> 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/canonical-forms-and-sweeps.md).

# Canonical forms and sweeps

A tensor network can be rewritten in many equivalent ways. Canonical forms pick one that is numerically stable and makes truncation easy to control. This page explains the idea and how sweeps use it.

## Gauge freedom

Inserting a matrix and its inverse on a bond does not change the network's product. The cores change, but the represented tensor does not. This is the gauge freedom of tensor networks.

## Canonical form

A chain is in canonical form when every core left of a chosen position is a left isometry (its contraction with itself on the physical and left indices gives the identity), and every core right of it is a right isometry. The position is the centre, and the norm of the whole network sits in the centre core.

## Why it helps

* Contractions stay numerically stable, because the norm is not spread across the chain.
* At the centre, the singular values of the bond to either side are the Schmidt coefficients, so truncation error can be read off directly.

## Sweeps

Moving the centre one bond at a time is called a sweep. Each step:

1. Moves the centre to the next core using an SVD.
2. Truncates the bond if the error budget allows.
3. Updates the centre core.

A sweep across the whole chain visits every bond once. Quantaria's planned DMRG-style sweeps use this structure during optimization. See [DMRG-style sweeps](/deep-dives/dmrg-sweeps.md).

## Quantaria's use

Hilbert-1 keeps its cores in canonical form for stable contraction. See [Canonical-form gauging](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/tensor-network-weights.md#canonical-form-gauging). Sweep-based optimization is a research direction, not part of the live models.

## Related

* [SVD and truncation](/deep-dives/svd-and-truncation.md)
* [Matrix product states from scratch](/deep-dives/matrix-product-states.md)


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