I have sworn on the Qur'an to something that is not true but not as a witness.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. If you had sworn with the Qur'an by saying, "By the Qur'an" or similar words, then, swearing by the Qur'an is like swearing by Allah, because the Qur'an is the word of Allah, which is one of His Attributes. Therefore, your swearing by the Qur'an falsely is an objectionable act that does not befit a Muslim. That is why the Messenger of Allah counted false swearing among the major sins. Al-Bukhari reported on the authority of Abdullah bin 'Amr that the Prophet (Sallallahu Alaihi wa Sallam) said, "The major sins are: associating partners with Allah in worship, being undutiful to parents, committing murder and making false swearing (Al-Yamaan Al-Ghamoos)" . The scholars said, 'this kind of swearing is called (Al-Ghamoos) because it plunges the person who makes it into the Hell-fire or into sins, regardless of its being made in a testimony or not' . According to many scholars, false swearing has no atonement except repentance and seeking forgiveness from Allah. However if you make an expiation for an unfulfilled oath in addition to repentance, you will be on the safe side. See Fatwa: 82288 for more. If you placed your hand on a copy of the Qur'an and you swore by it, that made your oath more emphatic. Anyway, you need to repent quickly and make an expiation in order to be on the safe side. Allah knows best.