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

Fabrizio Tamburini, Nicolò Leone, Matteo Sanna, Roberto Siagri

TL;DR

This work unifies Majorana’s infinite‑component mass–spin framework with paraparticle (Green) statistics by employing a minimal $\,\mathbb{Z}_2 \times \mathbb{Z}_2\,$-graded Lie algebra and $R$-matrix quantization, linking the Majorana tower to graded sectors that host paraboson and parafermion dynamics. It then translates this algebraic structure into a concrete photonic platform based on structured light (OAM/SAM) to realize a four‑mode ququart that behaves as a $\,\mathbb{Z}_2 \times \mathbb{Z}_2\,$ paraparticle, enabling deterministic one‑photon two‑qubit gates and universal quantum computation without nonlinear media. The paper also develops a fractional Nelson framework to model graded, non‑Markovian noise and proposes diagnostic and correction schemes that leverage the graded statistics to enhance fault tolerance in photonic paraparticle circuits. By bridging relativistic paraparticle theory, braided Hopf algebras, and photonic hardware, the work opens pathways to paraparticle‑based quantum information processing and broader exploration of exotic statistics beyond fermions and bosons. The results provide a rigorous mathematical foundation for para-Majorana modes in SAM–OAM photonics, outline explicit gate sets with proven universality, and chart a practical route toward scalable, room‑temperature quantum computing using structured light. Overall, this framework suggests a versatile platform for exploring new quantum statistics, error correction methods, and potentially topological computation within integrated photonics.

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.

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

TL;DR

This work unifies Majorana’s infinite‑component mass–spin framework with paraparticle (Green) statistics by employing a minimal -graded Lie algebra and -matrix quantization, linking the Majorana tower to graded sectors that host paraboson and parafermion dynamics. It then translates this algebraic structure into a concrete photonic platform based on structured light (OAM/SAM) to realize a four‑mode ququart that behaves as a paraparticle, enabling deterministic one‑photon two‑qubit gates and universal quantum computation without nonlinear media. The paper also develops a fractional Nelson framework to model graded, non‑Markovian noise and proposes diagnostic and correction schemes that leverage the graded statistics to enhance fault tolerance in photonic paraparticle circuits. By bridging relativistic paraparticle theory, braided Hopf algebras, and photonic hardware, the work opens pathways to paraparticle‑based quantum information processing and broader exploration of exotic statistics beyond fermions and bosons. The results provide a rigorous mathematical foundation for para-Majorana modes in SAM–OAM photonics, outline explicit gate sets with proven universality, and chart a practical route toward scalable, room‑temperature quantum computing using structured light. Overall, this framework suggests a versatile platform for exploring new quantum statistics, error correction methods, and potentially topological computation within integrated photonics.

Abstract

We present a theoretical framework that integrates Majorana's infinite-component relativistic equation within the algebraic structure of paraparticles through the minimal nontrivial --graded Lie algebras and -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 --graded algebras and -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.

Paper Structure

This paper contains 56 sections, 1 theorem, 126 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Statistics of Parabosons and Parafermions
  3. Large $p$ values
  4. Photonic OAM Majorana Quasiparticles and Paraparticle Statistics
  5. Embedding Majorana Tower Particles into the Paraparticle Framework
  6. The Hilbert space.
  7. $\mathbb{Z}_2 \times \mathbb{Z}_2$-Graded Sector Decomposition
  8. Graded Projectors and Exchange Matrices $X_{(a,b)}$ for the Majorana Tower
  9. Quantum Computing with OAM Photons and paraparticle formulation
  10. Paraparticles OAM Photonic quasiparticles and Majorana states in Quantum Circuits
  11. OAM Photonic Qudits and Parastatistics in Quantum Circuits
  12. Deterministic single-photon 2-qubit gates:
  13. Deterministic 1-photon 2-qubit deterministic gates in photonic platforms
  14. Photonic realization of the Z2×Z2 Green algebra
  15. Guided--mode basis
  16. ...and 41 more sections

Key Result

Lemma 3.1

Let $\{\gamma^\mu\}_{\mu=0}^{3}$ be the Dirac matrices on a spinor space $\mathcal{S}$, satisfying $\{\gamma^\mu ,\gamma^\nu\} = 2\eta^{\mu \nu} \mathbf{1}_{\mathcal{S}}$ with $\eta= diag(+,-,-,-)$. Let $\{P_{(a,b)}\}$$(a,b\in\{0,1\})$ be the four central idempotents of the group algebra $\mathbb C[ On the Hilbert space $\mathcal{H}=\bigl(\bigoplus_{a,b}\mathbb C_{(a,b)}\bigr)\! \otimes \mathcal{S

Figures (4)

  • Figure 1: Schematic of a deterministic CNOT gate
  • Figure 2: Toffoli Gate with path entanglement and graded SAM-OAM modes
  • Figure 3: Jordan--Wigner transformation conversion scheme
  • Figure 4: Schematic decomposition of the $\mathbb{Z}_4$ ququart system into two qubits $A$ and $B$. The clock operator $\sigma$ is decomposed into $P = \sigma_z \otimes I$ and $Q = I \otimes \sigma_z$. The cyclic shift $\tau$ is implemented by a Hadamard $H$ on qubit $A$, a phase gate $S$ on qubit $B$, and a controlled-NOT ($\mathrm{CNOT}$) from $A$ to $B$.

Theorems & Definitions (2)

  • Lemma 3.1: graded--Clifford closure
  • proof