On MPS and PEPS…
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Transcript of On MPS and PEPS…
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