chapter 7. Finite-State Machines
Figure 7.12: Initial example FSM that is not further compressible via deparallelization for the subsequent compression via delinearization. (Left) the graph representation, (right) the state-transition table. The linear dependence of column C from the columnsA andB can be read off directly from the state-transition table.
Figure 7.13: Fully compressed FSM. (Left) the graph representation, (right) the state-transition table.