Introduction to Tensor Network States Sukhwinder Singh Macquarie University (Sydney)

Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Quantum many body system in 1-D

H ow many qubits can we represent with 1 GB of memory? Here, D = 2. To add one more qubit double the memory.

But usually, we are not interested in arbitrary states in the Hilbert space. Typical problem : To find the ground state of a local Hamiltonian H,

Ground states of local Hamiltonians are special

1)Gapped Hamiltonian  2)Critical Hamiltonian  Properties of ground states in 1-D

We can exploit these properties to represent ground states more efficiently using tensor networks.

Ground states of local Hamiltonians

Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Multidimensional array of complex numbers Tensors

Contraction a bc a d =

Contraction a bc a d =

Contraction a bc a d = a c

Trace = = a

Tensor product

Decomposition a bc a d = = =

Decomposing tensors can be useful = Number of components in M = Number of components in P and Q = Rank(M) =

Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Many-body state as a tensor

Expectation values

Correlators

Reduced density operators

Tensor network decomposition of a state

Essential features of a tensor network 1)Can efficiently store the TN in memory 2) Can efficiently extract expectation values of local observables from TN Total number of components = O(poly(N)) Computational cost = O(poly(N))

Number of tensors in TN = O(poly(N)) is independent of N

Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Matrix Product States

Recall!

Expectation values

But is the MPS good for representing ground states?

Claim: Yes! Naturally suited for gapped systems.

Recall! 1)Gapped Hamiltonian  2)Critical Hamiltonian 

In any MPS Correlations decay exponentially Entropy saturates to a constant

Recall!

Correlations in a MPS

Entanglement entropy in a MPS

1.Variational optimization by minimizing energy 2. Imaginary time evolution MPS as an ansatz for ground states

Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Summary The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

Thanks !