The Algebra of Light: Chapter 3
- Quantum Quill
- Jun 26
- 2 min read
Updated: 4 days ago
From Total Angular Momentum to Logic
How the Geometry of Light Becomes Our Native Hardware
Why Look Beyond Qubits?
Superconducting transmon grids, trapped-ion chains, neutral-atom arrays (Rydberg-excited), silicon CMOS electron-spin qubits, diamond NV-centre spins, and polarization-encoded photons all encode quantum information by selecting two-level subspaces from their underlying physical manifolds. While this qubit-based approach provides a robust foundation for quantum computation, it necessitates scaling through increasing qubit counts while simultaneously addressing the fundamental challenges of decoherence, inter-qubit crosstalk, and control field noise.
Our path is different: we use a single structured photon whose total angular momentum
J=SAM+OAM
already lives in a high-dimensional Hilbert space. Instead of stacking more qubits, we widen the alphabet itself.
What Exactly Is J?
Component Physical meaning Typical label
SAM Circular polarisation (spin) ± ℏ
OAM Helical twist of the wavefront… ℓ ℏ (ℓ = 0, ±1, …)
By selecting and coupling SAM and OAM, one photon becomes a ququart (4-level) or higher-dimensional qudit—exactly matching the four sectors of our Z₂ × Z₂-graded algebra.
Geometry as a Logic Gate
State preparation A programmable integrated photonic circuit (phase shifters + mode converters) writes the desired SAM/OAM pair—no bulk waveplates or q-plates.
Deterministic operations Waveguide geometries impose precise phase shifts; the graded-algebra sign (-1)^a·b tells us whether two operations commute, anticommute, or braid.
Read-out Mode sorters and polarisation analysers project the final J state onto single-photon detectors—no probabilistic post-selection.
Result: CNOT and Toffoli gates on demand at room temperature, using one photon and linear optics.
Why This Shifts the Paradigm
Criterion Qubit array (add more two-level units) Single SAM + OAM photon
Phys. footprint: Many individual qubits wired in an array one high-dimensional qudit
Op. temperature: Often milli-kelvin (superconducting) Room temperature
Gate model: Probabilistic or feed-forward corrections Geometry-protected,deterministic
Native dim. per carrier: 2 4+
From Geometry to Paraparticles
Because the four J-sectors map one-to-one onto the Z₂ × Z₂ grading, a single photon can emulate bosonic, fermionic, and the two paraparticle statistics first hinted at by Majorana’s infinite-spin tower. In plain terms: one photon can act like four different species—on command—making paraparticle logic a laboratory reality.
