Introduction to MERA Sukhwinder Singh Macquarie University

Multidimensional array of complex numbers Tensors

Cost of Contraction a bc a d =

Made of layers

Disentanglers & Isometries

Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

Coarse-graining transformation Length Scale

Coarse-graining transformation

Layer is a coarse-graining transformation

Coarse graining of operators

Scaling Superoperator

MERA defines an RG flow Wavefunction on coarse-grained lattice with two sites

Types of MERA

Binary MERATernary MERA

Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

Expectation values from the MERA

“Causal Cone” of the MERA

But is the MERA good for representing ground states? Claim: Yes! Naturally suited for critical systems.

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

In any MERA Correlations decay polynomially Entropy grows logarithmically

Correlations in the MERA

Entanglement entropy in the MERA

Therefore MERA can be used a variational ansatz for ground states of critical Hamiltonians

Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

Time Space

Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

Figure Source: Evenbly, Vidal 2011

MERA and spin networks

(Wigner-Eckart Theorem)

MERA and spin networks

Summary – MERA can be seen as.. 1.As defining a RG flow. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.Specific realization of the AdS/CFT correspondence.