Each column in the table corresponds to a different gate. Just start by playing the animations.
The dissipation coefficient is set to 1 in all simulations bellow. Therefore the time is dimensionless.
| Two Blobs | Three Blobs | ||||
|---|---|---|---|---|---|
| Phase Gate | Population Gate | Phase Gate | 2-Population Gate | 3-Population Gate | |
| Description | The right blob goes on a circle. | The two blobs collide. | The right blob goes on a circle. | Two of the three blobs collide. | The three blobs collide. |
| Initial State | \(\left| \alpha_0 \right\rangle + \left| -\alpha_0 \right\rangle\) | \(\left| \alpha_0 \right\rangle\) | \(\left| \alpha_0 \right\rangle + \left| \scriptstyle\sqrt[3]{-1} \displaystyle\alpha_0 \right\rangle + \left| \scriptstyle\sqrt[3]{-1}^2 \displaystyle\alpha_0 \right\rangle\) | \(\left| \alpha_0 \right\rangle + 2\left| \scriptstyle\sqrt[3]{-1} \displaystyle\alpha_0 \right\rangle\) | \(\left| \alpha_0 \right\rangle\) |
| Animation for \(\alpha_0=2\) | |||||
| Scaling of the Purity wrt \(\alpha_0\) and \(T\) |
\(1-P\propto\alpha_0^{-1.815}T^{-0.976}\) 
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\(1-P\propto\alpha_0^{-3.945}T^{-0.984}\) 
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\(1-P\propto\alpha_0^{0.967}T^{-0.988}\) 
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| Scaling of the Purity wrt \(\alpha_0\) and \(v\)(the phase-space "speed" of the blobs) |
\(1-P\propto\alpha_0^{-1.815}v^{0.976}\) \(v\propto 1/T\) |
\(1-P\propto\alpha_0^{-3.945}v^{0.984}\) \(v\propto 1/T\) |
\(1-P\propto\alpha_0^{-0.020}v^{0.988}\) \(v\propto\alpha_0/T\) |
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| The plots from the animation |
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| Other remarks | Only the area matters for the induced phase. The loss in purity also seems to scale with the area, given that the results for a single circle of area A and for two circles each of area A/2 were the same (the total area was the same, the path taken was √2 bigger). See the next table. | For a fixed \(\alpha_0\) the worst imperfections induced during the evolution are closer to the center. | Strange artefacts for the outer most blob, maybe due to cutoff dimension being too low. | For a fixed \(\alpha_0\) the imperfections are induces mainly on the way into and out of the collision (not during the collision itself). | |
| Technical Details | Various, see the code. 15000 samples, N=80 cutoff for the (1-P) log log plot. 20000 samples, N=40 for the comparisons in the next table, 15000 samples, N=40 for the animation. | 60000 samples, N=40 cutoff | 200000 samples, N=55 cutoff (which is insufficient for \(\alpha_0 \gt 2.6\)) | 680000 samples, N=55 cutoff (possibly insufficient for these plots and animations) | 680000 samples, N=55 cutoff |
| Computer Code Files | phase gate, population gate, animation | visualization, simulation | visualization, simulation | visualization, simulation | |
| Two Blobs, Only the Right One Moving | Two Blobs Moving in Opposite Directions | |
|---|---|---|
| Plots and animations for various radiuses of the circles. | ||
| Description | The right blob goes on a circle. | Both blobs go on circles. The total area of the circles is the same as in the one-blob case. |
| Scaling of the Purity wrt \(\alpha_0\) and \(T\) |
\(1-P\propto\alpha_0^{-1.815}T^{-0.976}\)
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\(1-P\propto\alpha_0^{-1.797}T^{-0.974}\) 
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| The final relative phase between the blobs |
Total time T on the x axis |
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| The final purity of the system |
Total time T on the x axis |
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| The contrast between the final purities for both approaches |
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