1. Conceptual Framework: Geometrical Architecture
When tracking a quantum algorithm, we map where multi-qubit configuration states live inside Hilbert space.
- Dimensional Scaling Laws: Each added qubit expands the vector arena exponentially. A single qubit uses a 2D space, while a two-qubit configuration requires a 4D complex Hilbert space.
- Allowed Dimensions vs. State Occupation: Distinguish between the total available tracks ($2^n$ dimensions) and the specific tracks currently occupied by nonzero coefficients.
2. Matrix Track Tracing: The $H \otimes I$ Implementation
Consider applying Hadamard to qubit 0 while qubit 1 remains unchanged (identity), with initial state $|00\rangle$.
Step 1: Synthesize the Unified Circuit Operator
$$U = H \otimes I = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \otimes \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
$$U = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} & 1 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\
1 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} & -1 \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\end{bmatrix}
= \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1
\end{bmatrix}$$
Step 2: Vector Matrix Transformation
$$U|00\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & -1 & 0 \\
0 & 1 & 0 & -1
\end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}
= \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}
= \begin{bmatrix} 1/\sqrt{2} \\ 0 \\ 1/\sqrt{2} \\ 0 \end{bmatrix}$$
In Dirac notation, this becomes:
$$|\psi_{\text{final}}\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|10\rangle$$
3. State Tracking and Amplitude Verification
- State Equivalence: The system is split equally between $|00\rangle$ and $|10\rangle$.
- Basis Separation: Because the gate acts on qubit 0, coefficients for $|01\rangle$ and $|11\rangle$ remain $0$.
- Normalization Check: The total probability remains 1:
$$\sum_i |c_i|^2 = \left|\frac{1}{\sqrt{2}}\right|^2 + |0|^2 + \left|\frac{1}{\sqrt{2}}\right|^2 + |0|^2 = \frac{1}{2} + 0 + \frac{1}{2} + 0 = 1$$