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

Relation between unitaries and MPS Generation of States C. Schön, E. Solano, F. Verstraete, J.I. Cirac and M.M. Wolf, PRL 95, 110503 (2005) MPS A decoupled Relation between unitaries and MPS Canonical form

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

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. 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. MPS M N-M

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.

Examples V The dimension of the ancilla grows linearly 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 R C # outgoing bonds in R # vertices inside R Boundary condition

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 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 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

Energy gap Classical system Same correlations PEPS !!!

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