John Baez Tensor Network States

About

Tensor network states are used to describe physical states and processes in many-body physics and related disciplines. An explicit connection between these current methods and the diagrammatic methods known in category was made in the following paper.

• Jacob D. Biamonte, Stephen R. Clark and Dieter Jaksch, Categorical Tensor Network States (2010).

Jacob Biamonte later taught a course explaining these results and exploring tensor networks in more detail.

• Jacob Biamonte, Lectures on Tensor Network States, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (2011), http://www.qubit.org/iqc2011.

Background reading

Here are some review articles on tensor network states:

• F. Verstraete, J.I. Cirac and V. Murg, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008), 143.

• J. I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A 42 (2009), 504004.

• Guifre Vidal, Entanglement renormalization: an introduction.

Vidal has written extensively about tensor networks, including:

Fermionic Tensor Network States

Fermionic MERA:

• http://arxiv.org/abs/0904.4151, http://arxiv.org/abs/0907.3184

Fermionic PEPS:

• http://arxiv.org/abs/0904.4667, http://arxiv.org/abs/0912.0646

Fermionic quantum circuits:

• http://arxiv.org/abs/0905.0669, http://arxiv.org/abs/0907.3689

Anyonic tensor network states

• http://arxiv.org/abs/1006.2478, http://arxiv.org/abs/1006.3532

Connection to spin networks and category theory

Vidal pointed out in

• http://arxiv.org/abs/0907.2994

that tensor network states made of symmetric tensors were nothing but spin networks as used in loop quantum gravity. He suggests that this could lead to new interactions between different subjects. See also.

• Ville Bergholm and Jacob D. Biamonte, Categorical Quantum Circuits, J. Phys. A: Math. Theor. 44, 245304 (2011).