Graded Paraparticle Algebra of Majorana Fields for Multidimensional Quantum Computing with Structured Light

TL;DR

This paper develops a theoretical framework combining Majorana fields, paraparticle algebra, and structured light to enable advanced quantum computing with photonic qubits and error correction.

Contribution

It introduces a novel algebraic approach integrating Majorana equations with paraparticles for quantum computing applications in photonics.

Findings

Derived a Majorana mass-spin relation for structured light quasiparticles.

Extended quantum gate concepts to photonic platforms using paraparticle algebra.

Proposed error correction methods leveraging fractional statistics in graded qudits.

Abstract

We present a theoretical framework that integrates Majorana's infinite-component relativistic equation within the algebraic structure of paraparticles through the minimal nontrivial $\mathbb{Z}_2 \times \mathbb{Z}_2$--graded Lie algebras and $R$-matrix quantization. By mapping spin-dependent mass spectra to graded sectors associated with generalized quantum statistics, we derive an equation embodying Majorana's mass-spin relation describing Majorana quasiparticles of structured light carrying spin and orbital angular momentum. These quanta in the $\mathbb{Z}_2 \times \mathbb{Z}_2$--graded algebras and $R$-matrix formulations extend the previous results from superconducting qubits to photonic platforms and set up deterministic 2-photon gates involving at least two qubits encoded in a single photon without nonlinear effects. This makes feasible general quantum computing pathways exploiting fractional statistics through Nelson's quantum mechanics and implement a novel procedure for error correction in photonic platforms. Furthermore, this approach makes possible to set paraparticle-based quantum information processing, beyond fermions and bosons, using graded qudits.