> For the complete documentation index, see [llms.txt](https://docs.quantaria.xyz/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.quantaria.xyz/deep-dives/matrix-product-operators.md).

# Matrix product operators as weight matrices

A matrix product operator (MPO) is the matrix version of an MPS. Quantaria's layers are MPOs, so this page explains how a dense weight matrix becomes a chain of cores.

## From a matrix to a chain

A dense layer computes y = Wx, where W is an M × N matrix. Reshape the input and output indices into multi-index form:

* x has indices (j₁, …, j\_d), with the j-th dimension size nₖ
* y has indices (i₁, …, i\_d), with the i-th dimension size mₖ

Then W is written as a product of cores:

W\[(i₁…i\_d), (j₁…j\_d)] = G¹\[i₁, j₁] · G²\[i₂, j₂] · … · Gᵈ\[i\_d, j\_d]

Each core Gᵏ has size χₖ₋₁ × mₖ × nₖ × χₖ. The bond dimensions χₖ connect neighbouring cores.

## Worked example

The layer in the [Quantaria Overview](https://github.com/bhavy2908/dataai_docs/tree/main/overview/quantaria-overview.md) is a 4096 × 4096 matrix, so it has 16,777,216 dense parameters. Split into d = 4 cores, with mₖ = nₖ = 8 and χ = 64 on the internal bonds, the core sizes add up to 532,480 parameters. That is about 31.5× fewer.

The example is illustrative. The per-layer setting of χ determines the actual size.

## Applying the layer

The layer is applied by contracting the input vector with the chain one core at a time. The cost scales with the bond dimension, not with the full matrix, which is why a well-chosen χ can make the layer both smaller and cheaper to compute.

## Limits

* An MPO can only represent matrices whose correlations fit within the chosen bonds. A small χ is cheaper but may lose accuracy.
* Reordering the index splits changes the result. The grouping of dimensions into cores is a design choice.

## Quantaria's use

Every weight layer in Hilbert-1 is an MPO. The factorization is trained from the first step, not applied to a dense model afterwards. See [Native, not compressed](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/tensor-network-weights.md#native-not-compressed).

## Related

* [Tensor-train decomposition of a weight matrix](/deep-dives/tensor-train-decomposition.md)
* [Parameter counting for tensor-network layers](/deep-dives/parameter-counting.md)


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