Elaborating somewhat on `Mithrandir24601`

's response —

The feature you're worried about, that a quantum computer might produce a different answer on the next run of the computation, is also a feature of randomised computation. It is good in some ways to be able to obtain a single answer repeatably, but in the end it is enough to be able to obtain a correct answer with high enough confidence. Just as with a randomised algorithm, what is important is that you can be sure of the chances of getting the correct answer in any given run of the computation.

For instance, your quantum computer might give you the correct answer to a YES / NO question two times out of every three. This might seem like a poor performance, but what this means is that if you run it many times, you can simply take the majority answer and be *very* confident that the majority rule gives you the correct answer. (The same is true for normal randomised computation as well.) The way that the confidence increases with the number of rune, means that so long as any one run gives an answer which has significantly more than *just a 50% chance* of being correct, you can make your confidence as high as you like just by doing a modest number of repeated runs (though more runs are required, the closer the chances of a correct answer in any one run are to 50%).

In theoretical terms, we give the name **BQP** to the collection of problems which are solvable in $\mathrm{poly}(n)$ computational steps by a quantum computer, for input sizes which can be specified by an $n$-bit string, where the answer is correct with probability at least 2/3; by the argument above, the exact same set of problems is given if you demand that the answer be correct with probability 999/1000, or (1 − 1e-8).

For problems which have more elaborate answers than YES / NO questions, we can't necessarily assume that the same answer will be produced more than once so that we can take a majority vote. (If you are using a quantum computer to sample from an exponential number of outcomes, it is possible that there are some smaller but still exponentially many quantity of answers which are correct and useful!) Suppose that you are trying to solve an optimisation problem: it might not be easy to verify that you have found the optimal solution, or a nearly-optimal solution — or that the answer that you've gotten is even the best that the quantum computer can do (what if the next run gives you a better answer by chance?). In this case, what is important is to determine what you know about the problem, whether there is an independent way to verify a solution (is your problem in **NP**, meaning that you can in principle efficiently check any answer you're given?), and what quality of solution you would be happy with.

Again, this is all true for randomised algorithms as well — the difference being that we expect quantum computers to be able to solve problems that a randomised computer alone could not easily solve.

3It doesn't make much sense to say that "

a quantum computers belong to X complexity class". That's like saying "a (classical) computer belongs to the Y complexity class". A (quantum) computer is a device on which you run (quantum) algorithms, such algorithmsmaybelong to a given computational class. You can just as well try and solve P or PP problems on a quantum computers. Also, quantum algorithms do not have to be probabilistic. – glS – 2018-03-12T23:26:31.093@glS Fair points, so I've edited to fix/clarify this - the only thing is that non-probabilistic algorithms still have a bounded error, in that the failure rate is 0, so probabilistic is just a generalisation of deterministic – Mithrandir24601 – 2018-03-13T00:18:30.033