Tensor Networks

Introduction

Condensed matter physics studies the effects of macroscopic and microscopic changes in matter caused by strong interactions among present particles (atoms, electrons, etc.). A typical problem involves the mathematical modeling of induced correlation impacts and the study of phase transitions, non-trivial magnetic orders, spin-textures, fractional quantum Hall effects, and quasiparticle interferences each resulting from many-body interactions within high-temperature superconductors, extremely cold atom gases, or liquids.
From the analytical and numerical perspective the analysis of these systems is chalanging due to the exponential scaling of the problems size. In this context, tensor tetworks provide an effient way of simulating sroungly correlated quantum systems while avoiding the curse of dimensionalty.

Statistical Mechanics vs Quantum Mechanics

Hilbert Space of Pauli Matrices

An important class of Hamiltonians is given on the product space of local Hilber-Spaces formed by the so called Pauli Matrices

$I = \begin{pmatrix} 1 &0\\ 0 &1 \end{pmatrix}, \quad \sigma_x = \begin{pmatrix} 0 &1\\ 1 &0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 &-i\\ i &0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 &0\\ 0 &-1 \end{pmatrix},$

that are unitary and Hermitian with respect to the innner product $\langle A,B \rangle = \frac{1}{d}\text{Tr}(A^{\ast}B)$

Principle of Entanglement

We start with a simple illustration of entanglement and consider Bell states which are quantum state given as superposition of two qubits $\ket{00}$ and $\ket{11}$

$\begin{aligned}\ket{\phi}^{\pm} =\frac{1}{\sqrt{2}}\left( \ket{01}\pm \ket{10} \right),\quad \ket{\psi^{\pm}} = \frac{1}{\sqrt{2}}\left( \ket{00}\pm \ket{11} \right),\end{aligned}$

where we made an identification $\ket{00} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \otimes \begin{pmatrix} 1\\ 0 \end{pmatrix}, \quad \ket{11} = \begin{pmatrix} 0\\ 1 \end{pmatrix} \otimes \begin{pmatrix} 0\\ 1 \end{pmatrix}$. More generally, a physical system of two qubits $\ket{0}$ and $\ket{1}$ is represented by

$\begin{aligned} \hspace{3cm} a_0\ket{0}+a_1\ket{1}, \qquad |a_0|^2+|a_1|^2 = 1 \end{aligned}$

We next inspect inspect the partial trace of $\ket{\psi}\bra{\psi}$ with respect to system $B = \{\ket{0},\ket{1}\}$

$\begin{aligned} \text{Tr}_{B}(\ket{\psi}\bra{\psi}) &= \frac{1}{2} (I_A\otimes \bra{0} \ket{\psi}\bra{\psi})(I_A\otimes \ket{0})+(I_A\otimes \bra{1} \ket{\psi}\bra{\psi})(I_A\otimes \ket{1}) \\ &= \frac{1}{2}\Big((I_A\otimes \bra{0}) \frac{1}{2}(\ket{00} \bra{00} +\ket{00} \bra{11}+\ket{11} \bra{00}+\ket{11} \bra{11}) (I_A\otimes \ket{0})\\ &\hspace{1cm}+ (I_A\otimes \bra{1}) (\frac{1}{2}(\ket{00} \bra{00} +\ket{00} \bra{11}+\ket{11} \bra{00}+\ket{11} \bra{11})) (I_A\otimes \ket{1})\Big)\\ &= \frac{1}{4}\left((\ket{0}\otimes\bra{00}+\ket{1}\otimes\bra{11}) (I_A\otimes \ket{0} )+(\ket{0}\otimes\bra{00}+\ket{11}\otimes\bra{1})(I_A\otimes \ket{1})\right)\\ &= \frac{1}{2}(\ket{0}\bra{0}+\ket{1}\bra{1}) \end{aligned}$

which is an operator written is a maximally mixed state on system $A$. In other words, if both Alice and Bob had each one qubit $\ket{0}$ and $\ket{1}$ respectivaly there is no way how Alice could infer the hidden quantum state by performing measurements only in her system. Moreover performing the partial trace yields the density matrix $\rho_A$. This means that if Alice measure her qubit along any axis, the result is completely random – each with probabilty $\frac{1}{2}$.

Area Laws

Canonical Form Tensor Networks

Local Environment and Quasi-Canonical Form

Approximating Low Energy States of Hamilitonians

Imaginary Time Evolution

Density Matrix Normalization Group

References

• Ulrich Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics, Volume 326, Issue 1, 2011, Pages 96-192, ISSN 0003-4916

• Román Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics, Volume 349, 2014, Pages 117-158, ISSN 0003-4916