> 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/parameter-counting.md).

# Parameter counting for tensor-network layers

This page shows how to count the parameters of a tensor-network layer, so you can check the sizes quoted in these docs.

## The general formula

For an MPO with d cores, where core k maps an input size nₖ to an output size mₖ and has bond dimensions χₖ₋₁ and χₖ, the parameter count is

params = Σₖ χₖ₋₁ · mₖ · nₖ · χₖ

with the boundary bonds χ₀ = χ\_d = 1.

If every internal bond is χ and every mₖ = nₖ = m, this simplifies to roughly d · m² · χ² for the internal cores.

## The worked example

The layer used in the docs is a 4096 × 4096 matrix, with d = 4, mₖ = nₖ = 8 and χ = 64 on the internal bonds. The dense layer has 4096 × 4096 = 16,777,216 parameters.

Counting the cores:

* Core 1 (χ₀ = 1, χ₁ = 64): 1 · 8 · 8 · 64 = 4,096
* Core 2 (64 to 64): 64 · 8 · 8 · 64 = 262,144
* Core 3 (64 to 64): 64 · 8 · 8 · 64 = 262,144
* Core 4 (64 to 1): 64 · 8 · 8 · 1 = 4,096

Total: 4,096 + 262,144 + 262,144 + 4,096 = 532,480 parameters.

The ratio is 16,777,216 / 532,480, about 31.5×.

## Why the internal cores dominate

The internal cores carry χ² and so account for most of the cost. Reducing χ has a much bigger effect than reducing the physical dimensions. This is the main reason rank-adaptive training targets the bonds.

## Checking a quoted number

To check a figure in the docs, you need the shape, the number of cores, and the bond dimensions. Without those, a quoted reduction cannot be reproduced.

## Related

* [Matrix product operators as weight matrices](/deep-dives/matrix-product-operators.md)
* [Tensor-Network Weights](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/tensor-network-weights.md)


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