Introduction to Tensor Network States Sukhwinder Singh Macquarie

Contents • The quantum many body problem. • Diagrammatic Notation • What is a

How many qubits can we represent with 1 GB of memory? Here, D =

But usually, we are not interested in arbitrary states in the Hilbert space. Typical

Ground states of local Hamiltonians are special

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

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

Tensors Multidimensional array of complex numbers

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

Essential features of a tensor network 1) Can efficiently store the TN in memory

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

But is the MPS good for representing ground states?

But is the MPS good for representing ground states? Claim: Yes! Naturally suited for

Recall! 1) Gapped Hamiltonian 2) Critical Hamiltonian

In any MPS Correlations decay exponentially Entropy saturates to a constant

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

Summary • The quantum many body problem. • Diagrammatic Notation • What is a

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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

How 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

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

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

Tensors Multidimensional array of complex numbers

Contraction =

Trace = =

Tensor product

Decomposition = = =

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

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 Total number of components = O(poly(N)) 2) Can efficiently extract expectation values of local observables from TN 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?

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

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

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 !