1. 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.

2. Part I: Sequential generation of unitaries.

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

4. Relation between unitaries and MPS
Generation of States C. Schön, E. Solano, F. Verstraete, J.I. Cirac and M.M. Wolf, PRL 95, (2005)
MPS A
decoupled
Relation between unitaries and MPS
Canonical form

5. MPS canonical form (G. Vidal, PRL 2003)
Canonical unique MPS representation:
Canonical conditions

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

7. Pushing forward. Generation of isometries.
MPS M
N-M

8. Results. A dichotomy. M=N (Unitaries). M=1
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.

9. Examples V The dimension of the ancilla grows linearly
Optimal cloning. V
The dimension of the ancilla grows linearly
<< exp(N) (worst case)

10. 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)

11. Part II: PEPS as unique GS of local Hamiltonians.

12. Summary
PEPS
Injectivity
Parent Hamiltonians
Uniqueness
Energy gap.

13. 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)

14. PEPS
Physical systems

15. PEPS Working in the computational basis Hence
Contraction of tensors following the graph of the PEPS v v

16. Injectivity R R C # outgoing bonds in R # vertices inside R
Boundary condition

17. 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

18. Parent Hamiltonian Notation: For sufficiently large R
For each vertex v we take and

19. Parent Hamiltonian By construction R PEPS g.s. of H H frustration free
Is H non-degenerate?

20. 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.

21. Energy gap Injectivity Unique GS 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

22. Energy gap
Classical system
Same correlations
PEPS !!!

23. Energy gap. Classical 2D Ising at critical temp. No gap
PEPS ground state of gapless H.
Power low decay
It is the unique g.s. of H
Injective
Non-injective