#import "./poster.typ": *

#show: poster.with(
  size: "594x841",
  title: "Projected Entangled Pair States (PEPS)",
  authors: "<NAME>, <NAME>, <NAME>",
  departments: "",
  // univ_logo: "./images/logo_detail_copenhagen_2x.png",
  // univ_logo: "./images/CPHScience.svg",
  univ_logo: "./images/Ku-ucph-logo-svg.svg",
  footer_text: "4EU+ Quantum Information and Quantum Many-Body Theory",
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#set text(size: 22pt)

#let box_color = luma(200)
#let block_stroke = 2pt
#let peps_fig_scale = 50%

= Definition
Projected Entangled Pair States (PEPS) are a generalization of MPS to one and higher dimensions. They are explicitly constructed such as to mimic the entanglement structure of ground states of local quantum Hamiltonians: they have strong local correlations, possess all the symmetries of the problem, and obey an area law of their entanglement entropy.

#align(horizon)[
#grid(
  columns: (8fr, 4fr),
  rows: 4,
  [
    - Given a physical sytem composed of local sites with site-Hilbertspaces $cal(H)_i equiv bb(C)^(d_i)$ situated on a graph with edges $E= {e_(i,j)}$ and vertices $V = {v_i}$.
  ],
  [
    #figure(
      image("./images/graph.svg", 
            width: peps_fig_scale),
    )
  ],
  [
    - For each vertex $v_i$, and for each edge $e_(i,j)$ connected to $v_i$ (and $v_j$), associate an ancilla degree of freedom $a_(i,j) in cal(H)_(i,j) equiv bb(C)^(d_(i,j))$.
  ],
  [
    #figure(
      image("./images/ancillae.svg", 
            width: peps_fig_scale),
    )
  ],
  [
    - Maximally entangle ancillae $a_(i,j)$ and $a_(j,i)$ associated with each edge $e_(i,j)$: $ |Phi_(i,j)) = sum^(d_(i,j))_(l=1) |l angle.r times.circle |l angle.r $
  ],
  [
    #figure(
      image("./images/entangled-ancillae.svg", 
            width: peps_fig_scale),
    )
  ],
  [
    - Apply a linear map $cal(P)_i: times.big.circle_(j_i) cal(H)_(a_(i,j)) arrow.r cal(H)_i$ to the ancillae of site $i$, to obtain the projected entangled pair states (PEPS) $|Psi angle.r = times.big.circle_(e in E) cal(P)_e |Phi_e) $.
  ],
  [
    #figure(
      image("./images/projected-entangled-ancillae.svg", 
            width: peps_fig_scale),
    )
  ],
  [
    - The final PEPS is a tensor network state that has the same connectivity as the original graph and that lives in the total Hilbert space $cal(H) = times.big.circle_i cal(H)_i$.
    
  ],
  [
    #figure(
      image("./images/peps-network.svg", 
            width: peps_fig_scale),
    )
  ]
)
]
*[Cirac et. al. 2021]*
= Entanglement Area Law
#block(
  fill: box_color,
  inset: 10pt,
  radius: 8pt,
  stroke: block_stroke,
  [*Entanglement Area Law*
  
  The entanglement entropy of a region $cal(A)$ of quantum state with finite (local) entanglement scales as $partial cal(A)$, the boundary of $cal(A)$.
  $ S_(cal(A)) #h(10pt)~#h(10pt) partial cal(A) $
  ],
)
This is in constrast with the volume law  Haar-distributed states follow.

Given the Schmidt decomposition of a state $|Psi angle.r$ across a bipartition of the system into the "In" system $cal(A)$ and the "Out" system $cal(B)$ (where $sum_i lambda^2_i = 1$):

$ |Psi angle.r = sum_i lambda_i |I_i angle.r times.circle |O_i angle.r $

The entanglement entropy of the region $cal(A)$ is given by:
$ S_(cal(A)) = - sum_i lambda^2_i log(lambda^2_i) $
The maximal value this can take is when all the $lambda_i = 1/N_S$, with $N_S$ the Schmidt rank.
For finitely entangled systems, $S_(cal(A))$ is thus bounded by the finite Schmidt rank $N_S$.
$ S_(cal(A)) <= log(N_S) $

It turns out that PEPS satisfy an *area law* for their entanglement entropy *by construction*. This property makes PEPS an efficient representation of quantum states with finite (local) entanglement.

#figure(
  image("./images/area-law.svg", 
        width: 40%),
)

// #grid(
//   columns: (8fr, 5fr),
//   [
// The entanglement entropy of the shaded area $cal(A)$ is the sum of the entanglement entropy across the cut virtual bonds, which by construction all have a finite bond dimension (and Schmidt rank) $chi$.
// #v(10pt)
// The entanglement entropy of a single cut is bounded by $log(chi)$. The total entanglement entropy of the region $cal(A)$ is thus proportional to the number of cut virtual bonds times $log(chi)$.
//   ],
//   [
// #figure(
//   image("./images/area-law.svg", 
//         width: 90%),
// )
//   ],
// )
The entanglement entropy of the shaded area $cal(A)$ is the sum of the entanglement entropy across the cut virtual bonds, which by construction all have a finite bond dimension (and Schmidt rank) $chi$.

