> 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/cayley-unitary-adapters.md).

# The Cayley transform and unitary adapters

A unitary matrix preserves norms, which is what a quantum gate must do. Keeping a trained matrix unitary during gradient descent is hard. The Cayley transform is one way to parameterize unitaries so that training stays on the unitary manifold. This page explains the method.

## Unitary matrices

A matrix U is unitary if U†U = I. Quantum gates are unitary, and so are the adapters Quantaria's Dirac line plans to use.

## The transform

For any skew-Hermitian matrix A (A† = −A), the matrix

U = (I − A)(I + A)⁻¹

is unitary. The map from A to U is the Cayley transform. Because A can be any skew-Hermitian matrix, training A with ordinary gradient steps keeps U unitary without any projection step.

## Why this helps

Without a parameterization like this, gradient steps push the matrix off the unitary manifold, and an extra step is needed to bring it back. The Cayley form removes that step.

## Limits

* Inverting I + A has a cost, and the inversion can be ill-conditioned for some A.
* A unitary adapter only preserves norms. It does not by itself make a layer useful. The training task still decides what the adapter learns.

## Quantaria's use

The Dirac line plans Cayley-parameterized unitary adapters at the boundary between circuit and tensor-network sections. The published work reports that they lowered perplexity by up to 1.6% in one setting ([arXiv:2410.17397](https://arxiv.org/abs/2410.17397)). That is the paper's result, not a Quantaria measurement. See [Quantum Path](https://github.com/bhavy2908/dataai_docs/tree/main/architecture/quantum-path.md).

## Related

* [MERA and disentanglers](/deep-dives/mera-and-disentanglers.md)


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