# Question on last tasks on "Modeling the toric code on a quantum computer" demo **URL:** \ **Category:** Demos\ **Created:** [April 11, 2025, 9:14am UTC](https://discuss.pennylane.ai/t/question-on-last-tasks-on-modeling-the-toric-code-on-a-quantum-computer-demo/8386 "2025-04-11T09:14:24Z")\ **Posts on this page:** 4\ **Page:** 1
**Author:** ![jag](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/jag/32/3705_2.png) [@jag](https://discuss.pennylane.ai/u/jag)\ **Post date:** [April 11, 2025, 9:14am UTC](https://discuss.pennylane.ai/t/question-on-last-tasks-on-modeling-the-toric-code-on-a-quantum-computer-demo/8386/1 "2025-04-11T09:14:24Z")
Hi, At the end of notebook [Modeling the toric code on a quantum computer | PennyLane Demos](https://pennylane.ai/qml/demos/tutorial_toric_code), there are these tasks which I’m trying to do. ```auto Note I encourage dedicated readers to calculate the phase accumulated by exchanging: 1. A Z Group excitation and a Z Group Excitation 2. An X Group excitation and an X Group Excitation 3. A combination Ψ particle and an X Group excitation 4. A combination Ψ particle and a Z Group excitation 5. A combination Ψ particle with another Ψ particle The combination particle should behave like a standard fermion. You can create and move combination particles by applying PauliY operations. ``` For task 1, i.e. A Z Group excitation and a Z Group Excitation, this is my code ```python prep1 = [(1, 1), (2, 1)] prep2 = [(1, 3)] loop1 = [(1,2),(2,2),(3,2),(3,1),(2,1),(2,2)] # WITHOUT LOOPING expvals = excitations(prep1 + prep2, []) x_expvals, z_expvals = separate_expvals(expvals) fig, ax = excitation_plot(x_expvals, z_expvals) ax.plot(*zip(*(prep1)), color="maroon", linewidth=10) ax.plot(*zip(*(prep2)), color="maroon", linewidth=10) display(plt.show()) z_around_z = hadamard_test(prep1 + prep2, [], [], []) print("WITHOUT LOOPING - Mutual exchange statistics - Move z excitation around z excitation: ", z_around_z) # WITH LOOPING expvals = excitations(prep1 + prep2 + loop1, []) x_expvals, z_expvals = separate_expvals(expvals) fig, ax = excitation_plot(x_expvals, z_expvals) ax.plot(*zip(*(prep1)), color="maroon", linewidth=10) ax.plot(*zip(*(prep2 + loop1)), color="maroon", linewidth=10) display(plt.show()) z_around_z = hadamard_test(prep1 + prep2, [], loop1, []) print("WITH LOOPING - Mutual exchange statistics - Move z excitation around z excitation: ", z_around_z) ``` I did with and without looping to compare the output of the layout and hadamard test. ![z_around_z](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/a/a23d6768ae41de0afca22d412f0cfb5b030fff42.png) Question 1: Without any looping, I’m getting +1 from the Hadamard test and 0.0 with looping. Can I know which one is the correct output? Is my looping path correct since I see some “extra” excitations in my loop path? Question 2: I modified the excitation and Hadamard test code to work with combination Ψ particle using PauliY operator. Then, I did the task 3, i.e. A combination Ψ particle and an X Group excitation. The output is as below (with and without looping). Similar to the previous question’s observations, without looping, I’m getting +1 from Hadamard test and 0.0 with looping. Can I know which one is the correct output? Is my looping path correct since I see some “extra” excitations in my loop path? ![y_around_x](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/a/a79ca604fb5106dfa5d0cfe006673fefc976750b.png) Thank you. ---
**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:** [April 14, 2025, 8:49pm UTC](https://discuss.pennylane.ai/t/question-on-last-tasks-on-modeling-the-toric-code-on-a-quantum-computer-demo/8386/2 "2025-04-14T20:49:14Z")
Hi @jag , thanks for your questions! I’m checking with the team. We’ll get back to you as soon as we have capacity. We may need a couple of days since we’re working on a new PennyLane version that will be released tomorrow! 😎 ---
**Author:** ![jag](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/jag/32/3705_2.png) [@jag](https://discuss.pennylane.ai/u/jag)\ **Post date:** [April 15, 2025, 4:07am UTC](https://discuss.pennylane.ai/t/question-on-last-tasks-on-modeling-the-toric-code-on-a-quantum-computer-demo/8386/3 "2025-04-15T04:07:04Z")
Hi @CatalinaAlbornoz , thank you for looking into my questions. I’m learning a lot from Pennylane especially the tutorials. Looking forward to the new Pennylane release too. 🙂 ---
**Author:** ![christina](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/christina/32/583_2.png) [@christina](https://discuss.pennylane.ai/u/christina)\ **Post date:** [April 23, 2025, 3:07pm UTC](https://discuss.pennylane.ai/t/question-on-last-tasks-on-modeling-the-toric-code-on-a-quantum-computer-demo/8386/4 "2025-04-23T15:07:16Z")
Thanks for the question @jag . When we are calculating the statistics, we need to make sure to bring the particles back to the same spot. If the excitations are in a different place, we are in a completely different state. When we bring the excitations back to the same spot, we still have the same probability distribution but the state just has a different phase. Since the excitations are in completely different places, the hadamard test gives us 0 because there’s no overlap. With this set of prep and loop: ```auto prep1 = [(1, 1), (2, 1), (0,2)] prep2 = [(1, 3)] loop1 = [(2,3), (3,3), (4,2), (4,1), (3,0), (2,0), (1,1), (1,2)] ``` ![Screenshot 2025-04-23 at 11.03.18 AM](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/7/73d55f8c360db06452e579d92c43ce9bcc3a489c.png) We keep the excitations in exactly the same locations at the end, but one of the excitations made a full circuit around the other. In this case, we can see that the hadamard test gives us a relative phase offset of `0`, as `cos(phi) = 1`. The X and Z excitations on their own are bosonic. It’s only when they interact that things get interesting. A similar approach needs to be taken for the composite particle too.