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WIREs Comput Mol Sci
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Low entanglement wavefunctions

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Abstract We review a class of efficient wavefunction approximations that are based around the limit of low entanglement. These wavefunctions, which go by such names as matrix product states and tensor network states, occupy a different region of Hilbert space from wavefunctions built around the Hartree–Fock limit. The best known class of low entanglement wavefunctions, the matrix product states, forms the variational space of the density matrix renormalization group algorithm. Because of their different structure to many other quantum chemistry wavefunctions, low entanglement approximations hold promise for problems conventionally considered hard in quantum chemistry, and in particular problems which have a multireference or strong correlation nature. In this review, we describe low entanglement wavefunctions at an introductory level, focusing on the main theoretical ideas. Topics covered include the theory of efficient wavefunction approximations, entanglement, matrix product states, and tensor network states including the tree tensor network, projected entangled pair states, and the multiscale entanglement renormalization ansatz. © 2012 John Wiley & Sons, Ltd. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

The Hilbert space of many‐electron wavefunctions. The independent particle and independent local subsystem limits lie in different regions of the Hilbert space. By increasing the excitation level or entanglement, respectively, in approximations built around these limits, they can be made to span the full Hilbert space.

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(Left) The original real space RG is associated with a tree structure for the wavefunction, where a layer of sites is subjected to successive coarse grainings (isometries, W). However, the coarse grainings within the blocks neglect to take into account entanglement at the boundaries of the blocks. (Right) This is rectified within the MERA by first performing a disentangling operation U between the block boundaries before coarse graining.

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The multiscale entanglement renormalization ansatz is constructed from two types of tensors: isometries W, which map the states of a block of sites to a single effective sites; and disentanglers U, which perform a unitary rotation of the basis of two or more sites.

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Naive contraction of the square lattice of tensors that arises from the projected entangled pair states (PEPS) norm evaluation is of exponential cost. Instead, we consider first a contraction of two columns, starting from the left edge. We then view the dangling bonds (pointing toward the right, here r1, r2, r3, r4) as the local state indices of a fictitious wavefunction . We can imagine approximating such a wavefunction by a matrix product state (MPS) of a small auxiliary dimension, . This effectively approximates the two column structure by a single‐column structure. Repeating this for the remaining columns of the ‐square lattice, we can contract the full PEPS from left to right efficiently. The final top‐to‐bottom contraction involves only a single column and is of the same complexity as an MPS norm evaluation.

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(Bottom) Projected entangled pair states (PEPS) norm evaluation can be represented by fusing a PEPS bra lattice (pointing downward) with a PEPS ket lattice (pointing upward). The result is a square lattice which must be contracted over the auxiliary indices. (Top) Each contraction of the local states |np〉 results in an tensor with four auxiliary indices, each of dimension M2.

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Projected entangled pair states (PEPS) are associated with arbitrary networks of entanglement. Shown here is a representation of the PEPS for a square lattice of sites. Each local state |np〉 is then associated with a tensor with four auxiliary indices l, r, u, d that point along the bonds of the square lattice. The PEPS approximation for the wavefunction coefficient is obtained by contracting all the auxiliary indices.

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(Left) For every local state |np〉, a tree tensor network (TTN) associates a tensor with z auxiliary indices. Here shown is a TTN where each tensor has three auxiliary indices. (Right) The wavefunction coefficient is obtained by contracting the auxiliary indices according to the tree connectivity

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(Bottom) A norm can be represented graphically by fusing the vertical indices of the matrix product state (MPS) bra (pointing downward) with the vertical indices of the MPS ket (pointing upward), denoting summation over all the state indices, n1, n2nk. (Top) Performing a state summation at each site leads to an intermedate matrix of dimension M2, and the norm evaluation is a product of the matrices. (Note that the leftmost and rightmost matrices are, in practice, row and column vectors.)

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Area laws express locality in physical systems. In a one‐dimensional system, the boundary area is independent of the overall system size, thus entanglement entropy in a physical ground state is expected to be independent of system size, unless at a quantum critical point. In two‐dimensional system, the boundary area can scale as the system width, and the entanglement entropy scales similarly, except for critical and topological corrections.

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Graphical representation of matrix product states. (Left) For every state of a site |np〉, there is a matrix . The indices of the matrix, ip − 1 and ip, are referred to as auxiliary indices and are of dimension M. Note that the leftmost and rightmost matrices are, in practice, row and column vectors. In this way, the product over all the matrices yields a scalar, the wavefunction coefficient . (Right) To obtain the matrix product state approximation to the wavefunction coefficient , we contract over all the auxiliary indices in the product . Graphically, the contraction is represented by joining together the auxiliary index lines.

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