# Computing Hilbert-Schmidt distance using quantum circuit

**URL:** <https://discuss.pennylane.ai/t/computing-hilbert-schmidt-distance-using-quantum-circuit/2499>\
**Category:** PennyLane Help\
**Created:** [February 3, 2023, 6:26pm UTC](https://discuss.pennylane.ai/t/computing-hilbert-schmidt-distance-using-quantum-circuit/2499 "2023-02-03T18:26:02Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![raven\_of\_asgard](https://avatars.discourse-cdn.com/v4/letter/r/848f3c/32.png) [@raven\_of\_asgard](https://discuss.pennylane.ai/u/raven_of_asgard)\
**Post date:** [February 3, 2023, 6:26pm UTC](https://discuss.pennylane.ai/t/computing-hilbert-schmidt-distance-using-quantum-circuit/2499/1 "2023-02-03T18:26:02Z")

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Say I have two density matrices, `[[0.7, 0.2 + 0.1j], [0.2 - 0.1j, 0.3]]` and `[0.5, 0.5j], [-0.5j, 0.5]]`.

Can anyone please help me with on how to compute the Hilbert-Schmdit distance between these these two matrices using a quantum circuit? It is my understanding that it can be implemented using SWAP test. However, I am not sure how to initialize the quantum circuit whose states is represented by the above density matrix.

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**Author:** ![CatalinaAlbornoz](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/catalinaalbornoz/32/1196_2.png) [@CatalinaAlbornoz](https://discuss.pennylane.ai/u/CatalinaAlbornoz)\
**Post date:** [February 6, 2023, 11:13pm UTC](https://discuss.pennylane.ai/t/computing-hilbert-schmidt-distance-using-quantum-circuit/2499/2 "2023-02-06T23:13:12Z")

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Hi @raven_of_asgard,

You can use PennyLane’s [qml.HilbertSchmidt](https://docs.pennylane.ai/en/stable/code/api/pennylane.HilbertSchmidt.html) template! In the [usage details](https://docs.pennylane.ai/en/stable/code/api/pennylane.HilbertSchmidt.html#usage-details) of the documentation for this template you can learn how to compare two unitaries.

You can probably combine this with [qml.QubitDensityMatrix](https://docs.pennylane.ai/en/stable/code/api/pennylane.QubitDensityMatrix.html) if you want to create a circuit from a density matrix.

Please let me know if you have any questions about this!

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**Author:** ![raven\_of\_asgard](https://avatars.discourse-cdn.com/v4/letter/r/848f3c/32.png) [@raven\_of\_asgard](https://discuss.pennylane.ai/u/raven_of_asgard)\
**Post date:** [February 7, 2023, 2:47pm UTC](https://discuss.pennylane.ai/t/computing-hilbert-schmidt-distance-using-quantum-circuit/2499/3 "2023-02-07T14:47:27Z")

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Thank you, @CatalinaAlbornoz. This is exactly what I was looking for.