1. The Geometry of Quantum State Space
In classical digital computing, the fundamental unit of information is the bit, existing strictly in one of two discrete states: $0$ or $1$. In quantum computing, the fundamental unit is the quantum bit (qubit). A qubit is a two-level quantum mechanical system whose state vector $|psi angle$ exists as a linear superposition in a two-dimensional complex Hilbert space $mathcal{H}_2$ spanned by the orthonormal computational basis vectors ${|0 angle, |1 angle}$:
$$|psi angle = alpha |0 angle + eta |1 angle, quad alpha, eta in mathbb{C}$$where $alpha$ and $eta$ are complex probability amplitudes satisfying the Born rule normalization condition:
$$|alpha|^2 + |eta|^2 = 1$$Because $alpha$ and $eta$ are complex numbers, each has two real parameters (magnitude and phase), totaling four real degrees of freedom. However, two physical constraints reduce these four degrees of freedom to exactly two:
- Unit Normalization ($|alpha|^2 + |eta|^2 = 1$): Eliminates one degree of freedom.
- Global Phase Invariance: Multiplying an entire state by an overall phase factor $e^{igamma}|psi angle$ leaves all measurement statistics and quantum mechanical expectation values completely unchanged ($langlepsi| mathbf{A} |psi angle = langlepsi| e^{-igamma} mathbf{A} e^{igamma} |psi angle$). Factoring out the global phase allows us to choose $alpha$ to be real and non-negative.
Applying these reductions yields the parameterized canonical representation of a pure qubit state:
$$|psi angle = cosleft( rac{ heta}{2} ight) |0 angle + e^{iphi} sinleft( rac{ heta}{2} ight) |1 angle$$where $ heta in [0, pi]$ is the polar colatitude angle and $phi in [0, 2pi)$ is the azimuthal phase angle. This maps the complex projective line $mathbb{C}P^1$ directly onto the surface of a unit sphere in $mathbb{R}^3$, termed the Bloch Sphere.
2. Coordinate Mapping: From Hilbert Space to $mathbb{R}^3$
The position of a quantum state on the Bloch sphere is defined by the 3D Cartesian coordinates of the Bloch vector $mathbf{r} = (u, v, w) in mathbb{R}^3$:
$$egin{aligned} u &= sin heta cosphi = 2 ext{Re}(alpha^* eta) = langlepsi| sigma_x |psi angle v &= sin heta sinphi = 2 ext{Im}(alpha^* eta) = langlepsi| sigma_y |psi angle w &= cos heta = |alpha|^2 - |eta|^2 = langlepsi| sigma_z |psi angle end{aligned}$$The components of the Bloch vector correspond physically to the expectation values of the three Pauli observables ($sigma_x, sigma_y, sigma_z$).
3. Standard Quantum Logic Gates as Bloch Sphere Rotations
Single-qubit quantum logic gates are represented by $2 imes 2$ unitary matrices $mathbf{U} in SU(2)$. Due to the double-cover Lie group homomorphism between $SU(2)$ and the 3D rotation group $SO(3)$, every single-qubit quantum gate corresponds to a pure spatial rotation of the Bloch sphere:
| Quantum Gate | Matrix Operator | Geometric Bloch Rotation | Canonical Transformation |
|---|---|---|---|
| Pauli-X (NOT) | $$egin{bmatrix}0&11&0end{bmatrix}$$ | $180^circ$ rotation around $X$-axis. | $|0 angle leftrightarrow |1 angle$ |
| Pauli-Y | $$egin{bmatrix}0&-ii&0end{bmatrix}$$ | $180^circ$ rotation around $Y$-axis. | $|0 angle o i|1 angle, quad |1 angle o -i|0 angle$ |
| Pauli-Z (Phase) | $$egin{bmatrix}1&0 &-1end{bmatrix}$$ | $180^circ$ rotation around $Z$-axis. | $|0 angle o |0 angle, quad |1 angle o -|1 angle$ |
| Hadamard ($H$) | $$ rac{1}{sqrt{2}}egin{bmatrix}1&11&-1end{bmatrix}$$ | $180^circ$ rotation around diagonal $(X+Z)/sqrt{2}$. | $|0 angle o |+ angle, quad |1 angle o |- angle$ |
| Phase ($S$) | $$egin{bmatrix}1&0 &iend{bmatrix}$$ | $+90^circ$ ($pi/2$) rotation around $Z$-axis. | $|+ angle o |+i angle, quad S^2 = Z$ |
| $pi/8$ Gate ($T$) | $$egin{bmatrix}1&0 &e^{ipi/4}end{bmatrix}$$ | $+45^circ$ ($pi/4$) rotation around $Z$-axis. | $T^2 = S, quad T^4 = Z$ |
4. Density Matrix & Pure vs Mixed States
The state of a qubit can be represented generally by its $2 imes 2$ density operator $ ho$. For a pure state $|psi angle$:
$$ ho = |psi angle langlepsi| = egin{bmatrix} cos^2( heta/2) & cos( heta/2)sin( heta/2)e^{-iphi} cos( heta/2)sin( heta/2)e^{iphi} & sin^2( heta/2) end{bmatrix} = rac{1}{2}left(mathbf{I} + u sigma_x + v sigma_y + w sigma_z ight)$$A pure state satisfies $ ho^2 = ho$ and $ ext{Tr}( ho^2) = 1$, placing the state vector directly on the unit sphere ($|mathbf{r}| = 1$). In real quantum computers (superconducting transmons, trapped ions), interaction with the environment causes quantum decoherence, shrinking the vector into the interior of the Bloch ball ($|mathbf{r}| < 1$). The center of the sphere $mathbf{r} = (0, 0, 0)$ represents the maximally mixed state $ ho = rac{1}{2}mathbf{I}$, where measurement yields equal $50%$ probability with zero phase coherence.
5. Worked Mathematical Example: Hadamard Gate on $|0 angle$
Initial State: $|0 angle implies heta = 0^circ, phi = 0^circ$. Bloch vector: $mathbf{r} = (0, 0, 1)$.
Step 1: Apply Hadamard Matrix
$$H |0 angle = rac{1}{sqrt{2}} egin{bmatrix} 1 & 1 1 & -1 end{bmatrix} egin{bmatrix} 1 0 end{bmatrix} = rac{1}{sqrt{2}} egin{bmatrix} 1 1 end{bmatrix} = rac{1}{sqrt{2}} |0 angle + rac{1}{sqrt{2}} |1 angle = |+ angle$$Step 2: Solve New Bloch Angles
$$cos( heta/2) = rac{1}{sqrt{2}} implies rac{ heta}{2} = 45^circ implies heta = 90^circ$$ $$e^{iphi} sin( heta/2) = rac{1}{sqrt{2}} implies e^{iphi} = 1 implies phi = 0^circ$$Step 3: New Bloch Coordinates & Measurement Probabilities
$$u = sin(90^circ) cos(0^circ) = 1.0, quad v = 0.0, quad w = cos(90^circ) = 0.0 implies mathbf{r} = (1.0, 0.0, 0.0)$$ $$P(|0 angle) = left| rac{1}{sqrt{2}} ight|^2 = 0.50 quad (50%), quad P(|1 angle) = left| rac{1}{sqrt{2}} ight|^2 = 0.50 quad (50%)$$The state has rotated from the North Pole down to the positive $X$-axis on the equator, creating an ideal equal superposition.