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The basic problem of much of condensed matter and high energy
physics, as well as quantum chemistry, is to find the ground state
properties of some Hamiltonian. Many algorithms have been invented
to deal with this problem, each with different strengths and
limitations. Ideas such as entanglement entropy from quantum
information theory and quantum computing enable us to understand the
difficulty of various problems. I will discuss recent results on
area laws and use these to prove that we can use matrix product
states to efficiently represent ground states for one-dimensional
systems with a spectral gap. I will also discuss recent results on
higher-dimensional matrix product states, in an attempt to extend the
remarkable success of matrix product algorithms beyond one dimension.
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