Editing Openai/6963b524-bf70-8010-b81c-e31f5de96ab9 (section)

=== Assistant: Short answer: nope, not real Shor. ===
Short answer: nope, not real Shor. Not even close.

Shor lives and dies by the Quantum Fourier Transform (QFT). And QFT is basically a festival of tiny phase rotations:
R2,R3,R4,…R_2, R_3, R_4, \dotsR2,R3,R4,… with angles like π/2, π/4, π/8, …
Those are not Clifford. Toffoli doesn’t help with phases at all — it’s a beefy classical logic gate in a quantum costume.

Clifford + Toffoli can do:
* all stabilizer stuff (Bell states, GHZ, teleportation, syndrome extraction, etc.)
* any reversible classical computation (modular multiplication, adders, control logic — all good)

But Shor needs:
* modular exponentiation (you can do this with Toffoli-heavy reversible circuits, so that part is fine)
* QFT with non-Clifford phase gates (this is where everything breaks)

Without those non-Clifford rotations, the interference pattern that reveals the period just… never forms. You measure noise, not number theory.

Now, tiny loophole that people sometimes bring up:
You can approximate QFT using Clifford + T (since T is non-Clifford), and Toffoli can be decomposed into Clifford+T anyway. So once you allow T gates, boom, universality unlocked, Shor is back on the menu.

So the scorecard:
* Clifford + Toffoli → ❌ not universal, ❌ can’t run Shor
* Clifford + T → ✅ universal, ✅ can run Shor
* Toffoli alone → king of classical, zero quantum mojo

In other words: you’ve got the calculator and the wiring, but not the phase wizardry. And Shor is basically a phase wizard in a hoodie doing number theory behind the scenes. 🧙‍♂️⚡

This ties straight into your MoE / fast-learning experiments too, by the way: interference is the secret sauce, and interference needs continuous phases, not just discrete flips.