# How to describe the phase detection operator of the adaptive GROVER algorithm mathematically?

**URL:** https://discuss.pennylane.ai/t/how-to-describe-the-phase-detection-operator-of-the-adaptive-grover-algorithm-mathematically/8517
**Category:** PennyLane Help
**Created:** [May 17, 2025, 3:29pm UTC](https://discuss.pennylane.ai/t/how-to-describe-the-phase-detection-operator-of-the-adaptive-grover-algorithm-mathematically/8517 "2025-05-17T15:29:35Z")
**Posts on this page:** 2
**Page:** 1

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### Author: ![RX1](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/rx1/32/698_2.png) [@RX1](https://discuss.pennylane.ai/u/RX1)
#### Post date: [May 17, 2025, 3:29pm UTC](https://discuss.pennylane.ai/t/how-to-describe-the-phase-detection-operator-of-the-adaptive-grover-algorithm-mathematically/8517/1 "2025-05-17T15:29:35Z")

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After reading document [Grover Algorithm for Multi-objective Searching with Iteration Auto-controlling | SpringerLink](https://link.springer.com/chapter/10.1007/978-3-319-11857-4_41), I have been curious about the mathematical expression of the phase detection operator (Fig. 3). I searched for articles related to quantum counting but still could no understand how to construct the phase detection unitary operator, and even could not write its oracle form.

 ![GP3ROHcQ](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/6/6bf8f95e3b9fe5a631a653669c5e73be6ae7c265.png)  
 ![TmMNdiJj](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/c/c91b7b82df638b942329b80e674233a2278e8236.png)

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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: [May 20, 2025, 4:28pm UTC](https://discuss.pennylane.ai/t/how-to-describe-the-phase-detection-operator-of-the-adaptive-grover-algorithm-mathematically/8517/2 "2025-05-20T16:28:35Z")

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Hi @RX1 ,

Have you checked out the [Codebook module on Grover’s algorithm](https://pennylane.ai/codebook/grovers-algorithm) and the [PennyLane demo on Grover’s](https://pennylane.ai/qml/demos/tutorial_grovers_algorithm)?

They both talk about the Oracle. The [Searching with Circuits Codebook topic](https://pennylane.ai/codebook/grovers-algorithm/searching-with-circuits) might be particularly interesting for you.

I hope this helps!