Qing is electro Alhaitham.
One can run 4p TF on Qing. But if you want to do a quicken build, 4p Gilded is better for the EM buff. Qing will not give herself EM like Cyno does, which means in general Qing will have less EM. This makes Gilded even more valuable for Qing.
Look for crit, atk substats. Qing does not need much ER given her 40 cost burst. Use ATK sands, electro goblet, crit helmet. EM sands may be viable for some quicken/quickbloom comps.
For weapons, Jade Cutter, Jingjin, Mist, Skyward Blade, Foliar are 5 star options. Festering, Black Sword, Dongfang Umbrella, Iron Sting, Harbinger of Dawn are also viable options.
See Cyno for example teams. Qing can technically be used in almost any team Cyno is played in. The difference is that Cyno will tend to have more dps, while Qing is more of a quickswap unit.
But note that C6 Bennett's pyro infusion will override Qing's electro infusion, whereas Cyno's electro infusion cannot be overridden.
There is also physical Qing. This involves doing physical attacks with Qing, running pale flame/bloodstain/physical goblet, using Aquila Favonia, using a cryo unit for superconduct, perhaps Mika too. However, the dps from physical Qing is generally less than playing electro Qing given that much of Qing's damage from E and Q is purely electro. As such, it is generally not recommended to play physical Qing when other physical dps like Eula exist.
The reason Qing has higher base atk (323) than Cyno (318) is because Genshin believes the Qing dynasty was more powerful than the Tang dynasty. The reason Scaramouche has higher base atk (328) than Cyno (318) is because Genshin believes the Han dynasty was more powerful than the Tang dynasty. That is their opinion. They were all pretty powerful dynasties. As for Alhaitham (313)... that probably has more to do with Genshin's opinion on the Sui dynasty.
One must ask if the base atk has any correlation with the total compensation for these characters...
Level 10 q has total scaling $$s_q = 158.4 + 8(43.2) + 339.84 = 843.84$$ It costs 40 energy and has 12s cd. Assuming 61.6 electro dmg bonus for Qing (e.g. 2tf) the dps is $$x_q = 1.616s_q/12 = 113.63712$$
Level 10 e has total scaling $$s_e = 90.72 + 302.4 = 393.12$$ This is for doing double e to get infusion. (If one does e then ca detonation scaling is \(90.72 + 2(151.2) = 393.12\) which is same as above). Note c1 adds extra instance of damage to double e, but we are considering c0 here. Her infusion can be overridden by Bennett c6 and Chongyun.
The e has a 7.5s cd. The dps is $$x_e = 1.616s_e/7.5 = 84.704256$$ She gains an electro infusion from this ability and can also plunge attack with it.
Total dps from e and q only is $$x_e + x_q = 198.341376$$
Usually, one spams electro-infused CA when using Qing as a dps. At level 10, CA has scaling \(151.81+170 = 321.81\) and N1 has scaling 81.09. Qing can do 4 N1C1 within her infusion duration (before refreshing with E again). The dps is: $$x_c = 1.616*4*(81.09+321.81)/7.5 = 347.24608$$ Most of Qing's dps is from her CA. This assumes she does not run out of stamina (in reality, the dps will be less to account for time spent recovering stamina).
Her CA and Q have ICD, so not every hit will be buffed by aggravate. For most sword/claymore users, ICD between NA and CA are shared (except Ayaka/Ayato). For most polearm/bow/catalyst users, ICD between NA and CA are separate (except Xiao).
If Qing does 12 hits with her N1C1 combo, she can proc 4 aggravates. Suppose her e procs 1 aggravate. Her q does 10 hits and can proc 4 aggravates.
How much EM can she get? If running 4p gilded, she can get say 180 em, but will have 46.6% electro dmg bonus. If running 2p gilded 2p TF, she can get say 80 em. Or we can assume 4p TF and 0 EM. The 4p TF effect will still buff quicken (but not as much as EM would).
We will assume 0 EM and 4p TF, which buffs aggravate by 20%. The flat damage increase for aggravate at 0 EM is 1663.56. Increased by 20%, this becomes 1996.272. Standardizing for 3k atk with Bennett, this is equivalent to 66.5% extra scaling per aggravate.
So for her CA combo with 4 aggravates, the dps becomes $$x_c = 1.616*4*(81.09+321.81+66.5)/7.5 = 404.6$$ This is less than a 20% increase in damage.
As for her Q with 4 aggravates, the dps becomes $$x_q = 1.616(s_q + 4*66.5)/12 = 149.5$$ This is around a 30% increase.
Keqing's dps with quicken is less than Cyno's dps without quicken. While quicken does boost Keqing's damage, it does not boost it as much as for some other characters, given that Keqing's base kit lacks EM scaling and does not have as much synergy with dendro. Perhaps her burst (which is multi-hit and can proc aggravate multiple times) has more synergy with dendro than other parts of her kit, but her burst is not where most of Keqing's damage is from.
As such it is recommended to play Qing as a hyperbloom driver and even build her full EM (with 4p Gilded, EM sword like Dongfang/Iron Sting). Teams such as Nahida-Qing-Jean/Yaoyao-Yelan can work. In this way, Qing's dps can increase further.