Implementing the sine transform of fermionic modes as a tensor network

Based on the algebraic theory of signal processing, we recursively decompose the discrete sine transform of the first kind (DST-I) into small orthogonal block operations. Using a diagrammatic language, we then second-quantize this decomposition to construct a tensor network implementing the DST-I for fermionic modes on a lattice. The complexity of the resulting network is shown to scale as $\frac{5}{4}nlogn$ (not considering swap gates), where $n$ is the number of lattice sites. Our method provides a systematic approach of generalizing Ferris' spectral tensor network for nontrivial boundary conditions.

Figure 1

Orthogonalization of the recursion relation Eq. (7a) for the DST-I of size $n=2^{k+1}-1$, shown here for $k=2$.

Figure 2

Orthogonalization of the recursion relation (8a) for the DST-III of size $n=2^{k+1}$. Again, we have $k=2$. The part in the dashed box in the right diagram is considered in Fig. 3.

Figure 3

Detail for the orthogonalization of the recursion relation (8a) for the DST-III of size $n=2^{k+1}$, shown here for $k=3$, to resolve all relevant details. This figure corresponds to the part in the dashed box in the right diagram of Fig. 2 with $m=2^k$. The dotted box in the diagram on the left-hand side of this figure indicates the part that remains to be reformulated in terms of orthogonal operations, as shown on the right-hand side.

Figure 4

Diagrammatic representation of ${\widehat{\mathcal{S}}}_{31}^{\mathrm{I}}$ made up entirely from orthogonal operations acting on only two components. The diagram is obtained from the recursion relations (14) and (15), together with the corresponding closing conditions (16).

Figure 5

Second quantization of the FFT diagram from Eq. (1b). First, we unravel the shorthand notation (3). Then each block is replaced by its second quantization via Eqs. (24) and (25). The overall structure of the diagrams is unchanged because of the functorial relations Eq. (23), however, whenever two lines cross, we have to insert the fermion swap gate ${\mathrm{\Gamma }}_S$, obtained in Eq. (26).