Density Matrix Renormalization Group and Periodic Boundary Conditions: A Quantum Information Perspective

We introduce a picture to analyze the density matrix renormalization group (DMRG) numerical method from a quantum information perspective. This leads to a variational formulation of DMRG which allows for dramatic improvements in the case of problems with periodic boundary conditions. The picture also explains some features of the method in terms of entanglement and teleportation.

Figure 1

Schematic picture of the DMRG method for the $B•B$ (a) and the $B••B$ (b) configurations. Horizontal lines represent maximally entangled states $|\varphi ⟩$, the ellipses and circles (squares) the operators $Q$ ($P$) which map the auxiliary system into the physical ones.

Figure 2

Proposed configurations for the case of PBC. One may also use two spins instead of one.

Figure 3

Left: comparison between DMRG (squares) [2] and the new method (circles) for PBC, and $N=28$. For reference the DMRG results [2] for the Heisenberg chain with OBC (triangles) are also shown. Inset: variation of the local bond strength from the average along the chain, calculated with the new method and $D=40$.

Figure 4

Converting a general 5-qubit state into a MPS via teleportation (see the text).