Driving the register by hand is one thing; compiling an arbitrary target into the native gates is the synthesis problem. X_d and F_d are Clifford and only generate a finite group — adding one non-Clifford T_d (a phase at the golden angle, the maximally irrational rotation, the zero-free-parameter choice that makes the set universal with no low-order resonances) makes {X_d, F_d, T_d} dense in SU(d). Finding a short word for a target is then a search; the bidirectional co-adaptive search (grow a prefix and a suffix together, each scored against the other) reaches lower mean error than a one-directional beam at a matched budget. This runs at the register's selected dimension — switch the size on the left (d = 2 → 10 → 12) to watch the co-adaptive advantage grow with d. (Full sweep at a larger budget: package benchmark bench_qudit_gate_synth.)

This is the architecture's first-class qudit: not a qubit with extra levels bolted on, but a native multi-state register carried by the rich internal state of a single atom. The same atom offers two register sizes, chosen by control-laser frequency alone: a 10-state mode and a 12-state mode. Each bar above is one state of the register; its height is the probability, its colour the quantum phase. The qubit (d = 2) is just the special case.

The gates are the generalized Weyl set. The shift X_d cycles the states, the clock Z_d winds a phase ω^j = e^2πij/d onto each, the Fourier F_d spreads a single state into an equal mix of all d (watch one bar become d equal bars), and the phase gate P(θ) is clean: it imprints exactly the angles you ask for, with no distortion. Measure draws samples; the histogram overlays the live probabilities.

The lower panel weaves two qudits: a Fourier on the first, then a SUM_d (the qudit version of a CNOT) onto the second, builds the maximally entangled pair. All the probability lands on the diagonal a = b (perfect correlation). Its entanglement is not asserted but computed: the entropy of one qudit's state comes out to exactly ln d. Everything here runs the real φ-quantum qudit engine, compiled to WebAssembly, live as you click.