Operators and Hamiltonians
Just as a matrix product state factorises a wavefunction into a chain of local tensors, an operator on a one-dimensional system can be factorised in exactly the same way. The result is a matrix product operator (MPO): the operator analogue of an MPS. This page explains what that factorisation is, why the local tensors are not unique, and — above all — the particular upper-triangular Jordan-block structure that lets a sum of local terms be written as a single MPO. It is about understanding rather than construction: for how to build a Hamiltonian see Building Hamiltonians, and for the full type signatures see the Operators reference.
What an MPO is
An MPO is a collection of local MPOTensor objects contracted along a line. Where an MPS site tensor has one physical leg and two virtual legs, an MPO site tensor has two physical legs — one incoming and one outgoing — because it maps states to states, and again two virtual legs that thread the operator together along the chain. 
The diagram shows an MPO as a row of site tensors, each carrying a pair of physical indices (one in, one out) and linked to its neighbours through virtual bonds.
As with states, the construction comes in a finite and an infinite flavour. A FiniteMPO is a plain vector of MPOTensor objects with trivial (dimension-one) virtual spaces at the two ends, so that the network describes a genuine operator on a finite chain. An InfiniteMPO instead repeats a finite unit cell periodically, and is therefore stored as a periodic array of MPOTensor objects rather than an ordinary vector.
Gauge non-uniqueness
The local tensors of an MPO are not uniquely determined by the operator they encode. Exactly as for an MPS, an invertible gauge transformation can be inserted on any virtual bond and reabsorbed into the two neighbouring tensors without changing the contracted network. The individual site tensors are therefore defined only up to this virtual-space gauge freedom.
Element-wise comparison is unsafe
Because two different sets of local tensors can represent the very same operator, comparing MPOs tensor-by-tensor is not meaningful. Test for equality through gauge-invariant quantities instead.
Products and sums grow the virtual dimension
MPOs support the usual linear-algebra operations — addition, subtraction, and multiplication, either among themselves or acting on an MPS. Each such operation combines the virtual spaces of its operands, so the virtual dimension of the result is (generically) the product or sum of the input dimensions rather than staying fixed. Composing operators naively therefore makes the representation grow, and the growth compounds under repeated multiplication. <!– REVIEW: the claim that MPO×MPO multiplies virtual dimensions and MPO+MPO adds them (so repeated products grow the bond, up to the rank ceiling set by full-rank tensors) is the standard MPO-arithmetic picture carried over from the legacy man/operators.md; confirm the exact growth law and the full-rank cap MPSKit enforces. –> This growth is precisely what motivates the approximate algorithms that re-express a product or sum within a bounded virtual dimension; see the algorithm landscape for where those methods fit.
MPO Hamiltonians and the Jordan-block form
A quantum Hamiltonian is a sum of local terms rather than a single dense operator, yet it too can be written as one MPO. The trick is a characteristic upper-triangular block structure, so distinctive that the resulting object is usually called a Jordan-block MPO. In MPSKit this is the MPOHamiltonian family — FiniteMPOHamiltonian and InfiniteMPOHamiltonian — and it is what the Hamiltonian constructors assemble under the hood.
In its most general form, the per-site block matrix
where the corner entries 1 are identity operators and
A finite-state automaton
The upper-triangular shape makes 1), it may finish immediately by placing a single-site term through
alone generates the single-site terms, and in general
<!– REVIEW: the finite-state-automaton reading of W (v_L enters the identity state; D emits a 1-site term; C starts / A propagates / B closes an interaction so that C·A^k·B spans k+2 sites) is the standard interpretation carried over from the legacy page; confirm it matches MPSKit's index/row-column convention. –>
The repeated
The transverse-field Ising Hamiltonian
For the transverse-field Ising model,
the block matrix specialises to
Here
Verifying the expansion symbolically
Because
using Symbolics
L = 4
# generate W matrices, one per site
@variables A[1:L] B[1:L] C[1:L] D[1:L]
Ws = map(1:L) do l
return [1 C[l] D[l]
0 A[l] B[l]
0 0 1]
end
# left and right boundary vectors
Vₗ = [1, 0, 0]'
Vᵣ = [0, 0, 1]
# expand the contraction H = V_L W^{⊗L} V_R
expand(Vₗ * prod(Ws) * Vᵣ)D[1] + D[2] + D[3] + D[4] + B[2]*C[1] + B[3]*C[2] + B[4]*C[3] + A[2]*B[3]*C[1] + A[3]*B[4]*C[2] + A[2]*A[3]*B[4]*C[1]Reading off the result, the lone
Sparse and block structure
Because an MPOHamiltonian is an MPO with the extra Jordan-block structure, its virtual space is not a single space but a direct sum of spaces, one for each row (or column) of the block matrix BlockTensorMap objects rather than ordinary dense tensor maps, with each block occupying one cell of the JordanMPOTensor for exactly this layout.
Sparsity is what keeps it efficient
Most cells of
The JordanMPOTensor type and its internal accessors are implementation detail and may change; treat them as unstable and prefer the public constructors and @ref-documented interface.
Beyond nearest-neighbour, 1D chains
The same machinery is not limited to nearest-neighbour couplings or to strictly one-dimensional systems: quasi-1D cylinders and 2D lattices are obtained by snaking the MPO through a multi-dimensional array of physical spaces, and longer-range interactions slot into the
Where to go next
To build Hamiltonians from local terms, in 1D or on lattices, see Building Hamiltonians.
For the full type signatures and docstrings, see the Operators reference.
For the approximate algorithms that keep MPO products and sums bounded, see the algorithm landscape.