In Praise of Qudits: Why Higher-Dimensional Quantum Systems Deserve Centre Stage

Abstract

The qubit has become so dominant in quantum computing that alternatives are rarely discussed, yet it is a historical accident: two-level systems were simply the easiest to isolate experimentally. This paper argues that qudits — $d$-dimensional quantum systems with $d > 2$ — offer strictly more across every metric: error-correction headroom, richer topological structure, sharper computational phase transitions, and a natural home for the exceptional algebras that govern the deepest structures in mathematics and physics. Two values of $d$ are mathematically privileged: $d=4$ (spacetime spinors, two-qubit entanglement, quaternionic structure) and $d=8$ (octonions, $G_2$, the Fano plane). These are consequences of Hurwitz's theorem (1898), which identifies $d \in {1,2,4,8}$ as the only dimensions supporting normed division algebras. The paper covers: the $\mathrm{SU}(4)$ containment that makes two-qubit logic native to a single ququart; Dirac spinors as the natural $d=4$ Hilbert space; the $d=8$ octonion layer and its $G_2$ symmetry; topological advantages of higher $d$ including Hopfion braiding and non-Abelian statistics; and the Hopf fibration tower $S^1 \to S^3 \to S^2 \to \cdots \to S^7 \to S^{15} \to S^8$ as a construction manual for building qudits from qubits. v2.0 adds five new developments: The padit tower. The $p$-adic qudit (padit) in $L^2(\mathbb{Z}/p^n\mathbb{Z})$ extends Hurwitz to all prime powers $d = p^n$. Qubits are the $p=2$, $n=1$ special case. The Number-Theoretic Transform is the exact $p$-adic QFT (no floating-point error). The Fano plane is the $p=2$, $n=3$ padit — the natural home of the 3-qubit Origami ISA and the Steane $[![7,1,3]!]$ code. Bredon cohomology cascade. Equivariant cohomology of the Weyl chamber proves $\chi_\mathrm{Bredon} = 2$ (2-qubit, one topological class distinguishing CNOT from iSWAP) $\to 0$ (3-qubit, equivariantly contractible) $\to$ non-trivial (7-qubit Fano). The 3-qubit native gate architecture (TriQ) adds no topological value; the correct jump is directly from 2-qubit to Steane-7. FeMoco as a room-temperature $d=8$ computer. The iron-molybdenum cofactor of nitrogenase contains 7 Fe atoms corresponding to the 7 points of $\mathrm{PG}(2,2)$. The $\mathrm{N_2}$ fixation reaction is a 14-opcode Fano programme running at 300 K, decoherence-free in the $G_2$-symmetric Clifford sector. Nature builds $d=8$ qudit computers; we merely need to recognise them. Shor's algorithm has zero magic. The magic resource theory shows that Shor's algorithm achieves exponential speedup with mana $= 0$: all gates lie in the Clifford sector. Qudits have more Clifford structure, not more overhead. The standard objection — that higher $d$ requires more magic-state distillation — is precisely backwards. The architecture ladder. The Bredon cascade, the Hopf fibration tower, and the padit tower converge on the same four-level stack: native 2-qubit gates ($d=4$, $\chi_\mathrm{Bredon}=2$) $\to$ Steane $[![7,1,3]!]$ logical qudits ($d=8$, Fano) $\to$ compiled Toffoli and above. The open problem of v1 (design a $G_2$-symmetric Fano QEC code) is resolved: the Steane code is that code. Third, molecular evidence: the FeMo-cofactor [Mo–7Fe–9S–C–homocitrate] of nitrogenase is a 7-qubit Fano computer running at 300 K, room temperature, decoherence-free in the $G_2$-symmetric Clifford sector. The 7 Fe atoms occupy the 7 points of $\mathrm{PG}(2,2)$; the molybdenum apex and the interstitial carbide carbon (Spatzal et al. 2011, Science) provide structural scaffolding but do not carry Fano-register qubits. Nature builds $d=8$ qudit computers; we merely need to recognise them.