The entanglement entropy of a single cut is bounded by $log(chi)$. The total entanglement entropy of the region $cal(A)$ is thus proportional to the number of cut virtual bonds times $log(chi)$.

$ S_(cal(A)) #h(10pt) ~ #h(10pt) log(chi) dot.c partial cal(A) $

#colbreak()
= Parent Hamiltonians
#block(
  fill: box_color,
  inset: 10pt,
  radius: 8pt,
  stroke: block_stroke,
  [*Parent Hamiltonians for PEPS*

    Every injective PEPS is the unique ground state of a _local_, _frustration free_ parent Hamiltonian.
  ]
)

Consider the same shaded region $cal(A)$ as before. The support space $cal(S)_(cal(A))$ of the reduced density matrix $rho_(cal(A))$ grows as $partial cal(A)$ while the Hilbert space $cal(H)_(cal(A))$ grows as $cal(A)$.

Therefore we can always find a sufficiently large region such that $rho_(cal(A))$ has a non-trivial kernel $ker(rho_(cal(A))) = cal(H)_(cal(A)) \\ cal(S)_(cal(A))$.

It is thus the groundstate of every Hamiltonian h with:
- $h >= 0$
- $ker(h) = cal(S)_(cal(A))$

since by construction $tr(rho_(cal(A))h) = 0$.

The Hamiltonian can be extended to the entire lattice by assigning a region $cal(R)_v$ to every vertex $v$ of the lattice and defining
$ H = sum_v h_v times.circle bb(1) $

The Hamiltonian is _local_ since it is the sum of local terms $h_v$ acting on bounded regions $cal(R)_v$, and it is _frustration free_ since every term $h_v$ is minimized by the PEPS.

*Examples*

Many examples exist of the rather formal Parent Hamiltonian theorem. The groundstates of the following Hamiltonians can all be exactly expressed as PEPS.
+ The toric code
+ The 2d AKLT model
[*Perez-Garcia et. al. 2007*, *Orus 2014*]

// = Computation of local expectation values
// #block(
//   fill: box_color,
//   inset: 10pt,
//   radius: 8pt,
//   stroke: block_stroke,
//   [
//     *Computation of local expectation values*

//     Calculating expectation values of local observables is exponentially hard. Therefore approximate contractions are commonly used. 
//   ]
// )
// The computation of local e

= Computational complexity of PEPS contractions
#block(
  fill: box_color,
  inset: 10pt,
  radius: 8pt,
  stroke: block_stroke,
  [
    *Exact PEPS contraction is \#P-Hard*

    The exact calculation of the scalar product between two PEPS is an exponentially hard problem.
  ]
)

#grid(
  align: horizon,
  columns: (8fr, 3fr),
  column-gutter: 10pt,
  [For two arbitrary PEPS of N sites, it will always take a time $cal(O)(exp(N))$, no matter the contraction order.

Because in general _no canonical forms exist for PEPS_, calculating expecation values efficiently requires an approximate approach.

There exist many strategies for the approximate calculation of the contraction of PEPS which differ for finite and infinite systems.
],
  figure(
    image("./images/expecation-value.svg", 
          width: 100%),
  )
)

*Finite Systems*

Boundary Matrix Product State (MPS) methods are an efficient technique for contracting finite Projected Entangled Pair States (PEPS), transforming the 2D contraction problem into a sequence of 1D contractions.

*Infinite Systems*

There exist several methods for the contraction of infinite PEPS. These include:
== Boundary MPS methods
Boundary MPS methods transform the 2D contraction problem into a series of 1D contractions by representing the boundary of the PEPS as a Matrix Product State (MPS). The time complexity typically scales as $cal(O)(chi^(2N) + chi^3 L)$, where $L$ is the boundary length. These methods strike a balance between accuracy and computational feasibility, making them suitable for relatively large systems.

== Corner Transfer Matrix methods (CTM)
CTM methods approximate the environment tensors using corner and edge transfer matrices. These matrices are iteratively refined, with a typical time complexity of $cal(O)(chi^10)$. CTM methods are particularly effective for large systems and are widely used to study critical phenomena.

== Tensor Renormalization Group methods (TRG)
Coarse-graining methods, such as Tensor Renormalization Group (TRG), reduce the tensor network size while preserving essential features. These methods involve iterative tensor decompositions and contractions, generally scaling as $cal(O)(chi^6)$. They are particularly useful for extracting long-range properties in large-scale systems.

[*Orus 2014*]
// #block(
//   fill: box_color,
//   inset: 10pt,
//   radius: 8pt,
//   stroke: block_stroke,
//   [
//     *Effective environments*

//     All the methods to compute expectation values of 2d systems are based on techniques to compute effective environments of different sites.
//   ]

// )
