# Matrix product states

Matrix product states are one possible way of describing a quantum state. In general, this description must be inefficient (i.e. the size of the description scales exponentially with the number of qubits in the system). However, there are several special cases, such as cluster states and the AKLT model which have exact decompositions in terms of these states, and other cases, such as translationally invariant states which can be efficiently described to arbitrary accuracy.

### Valence Bond States

From the quantum information perspective, matrix product states have a comparitively simple description - between each nearest-neighbour lattice site, we introduce a *D*-dimensional maximally entangled state, |\phi>=\frac{1}{\sqrt{D}}\sum_{i=1}^D|ii> Hence, at each lattice site there are two ends to these maximally entangled states (in the case of a one-dimensional geometry). We then apply a projector on each lattice site, reducing the dimensionality from *D*2 to *d*.

For valence bond states, this constitutes an exact description. Two well-known examples exist - the cluster state and the AKLT model. In the case of the AKLT model, *D* = 2 and *d* = 3. The projectors to be applied can be written as P=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) and we do not apply projectors at the end of the chain. For the cluster state, *d* = *D* = 2 and P=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).

### Translationally Invariant States

Translationally invariant states are the ground states of local Hamiltonians where each term that is applied to a particular qubit is the same as applied to all other qubits.

#### One-Dimensional Systems

If we are given a Hamiltonian, *H*, that is one-dimensional and tranlationally invariant, how do we generate the projectors required to describe the state in terms of this valence bond description? INSERT HERE.

#### Periodic Boundary Conditions and Arbitrary Geometries

The valence bond description of the matrix product formulation gives a very natural generalisation to periodic boundary conditions and arbitrary geometries - the concept of a nearest-neighbour persists, and we just include a maximally entangled state between all of these.

### General States

Any quantum state can be described by the valence bond description, but if we wish to describe an N-qubit system, the maximally entangled state between nearest-neighbours must have dimension *D* = 2⌊*N*/2⌋.