The logic of a quantum computer is completely different!
First, every function used in a quantum computer must be reversible, ie with the output it must be possible to regenerate the input. Most conventional computer algorithms have already broken this rule. For example the addition is not reversible! If you add 2 + 7 it will give 9, but it is impossible to do a function that takes 9 and returns 2 + 7 because some of the information was lost in the process.
So conventional addition, subtraction, multiplication, and division operations are not possible on a quantum computer. With this restriction it is also realized that any quantum function must have the same number of input bits and output bits. This is not a limitation, since reversible methods exist for all computable functions, but it requires a different way of thinking about solving a problem.
Another very different property is that in the middle of a quantum program no bit can be modified, copied or deleted. Such an operation would "ruin" the bits! Moreover, in a quantum function there can be no loops, no ifs, or any kind of conventional flow control: one operation runs after another sequentially. And quantum operations change all bits at once in a way that would require many normal operations to do so by forming a new set of basic instructions that is much more efficient for some types of problems.
If the quantum computer starts to be viable, the most accepted idea is that programming will continue in the same way, but with quantum functions! A conventional computer will write the input bits into the memory of a quantum computer and give a signal for it to start the function. Once the result is calculated, just read the output values. I imagine there will be a library of quantum functions to use in the code and this should be transparent to the programmer.
The quantum computer is extremely fast to make a discrete Fourier transform by solving it with complexity O (n ^ 2). In a conventional computer this operation has complexity O (n * 2 ^ n), much greater. Hence, in 1994 a mathematician named Peter Shor made a quantum algorithm with complexity O (n ^ 3) to factorize a number using the Fourier quantum transform. After that, the interest in quantum computers has grown a lot, because factoring a number with this speed would break many of today's cryptographic systems.
I tried to speak the most important, it is very difficult to explain this trying to be as basic as possible! A more detailed discussion can easily come in deep existential philosophy and the strangeness of the universe, which would not help much to answer the question :)