The advantage would be that you wouldn't need tightly controlled and hard to make materials like U-235 or Pu to make one.
I'm not in any way saying that using lasers would be a plausible route to such a weapon, since the NIF facility is huge, but if it turns out that the research needs to focus on how to get more output per shot, which I think it inevitably would since a typical conventional or nuclear power plant generates on the order of 1 GW thermal power (To match that with a 1 Hz repetition rate, likely a stretch for a MJ class laser, you would need to generate 1 GJ per shot, comparable to the energy in a ton of TNT.), it would probably be touching on areas that are highly classified.
Shiiiit... here I was thinking how cool it would be if they could miniaturize this, having somehow forgotten that my pet solution to the Fermi paradox is that a nigh-inevitable wrung on the ladder to interstellar presence involves discovering One Weird Trick to release a whole bunch of energy pretty easily, even on a DIY basis. Instant end of civilization. Even ant-like societies might have mutated members who'd go rogue and misuse the tech, and it wouldn't take many to ruin everything.
Basically it's a twist on the ice-9 solution to the paradox.
One use for such a 'laser initiated fusion' is the LLNL (Teller) proposed X-Ray laser satellite from the Star Wars program in the 80s. The proposed solution then was to use the X-rays from an exploding bomb at the heart of a satellite and amplify and focus them through a 'lasing' material. It turns out they never found a lasing material that would work, and it would be fairly easy to confuse defeat it, so the project died. What also killed it was the end of the testing program.
I'm not in any way saying that using lasers would be a plausible route to such a weapon, since the NIF facility is huge, but if it turns out that the research needs to focus on how to get more output per shot, which I think it inevitably would since a typical conventional or nuclear power plant generates on the order of 1 GW thermal power (To match that with a 1 Hz repetition rate, likely a stretch for a MJ class laser, you would need to generate 1 GJ per shot, comparable to the energy in a ton of TNT.), it would probably be touching on areas that are highly classified.