On MPS and PEPS

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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.

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Part I Sequential generation of unitaries.
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Summary

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

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Generation of StatesC. Schön, E. Solano, F.
Verstraete, J.I. Cirac and M.M. Wolf, PRL 95,
110503 (2005)
A
Relation between unitaries and MPS
Canonical form
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MPS canonical form (G. Vidal, PRL 2003)

• Canonical unique MPS representation

Canonical conditions
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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.

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Pushing forward. Generation of isometries.
MPS
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Results. A dichotomy.

• MN (Unitaries).
• No non-trivial unitary can be implemented
  sequentially, even with an infinitely large
  ancilla.
• M1
• Every isometry can be implemented sequentially.
• The optimal dimension of the ancilla is the one
  given in the canonical MPS decomposition of U.

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Examples

• Optimal cloning.

V
The dimension of the ancilla grows linearly
ltlt exp(N) (worst case)
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Examples

• Error correction. The Shor code.
• It allows to detect and correct one arbitrary
  error

It only requires an ancilla of dimension 4
ltlt 256 (worst case)
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Part II PEPS as unique GS of local Hamiltonians.
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Summary

• PEPS
• Injectivity
• Parent Hamiltonians
• Uniqueness
• Energy gap.

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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)

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PEPS
Physical systems
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PEPS
Working in the computational basis
Hence
Contraction of tensors following the graph of the
PEPS
v
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Injectivity
outgoing bonds in R
R
vertices inside R
Boundary condition
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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
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Parent Hamiltonian
Notation
For sufficiently large R
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Parent Hamiltonian
By construction
R
PEPS g.s. of H
H frustration free
Is H non-degenerate?
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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.
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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
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Energy gap
Classical system
Same correlations
PEPS !!!
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Energy gap.
No gap
Classical 2D Ising at critical temp.
PEPS ground state of gapless H.
Power low decay
It is the unique g.s. of H
Injective
Non-injective
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