PEPS and 2D Systems

Matrix product states are tailored to one dimension: their chain geometry matches the line of qubits, and a 1D area law keeps the bond dimension bounded. Two- dimensional systems need a tensor network whose geometry matches the lattice. That generalisation is the projected entangled pair state (PEPS).

From a chain to a grid of tensors

A PEPS places one tensor at each site of a 2D lattice. On a square lattice each tensor has one physical index (the qubit) plus four bond indices — up, down, left, right — connecting it to its four neighbours:

T^{\,s}_{u\,d\,l\,r} .

Contracting all the internal bonds over the whole grid reproduces the full $2^n$ -amplitude tensor of the 2D state. PEPS is the natural 2D analogue of the MPS chain: where MPS strings tensors along a line, PEPS tiles them across a plane.

Why PEPS respects the 2D area law

The point of matching geometry is that PEPS can capture the 2D area law efficiently in storage. A region of linear size $L$ on the lattice is cut from the rest by a boundary of length $\sim L$ , and the bonds crossing that boundary carry the entanglement. With bond dimension $D$ per bond, a PEPS supports entanglement entropy up to $S \sim L \log_2 D$ across such a cut — exactly the 2D area-law scaling. A constant $D$ therefore describes 2D ground states whose entanglement grows only with the boundary, which an MPS could only do by letting its single bond dimension blow up exponentially in $L$ .

The catch: exact contraction is intractable

For an MPS you can contract the chain exactly and cheaply, sweeping along the line. For a PEPS there is no such ordering: contracting a 2D grid forces intermediate tensors whose size grows exponentially with the lattice width. In fact, exactly contracting a general PEPS — even just to compute a single amplitude or a norm — is a #P-hard problem. So while PEPS stores a 2D state efficiently, reading observables out of it is the hard part, the opposite of the MPS situation.

Approximate contraction

Practical PEPS algorithms therefore use approximate contraction schemes:

Boundary-MPS methods sweep a row of the grid into the next, representing the growing boundary as an MPS and truncating its bond dimension at each step.
Corner transfer matrix and tensor-renormalisation methods coarse-grain the network, repeatedly merging and truncating blocks of tensors.

Both introduce a second, controlled approximation (on top of the finite $D$ ) and both cost a high power of $D$ — typically $O(D^{10})$ or worse for variational ground-state search. This steep cost is why 2D simulations reach far smaller bond dimensions than 1D ones.

What PEPS delivers in practice

Despite the cost, PEPS is a leading tool for 2D strongly correlated systems: frustrated magnets, the 2D Hubbard model, and topologically ordered phases that an MPS cannot represent efficiently. The trade-off is stark and worth remembering: PEPS buys you the correct 2D entanglement structure at the price of contraction that is exponentially harder than 1D. That price is the central reason two-dimensional quantum systems remain a frontier for classical simulation, and it sets up the broader limits we examine next.