10,616 research outputs found
Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
In this paper we explore the practical use of the corner transfer matrix and
its higher-dimensional generalization, the corner tensor, to develop tensor
network algorithms for the classical simulation of quantum lattice systems of
infinite size. This exploration is done mainly in one and two spatial
dimensions (1d and 2d). We describe a number of numerical algorithms based on
corner matri- ces and tensors to approximate different ground state properties
of these systems. The proposed methods make also use of matrix product
operators and projected entangled pair operators, and naturally preserve
spatial symmetries of the system such as translation invariance. In order to
assess the validity of our algorithms, we provide preliminary benchmarking
calculations for the spin-1/2 quantum Ising model in a transverse field in both
1d and 2d. Our methods are a plausible alternative to other well-established
tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in
2d. The computational complexity of the proposed algorithms is also considered
and, in 2d, important differences are found depending on the chosen simulation
scheme. We also discuss further possibilities, such as 3d quantum lattice
systems, periodic boundary conditions, and real time evolution. This discussion
leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of
corner transfer matrices and corner tensors. Our paper also offers a
perspective on many properties of the corner transfer matrix and its
higher-dimensional generalizations in the light of novel tensor network
methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on
some of the algorithms have been moved to appendices. To appear in PR
Lecture Notes of Tensor Network Contractions
Tensor network (TN), a young mathematical tool of high vitality and great
potential, has been undergoing extremely rapid developments in the last two
decades, gaining tremendous success in condensed matter physics, atomic
physics, quantum information science, statistical physics, and so on. In this
lecture notes, we focus on the contraction algorithms of TN as well as some of
the applications to the simulations of quantum many-body systems. Starting from
basic concepts and definitions, we first explain the relations between TN and
physical problems, including the TN representations of classical partition
functions, quantum many-body states (by matrix product state, tree TN, and
projected entangled pair state), time evolution simulations, etc. These
problems, which are challenging to solve, can be transformed to TN contraction
problems. We present then several paradigm algorithms based on the ideas of the
numerical renormalization group and/or boundary states, including density
matrix renormalization group, time-evolving block decimation,
coarse-graining/corner tensor renormalization group, and several distinguished
variational algorithms. Finally, we revisit the TN approaches from the
perspective of multi-linear algebra (also known as tensor algebra or tensor
decompositions) and quantum simulation. Despite the apparent differences in the
ideas and strategies of different TN algorithms, we aim at revealing the
underlying relations and resemblances in order to present a systematic picture
to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been
changed into the format of book; new sections about tensor network and
quantum circuits have been adde
Classical simulation versus universality in measurement based quantum computation
We investigate for which resource states an efficient classical simulation of
measurement based quantum computation is possible. We show that the
Schmidt--rank width, a measure recently introduced to assess universality of
resource states, plays a crucial role in also this context. We relate
Schmidt--rank width to the optimal description of states in terms of tree
tensor networks and show that an efficient classical simulation of measurement
based quantum computation is possible for all states with logarithmically
bounded Schmidt--rank width (with respect to the system size). For graph states
where the Schmidt--rank width scales in this way, we efficiently construct the
optimal tree tensor network descriptions, and provide several examples. We
highlight parallels in the efficient description of complex systems in quantum
information theory and graph theory.Comment: 16 pages, 4 figure
Classical simulation of quantum many-body systems with a tree tensor network
We show how to efficiently simulate a quantum many-body system with tree
structure when its entanglement is bounded for any bipartite split along an
edge of the tree. This is achieved by expanding the {\em time-evolving block
decimation} simulation algorithm for time evolution from a one dimensional
lattice to a tree graph, while replacing a {\em matrix product state} with a
{\em tree tensor network}. As an application, we show that any one-way quantum
computation on a tree graph can be efficiently simulated with a classical
computer.Comment: 4 pages,7 figure
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