Novel and improved algorithms for the contraction of 2D tensor networks
Author(s)
Lan, Wangwei
Advisor(s)
Evenbly, Glen
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Abstract
Tensor network algorithms are important numerical tools for studying quantum many-body problems. However, the high computational costs have prevented its applications in two-dimensional (2D) systems. In this thesis, we discussed our work on more efficient contractions of 2D tensor networks. In particular, for 2D statistical mechanics, we propose a modified form of a tensor renormalization group algorithm for evaluating partition functions of classical statistical mechanical models on 2D lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this new approach only scales as O(χ4) in terms of the bond dimension χ, significantly cheaper than the O(χ6) cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension χ. In 2D quantum mechanics, we propose a pair of approximations that allows the leading order computational cost of contracting an infinite projected entangled-pair state (iPEPS) to be reduced from O(χ3D6) to O(χ3D3) when using a corner-transfer approach. The first approximation involves (i) reducing the environment needed for truncation of the boundary tensors (ii) relies on the sequential contraction and truncation of bra and ket indices, rather than doing both together as with the established algorithm. Our benchmark results are comparable to the standard iPEPS algorithm. The improvement in computational cost enables us to perform large bond dimension
calculations, extending its potential to solve challenging problems.
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Date
2022-12-08
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Dissertation