The car exploded as it rammed into the front gate of the Aden security offices at dawn, a police official said. Several other soldiers were wounded in the blast.
Another police source said that two other attackers in an explosives-laden car attempted to ram through the gate after the first bombing, but were stopped by security forces.
The two "terrorist elements" are being questioned, he said.
Two police centres in different parts of the city simultaneously came under attack with rockets, a third source said, adding that there were no casualties.
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AQAP took advantage of the weakening of the central government in Sanaa during a 2011 uprising inspired by the Arab Spring, seizing swathes of territory in the south before being driven back in June 2012.
The group is still active in southern and eastern Yemen, and stages frequent attacks on security forces despite Yemeni military operations and US drone strikes against the jihadists.
Elsewhere in the country, two explosions late yesterday struck an oil pipeline that pumps around 10,000 barrels per day from the Ayadh fields, in the southern province of Shabwa, to the Nushaima terminal on the Gulf of Aden, a government official said.
The two bombs, planted under the pipeline in different locations, took the conduit out of service, the official added.
Attacks on oil and gas pipelines in Yemen are frequent, and Oil Minister Ahmad Dares said earlier this month that sabotage had cost the country USD 4.75 billion between March 2011 and March 2013.