The Algebra of Light- Building Logical States with SAM–OAM Qudits
- Quantum Quill
- Jul 31
- 2 min read
Updated: Sep 9
CHAPTER 8
How a Single Photon Becomes a Four-Level Processor
1 From Qubits to Qudits
Classical quantum hardware scales by adding more two-level qubits: millions of devices, control lines and cryogenics. A structured photon, by contrast, carries several orthogonal modes inside one particle. In our architecture, the photon’s total angular momentum
J=SAM+OAM
naturally supplies four distinct basis states—a ququart. One physical carrier now performs the logical work of multiple qubits, reducing hardware count while enlarging Hilbert space.
2 Choosing the Four-State Basis
We select the two polarisation handednesses (SAM = ±ℏ) and the first two orbital charges (ℓ = 0, +1). That yields the table below, mapping one-to-one onto the four Z₂ × Z₂ sectors introduced in Chapter 5:
Why these four? Any orthogonal set of m/ℓ pairs would work, but ℓ = 0 and +1 keep mode purity above 90 % in standard silicon photonics waveguides. Mode purity means that the launched field overlaps almost entirely with the intended eigen-mode (that is, the specific light pattern the waveguide guides cleanly), and leaks only minimally into unwanted neighbours. Higher |ℓ| values remain orthogonal in theory, yet couple more strongly to side-wall roughness, lowering purity and increasing logical error.
3 Writing the State On-Chip
The initial ququart is created without bulk optics. A trench-patterned waveguide array—essentially an integrated spatial-light modulator—imposes programmable phase shifts that set both SAM and OAM. Sub-wavelength heaters (or EO segments) tweak phases in micro-seconds, and shallow gratings strip away residual modes, delivering a clean four-state basis ready for logic.
4 Turning Geometry into Gates
Logical operations are deterministic detours through the waveguide mesh:
Phase-X rotates OAM by π, swapping |01⟩ ↔ |11⟩.
Polarisation-Z adds a π phase only to m = –1 sectors.
CNOT (SAM → OAM) is realised on-chip by routing the two SAM polarisations through different interferometric arms: the arm for m = –1 contains a short spiral segment that adds one unit of OAM, while the arm for m = +1 is straight. When the paths recombine, ℓ is flipped only for the m = –1 component—no bulk optics or separate phase plate required.
Toffoli (CCNOT) is built from two CNOTs plus a single π detour that acts only on the state (m = –1, ℓ = +1), exactly as detailed in Fig. 2 of our paper.
Because every gate is a fixed geometric path, small fabrication variations shift all four modes together; the logical transformation is preserved—an intrinsic topological robustness.
5 Scaling Strategies
Time-bin multiplexing clocks successive photons through the same chip, yielding tens of qudits per device.
Frequency combs duplicate the SAM–OAM rail at multiple telecom wavelengths, enabling massive parallelism.
Inter-die coupling with direction-selective gratings keeps ℓ intact when photons hop between chips.
With time-bin multiplexing, frequency rails, and inter-die coupling, a single shoe-box-sized module can reach a Hilbert-space volume comparable to today’s multi-kilo-qubit data-centre rigs—while compressing device count and footprint by roughly two orders of magnitude. In other words, the same logical power now fits into a SWaP-optimised package for true edge deployment: no cryogenics, no server-room racks—just structured-photon chips and a few laser drivers.
