How would AP Qiyana work?

The obvious answer is, "Not as well as AD Qiyana"

Sadly, in my experience, people typically downvote posts for even suggesting that people consider things like this. If you can manage to hold off on that and think outside the box with me, I would really appreciate it.

Thanks,

Her Passive and her W both have +30% AP ratios.

PASSIVE - ROYAL PRIVILEGE - Innate: Qiyana's next basic attack or damaging ability against an enemy units deals 15 − 83 (based on level) (+ 55% bonus AD) (+ 30% AP) bonus physical damage.

This effect cannot occur on the same target more than once every few seconds.

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W - TERRASHAPE - Active: Qiyana dashes X-units in the direction of a target location. If targets terrain, river or brush, Qiyana also gathers an Element in the process. If the gathered Element is different from the previously held, Royal Privilege's cooldown on all targets is Cooldown reduction icon reset.

While Qiyana holds an Element, she gains 20% bonus attack speed, and her basic attacks and basic abilities deal bonus magic damage. While near the corresponding Element's source and out of combat, Qiyana additionally gains bonus movement speed.

Bonus Magic Damage: 8 / 16 / 24 / 32 / 40 (+ 20% bonus AD) (+ 30% AP)

Theoretically these can stack, meaning if Qiyana holds an element, her first basic attack or basic ability against a target would deal 60% of her AP (half as magic and half as physical damage) to the target. Until she uses her Q, subsequent basic attacks and basic abilities would deal 30% of her AP as magic damage, and she will have +20% attack speed. After using her Q, she can reset her Passive and W-Passive by gathering a new (different from previous element) element with W. She can also reset her Q cooldown with E.

An AP on-hit build might work.

Otherwise, functioning more like a caster would allow her to use a big Q to targets below 50% health. Resetting her passive by grabbing Earth with W and resetting her Q with E ... I'm assuming that the passive damage and bonus damage from W is also increased by 60%, meaning her Q will apply 96% of her AP as bonus damage.

Edit: Looks like this last part isn't accurate. Only the Q damage is increased by 60% when the enemy is below 50% health. But, that at least means that you aren't missing out on as much scaling by not building AD. Resetting her passive with W then using the Wall Enchanted Q would still be her best burst combo.