the common art major physics notion is that "binary computers can think in yes and no, quantum computers can also think in maybe", that is total fantasy and fiction that implies that quantum computers will gain self awareness by virtue of being quantum.
first of all classifying current computer tech as "binary" is obtuse. operations are more complex than merely "yes / no"; The individual components of an individual logic gate are binary based... but that is because that is the most simple and direct way to build them, actual processing elements are built from those simple elements and are far more complex and versatile. And computer communication is commonly reduced to binary, but machine code has more flexibility then merely "yes/no" and and the actual storage often is in more then binary (SSD MLC for example, where each cell has 4 states, aka quadranery).
Quantom computing only gains in efficiency but does not somehow enable you to calculate functions that cannot be performed in current computers (just, perhaps much more slowly; it might make it more practical to perform certain calculations due to speedups).
the reason brains are so different than computers is because computers are constructs that do math (and do it well, and very very quickly), organic brains are not math processors but chemistry factories; we are very much deficient when it comes to math compared to a computer.