The ruler's ijtihad in matters open to ijtihad when he is qualified for it

Allah, glorified be His praise, says: "And (remember) Dawud and Sulayman (peace be upon them both), when they were judging concerning the field, when the people's sheep had grazed in it by night, and We were witness to their judgement." [Surah al-Anbiya: 78] "So We made Sulayman understand it, and to each We gave judgement and knowledge." [Surah al-Anbiya: 79]

Murrah narrates from Sayyiduna Ibn Mas'ud (may Allah be pleased with him) concerning the saying of Allah Almighty: "And Dawud and Sulayman, when they were judging concerning the field, when the people's sheep had grazed in it by night" [Surah al-Anbiya: 78], that it refers to grapevines. The grapes had grown and they (the sheep) ruined them, so Dawud (peace be upon him) gave the sheep to the owner of the vineyard. Sulayman (peace be upon him) said: O Prophet of Allah, the judgement is other than this. He said: What is it? He said: Hand the vineyard over to the owner of the sheep, and let him tend it until it is restored; and give the sheep to the owner of the vineyard, so that he may benefit from them. Afterwards the owner of the vineyard gets his vineyard back and the owner of the sheep takes back his sheep. Allah Almighty says: "So We made Sulayman understand it, and to each We gave judgement and knowledge" [Surah al-Anbiya: 79], meaning that We gave Sulayman understanding of the judgement. Allah Almighty prescribed the like of these judgements in such incidents when the Prophet (peace and blessings be upon him) judged concerning the she-camel of Sayyiduna al-Bara bin Azib (may Allah be pleased with him), when it entered a garden of the Ansar at night and ruined the garden. He (peace and blessings be upon him) said: "The garden owners are responsible for guarding by day, and the owners of animals must guard their animals by night."
Imam al-Shafi'i (may Allah have mercy on him) said: Al-Hasan bin Abi al-Hasan said: Were it not for this verse, I would have thought that the rulers had perished. But Allah Almighty praised this one (for his correctness) and praised that one for his effort.

Sayyiduna Amr bin al-As (may Allah be pleased with him) heard the Messenger of Allah (peace and blessings be upon him) saying: "When a ruler exercises his judgement (ijtihad) and judges correctly, he receives a double reward; and when he exercises his judgement and errs, he receives one reward."

Likewise (through another chain).

Sayyiduna Abu Hurayrah (may Allah be pleased with him) narrates from the Prophet (peace and blessings be upon him): "When a ruler exercises his judgement (ijtihad) and arrives at what is correct, he has a double reward; and when a ruler errs in his ijtihad, he receives one reward."
Only Ma'mar narrated it from Sufyan, and Abd al-Razzaq alone narrated it from him.

It is narrated from Sayyiduna Wathilah bin al-Asqa' (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) said: "Whoever attains knowledge will have two shares of reward, and if he could not attain knowledge, he will have one reward."

Nafi' narrates from Sayyiduna Abdullah (may Allah be pleased with him) that on returning from the Battle of the Confederates (al-Ahzab), the Messenger of Allah (peace and blessings be upon him) called out: "The Zuhr prayer is to be offered at Banu Qurayzah." The narrator said: Some Companions, fearing that the time would pass, offered the prayer before (reaching it), but others said: We will pray where the Prophet (peace and blessings be upon him) commanded us, even if the time passes. He (peace and blessings be upon him) did not rebuke either of the two groups.