It has been prompted following concerns by parents about the care given to their children.
Several families have met with Prof Sir Bruce Keogh, medical director of the NHS, who has asked Sir Ian Kennedy to lead the investigation.
In a statement, Bristol Children's Hospital said it was troubled that families continued to have concerns.
A recent inquest into the death of four-year-old Sean Turner, from Wiltshire, found that while there was no gross negligence, there were missed opportunities.
Steve Turner, Sean's father, said: "One Sunday morning I picked up the paper and read an horrific story with regard to the Bristol Children's Hospital.
"I tweeted Sir Bruce Keogh and asked him to pick up the reins and sort this mess out."
He agreed to meet with parents, including the mother of seven-year-old Cardiff boy Luke Jenkins, who died in 2012 following heart surgery at Bristol Children's Hospital.
Faye Valentine said the meeting with Dr Keogh was "promising".
"We feel quite pleased that something has come out of today and hopefully he'll look into our concerns," she said.
Dr Keogh said it was to be an inquiry by the parents but led by Sir Ian, who led the Bristol Royal Infirmary tragedy investigation.
"We've heard some very harrowing stories in a dignified way from the families," Dr Keogh said.
"Ian Kennedy is a highly respected individual, very ethical, got great integrity and the important thing is that he conducted the original Bristol Royal Infirmary inquiry.
"He understands the issues, he can hit the ground running.
"What I'm really after is getting to the bottom of these problems, finding a solution and getting the Bristol unit to be one of the very best in the country as quickly as possible."
A spokesman for the hospital said it had continued to make improvements, including better communication with parents.
An inquiry is to be held into the deaths of children following heart surgery at Bristol Children's Hospital.