====== Post 3 — Closed Loops and Berry Phase ======

This entry introduces closed loops on the Bloch sphere and the Berry phase, with examples using the [[https://github.com/erikkallman/hottbloch|accompanying Python code]].

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===== 1. Closed loops on the sphere =====

Certain unitary evolutions return a qubit state to its starting point.  
On the Bloch sphere these correspond to closed loops.  

Two canonical examples are:

* The **meridian loop**: a full $2\pi$ rotation about the y-axis, starting at $|0\rangle$.  
* The **equatorial loop**: a full $2\pi$ rotation about the z-axis, starting at $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$.

Both loops close after one full rotation.

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===== 2. The Berry phase =====

When a quantum state traces out a closed path, it accumulates a geometric phase.  
For a discretized path of states $\{\psi_k\}$, the Berry phase is estimated by

\[ \phi \approx -\operatorname{Im} \log \prod_k \langle \psi_k | \psi_{k+1} \rangle, \quad \psi_N = \psi_0. \]

For a qubit, this equals half the solid angle enclosed by the loop on $S^2$.

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===== 3. Using the code =====

The function **estimate_berry_phase** implements this calculation.  
Example:

<code python>
import hottbloch as h

# Meridian loop (rotation around y-axis)
states_a, loop_a = h.loop_meridian()
phi_a = h.estimate_berry_phase(states_a)
print("Berry phase (meridian loop):", phi_a)

# Equatorial loop (rotation around z-axis)
states_b, loop_b = h.loop_equator()
phi_b = h.estimate_berry_phase(states_b)
print("Berry phase (equatorial loop):", phi_b)
</code>

Typical output:

* Meridian loop: phase ≈ π.  
* Equatorial loop: phase ≈ –π.  

Signs depend on orientation, but magnitudes match the expected half solid angle.

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===== 4. Visualization =====

To plot both loops:

<code bash>
python hottbloch.py --out ./hott_outputs --loop meridian --theta 6.283185
python hottbloch.py --out ./hott_outputs --loop equator --theta 6.283185
</code>

Figures of the meridian and equatorial loops are saved in `./hott_outputs`.

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===== Conclusion =====

Closed loops on the Bloch sphere give rise to Berry phases, a geometric property of quantum evolution.  
This provides the foundation for studying homotopies — continuous deformations between loops — in the next post.