It seems to be generally accepted that you cannot touch the Quran without ablution. What about a copy of the Quran on the smartphone?
The generally-accepted view is that one can only touch a mus'haf (manuscript of the Qur'an) with ablution (free from minor and major impurities). When it comes to touching a smartphone when the mus'haf is on display, it is a matter of ijtihād where scholars differ in opinion. It is unlikely that there will be a definite answer on this topic.
Abdul-Rahman al-Barrak, a Saudi Salafi cleric, is of the view (see his fatwa, Arabic only) that it is permitted to recite the Qur'an from a smartphone without ablution since the requirement to be free from impurities apply only to touching a manuscript without a barrier. Saleh Al-Fawzan, an Islamic scholar and a member of the Council of Senior Scholars, shares the same view that it is permitted.
'Abdul-Karim Al-Khudair, a mufti and scholar and a member of the Council of Senior Scholars, is of the view that it is not permitted to read from a smart device without ablution. Muhammad Al-Munajjid, an Islamic scholar and the founder and supervisor of the website Islam Q&A, also ruled that ablution is a requirement (although you will find other views on Islam Q&A that says it is permissible).