On MPS and PEPS…

David Pérez-García.

Near Chiemsee. 2007.work in collaboration with F. Verstraete, M.M. Wolf and J.I. Cirac, L. Lamata, J. León, D. Salgado, E. Solano.

Part I: Sequential generation of unitaries.

Summary

Sequential generation of states. MPS canonical form. Sequential generation on unitaries

Generation of StatesC. Schön, E. Solano, F. Verstraete, J.I. Cirac and M.M. Wolf, PRL 95, 110503 (2005)

A

decoupled

MPS

Relation between unitaries and MPS

Canonical form

MPS canonical form (G. Vidal, PRL 2003)

Canonical unique MPS representation:

1

1

[1] [ ]1N

N

dN

i i Ni i

A A i i

[ ] [ ]†

[ ]† [ 1] [ ] [ ]

[ ]1

1 1

,

1

m mi i

i

m m m mi i

i

mi m m

N

A A

A A

A D D

D D

Canonical conditions

Pushing forward. Canonical form.D. P-G, F. Verstraete, M.M. Wolf, J.I. Cirac, Quant. Inf. Comp. 2007.

We analyze the full freedom one has in the choice of the matrices for an MPS.

We also find a constructive way to go from any MPS representation of the state to the canonical one.

As a consequence we are able to transfer to the canonical form some “nice” properties of other (non canonical) representations.

Pushing forward. Generation of isometries.

M N-M

MPS

Results. A dichotomy.

M=N (Unitaries). No non-trivial unitary can be

implemented sequentially, even with an infinitely large ancilla.

M=1 Every isometry can be implemented

sequentially. The optimal dimension of the ancilla is

the one given in the canonical MPS decomposition of U.

Examples

Optimal cloning.

V

The dimension of the ancilla grows linearly

<< exp(N) (worst case)

Examples

Error correction. The Shor code.

It allows to detect and correct one arbitrary error

It only requires an ancilla of dimension 4

<< 256 (worst case)

Part II: PEPS as unique GS of local Hamiltonians.

Summary

PEPS Injectivity Parent Hamiltonians Uniqueness Energy gap.

PEPS

2D analogue of MPS. Very useful tool to understand 2D

systems: Topological order. Measurement based quantum

computation (ask Jens). Complexity theory (ask Norbert).

Useful to simulate 2D systems (ask Frank)

PEPS

Physical systems

PEPS

Working in the computational basis

Hence

Contraction of tensors following the graph of the PEPS

v

v

Injectivity

R# outgoing bonds in R

# vertices inside R

Boundary condition

R

C

Injectivity

We say that R is injective if is injective as a linear map

Is injectivity a reasonable assumption?

Numerically it is generic. AKLT is injective.

Area Volume

Parent Hamiltonian

Notation:

For sufficiently large R

For each vertex v we take and

Parent Hamiltonian

By construction

R

C

R

PEPS g.s. of H

H frustration free

Is H non-degenerate?

Uniqueness (under injectivity)

We assume that we can group the spins to have injectivity in each vertex.

New graph. It is going to be the interaction graph of the Hamiltonian.

Edge of the graph

The PEPS is the unique g.s. of H.

Energy gap

In the 1D case (MPS) we have

This is not the case in the 2D setting. There are injective PEPS without gap. There are non-injetive PEPS that are

unique g.s. of their parent Hamiltonian.

Injectivity Unique GS Gap

Energy gap

Classical system

PEPS !!!

ji

jin hH,

1 ),(),...,(

)](exp[ HZ

nnH

Z...)],...,(2exp[

111

Same correlations

Energy gap.

No gapClassical 2D Ising at critical temp.

Power low decayPEPS ground state of gapless H.

It is the unique g.s. of H

Non-injective Injective