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kymki (kymki@undisclosed.example.com) — 2025-10-14T11:21:13+00:00

hottblockp5 - created

kymki (kymki@undisclosed.example.com) — 2025-09-23T10:26:37+00:00

Post 5 — The Hopf Fibration and Global Phase This entry introduces the Hopf fibration, which lifts Bloch sphere paths into SU(2), and shows how to visualize the accumulated global phase with the accompanying Python code. ---------- 1. Global phase and projective space $e^{i\phi}$$\mathbb{C}P^1$$S^2$\[ S^3 \longrightarrow S^2, \]$S^1$$S^3$$\mathbb{C}^2$$S^2$$S^3$$\langle \psi_k | \psi_{k+1} \rangle$$2\pi$$S^2$$S^3 \to S^2$

hottblockp4 - created

kymki (kymki@undisclosed.example.com) — 2025-09-23T10:24:57+00:00

Post 4 — Homotopies Between Loops This entry introduces homotopies as continuous deformations between loops on the Bloch sphere, with examples using the accompanying Python code. ---------- 1. Homotopy concept A homotopy is a continuous deformation from one loop $\gamma_0$ to another loop $\gamma_1$\[ H : [0,1] \times [0,1] \to S^2, \]\[ H(0,t) = \gamma_0(t), \quad H(1,t) = \gamma_1(t). \]$\pi_1(S^2) = 0$$p,q \in S^2$$s$\[ \gamma_s = \frac{\sin((1-s)\theta)}{\sin \theta} \, p + \frac{\sin(s\…

hottblockp3 - created

kymki (kymki@undisclosed.example.com) — 2025-09-23T10:22:39+00:00

Post 3 — Closed Loops and Berry Phase This entry introduces closed loops on the Bloch sphere and the Berry phase, with examples using the accompanying Python code. ---------- 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. $2\pi$$|0\rangle$$2\pi$$|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$$\{\psi_k\}$\[ \phi \approx -\operatorname{Im} \log \prod_k \langle \psi_k | \psi_{k+1} \rangl…

hottblockp2 - created

kymki (kymki@undisclosed.example.com) — 2025-09-23T10:20:21+00:00

Post 2 — Unitary Evolutions as Paths on the Sphere This entry introduces how unitary operations generate continuous paths on the Bloch sphere with the accompanying Python code. ---------- 1. SU(2) rotations A unitary rotation about a unit axis $\mathbf{n}$ through an angle $\theta$ is given by\[ U(\theta) = \cos\left(\tfrac{\theta}{2}\right) I - i \sin\left(\tfrac{\theta}{2}\right)(\mathbf{n} \cdot \boldsymbol{\sigma}). \]$I$$\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$$\mathbf{n}$$…

hottblockp1

kymki (kymki@undisclosed.example.com) — 2025-09-23T09:46:55+00:00

Post 1 — Qubits and the Bloch Sphere This first entry introduces the Bloch sphere representation of qubits and demonstrates how to obtain Bloch vectors with the accompanying Python code. ---------- 1. Qubit states and density matrices A pure qubit state is written as \[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1. \]\[ \rho = |\psi\rangle \langle \psi |. \]\[ \sigma_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \…

sierp.png - created

kymki (kymki@undisclosed.example.com) — 2025-03-27T07:24:12+00:00

qas-cos.pdf - created

kymki (kymki@undisclosed.example.com) — 2024-12-04T17:35:08+00:00