The attacks came as the Taliban announced a three-day cease-fire over the Eid-al-Fitr holiday, their first offer of its kind, following an earlier unliteral cessation of hostilities announced by the government.
But the Taliban say they will stop fighting for three days.
The Taliban established the office in Qatar ostensibly for peace talks with Western powers in an Islamic country they considered neutral, though the Persian Gulf kingdom has long allowed the US military to use its Al Udeid Air Base to launch aerial attacks on the Taliban and other militant groups in Afghanistan.
Also Saturday, a bomb blast in Nangarhar killed at least four civilians and wounded 16 others, said Attahullah Khogyani, spokesman for the provincial governor.
Pakistan today expressed support for peace efforts in Afghanistan, hours after the Taliban announced its first ceasefire in the war-torn country in almost 17 years.
President Ashraf Ghani welcomed the Taliban's move in a tweet from his official Twitter account. "They will kill more to make up for their ceasefire days". The cease-fire will coincide with Eid al-Fitr, the religious holiday that marks the end of Ramadan and begins June 16.
Mualvi Mir Hassain Shah Naseri, talking on behalf of Ulema Council, said that the council supported the ceasefire announced by the Taliban and the government and wanted lasting peace in the country.
Afghanistan faces insurgencies by the Taliban, Islamic State and the Taliban-linked Haqqani network.
The Taliban claimed responsibility for the twin suicide attacks, adding that it seized military equipment.
"We have been stopping fighting during the Eid days, but we would not announce it publicly due to security reasons", Mujahid said. But the movement of global forces has been limited in recent years under a curtailed advisory role, with a smaller counterterrorism mission that largely focuses on remnants of al-Qaeda and the Islamic State.
"The PML-N government had played a pivotal role in initiating steps for peace in Afghanistan, including hosting the first ever direct, face-to-face talks between the Afghan government and Afghan Taliban in Murree in July 2015".
President Donald Trump's administration has increased the number of USA troops in Afghanistan in an effort to give local forces, which continue to struggle despite more than a decade and a half of outside support, a battlefield advantage.
U.S. General John Nicholson, commander of U.S. Forces-Afghanistan and NATO-led Resolute Support mission, said the coalition forces will also honor the cease-fire.