BBC /Turing Pardon
Government rejects pardon request for Alan Turing
An online petition of over 23,000 signatures had requested the pardon.
Justice Minister Lord McNally dismissed the motion in the House of Lords.
"A posthumous pardon was not considered appropriate as Alan Turing was properly convicted of what at the time was a criminal offence," he said.
In 2009 former Prime Minister Gordon Brown issued an official apology to Mr Turing, labelling the treatment he had received as "utterly unfair" and "appalling".
Mr Turing was one of the key members of the staff at Bletchley Park that worked to crack Germany's Enigma codes, and Lord McNally acknowledged that in light of this work he had been treated harshly by the authorities.
"It is tragic that Alan Turing was convicted of an offence which now seems both cruel and absurd, particularly... given his outstanding contribution to the war effort," he said.
"However, the law at the time required a prosecution and, as such, long-standing policy has been to accept that such convictions took place and, rather than trying to alter the historical context and to put right what cannot be put right, ensure instead that we never again return to those times."
Mr Turing committed suicide in 1954, two years after his conviction.
2012 marks the centenary of his birth. The occasion is being marked by a series of events around the world including a commemorative postage stamp issued by Royal Mail.