I'd say a firm "no" at the moment. Most things in space flight have a lot of lead time, so if it takes a while to run a bunch of calculations and/or simulations to plot a burn or something, that doesn't usually matter. The majority of constraints with spaceflight are physical: how can we bring enough equipment, fuel and consumables to keep everyone alive for the duration of the mission and to do all the required manoeuvres, without making it impractible?

Having a better computer doesn't sidestep the 'tyranny of the rocket equation': the heavier your craft, the more fuel you'll need to burn to perform a manoeuvre; but bringing more fuel makes your craft heavier, so you'll need to burn even more...

Not really. There is a fairly small number of algorithms applicable to quantum computers and for anything else classical computers are better. Maybe there is an application in some specific signal processing problem for astronomy but the field of quantum computing isn't really related to space travel at all.

No.

Quantum computers excel at problems where the math is simple and the inputs are simple, but discerning a result requires exploring a bazillion possibilities - like solving sudoku or factoring large numbers.

Space travel is not this sort of problem - although there will certainly be heaps of small problems within space travel's massive problem stack that quantum computers could help with, such as finding new alloys and meta-materials.

Could be, but not currently nor in near future. If/when QC would be able to solve a whole lot of complex integrals (big if), this potentially could allow us to simulate different quantum equations and construct and check different universe models, which, if proven valid, could lead to some theory of everything, which then could lead to exploration of some weird physics that allows for FTL or at least near light speed travels.

But, as you can see, there's a whole lot of ifs, each one bigger than other.

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There might be advantages to using quantum computers to optimize solutions to n-body problems, but it shouldn’t make much difference normally given the length of mission planning.