I am a casual league player with multiple accounts with a peak rank at diamond IV and this is my first attempt at writing a guide for the so called AP Lucian. One of my accounts is a Hybrid Lucian only and ive played this way for the past few seasons. AP lucian is quite controversial when being played mostly because teammates in ranked feel that its a troll build. In fact, AP lucian is a painful champ to deal with because of his W, ardent blaze. Its high AP scaling can chunk large portions of health if the enemy is underestimating you. Along with his lightslinger passive and lich bane, Lucian becomes very deadly.

Pros
insane poker with his ardent blaze AP scaling
can easily chase down fleeing enemy champs
escape is relatively decent with the right usage of ardent blaze and relentess pursuit
ult is also an insane dps skill

Cons
requires a lot of skill
timing is hard to pull off
one wrong engage and death is near
your Q is practically useless as a damage factor until you have both gunblade and rageblade
miss you ardent blaze and youll never kill the enemy

Your Q is only needed to proc the lightslinger passive, hit enemy champs hiding behind minions, and minion farming
It is recommended to use E first because lightslinger reduces the cooldown of it,, maximizing the damage you can deal through autoattacks

Standard engaging:
E > AA > W >> AA > Q >> AA

(E sideways or diagonally to dodge skillshots and autoattack after each spell if possible)

Chasing:
E > AA > W > AA > R
under half health:
R > E

(Autoattack whenever you can and ult once under half health)

Disengage:
E > W > AA > R
or
W > E > AA > R

(depending on distance of the enemy champion, W if far enough to follow with an autoattack. E first and then W autoattack for a speed boost. If theres enough time and enemy is low in health, ult as youre running backwards)

Your ardent blaze is an effective skill for farming especially for those ranged minions. Q can easily clear the melee minions along with your passive. Save your ardent blaze for cannon minions since that skill does a higher amount of damage. Don't be afraid to use your skills and mana to farm the minions. Once you have lich bane it becomes much easier.

Stay in the back until all the crowd control skills have been used. Lucian is not a tanker and its best to avoid taking any damage at all. His long range high damage output W is an effective poker against grouped enemies. if theyre clustered well enough, a single ult can chunk huge portions of health or kill off a champion caught in all the bullets.