This is one of the important aspects of the “fundamental theorem of algebra“.
You can tell mathematicians think it’s important, they don’t call just anything the “fundamental theorem of whatever”.
Were I so equipped, I would have Preskill’s babies. So, for the not-faint-of-heart and the girded-of-loin, what follows is answer gravy: Deutsch’s problem: One of the classic examples of quantum computation is “Deutsch’s problem”.
Physicist: Yes, but they don’t fix problems the way the complex numbers do.
The nice thing about real numbers (which includes basically every number you might think of: 0, 1, π, -5/2, …) is that no matter how you add, subtract, multiply, or divide (other than 0) them together, you always get another real number. A mathematician would say “the real numbers are under addition, because any real numbers added together always give you another real number”. When you’re doing square roots the real numbers are not closed. For example, to find , you just answer the question, , and find that the answers are ? But if you try the same thing with you’ll be trying to answer the question , which doesn’t have any answers (try it).
So, after answering more than a couple of questions it start stabbing. Somehow you find out that either the oracle always says the same thing, or it says it likes exactly half of the numbers and hates half of the numbers.
If you give the Oracle a number it either says “Good number! So, for all the numbers 0 to 7 it could hate all of them, like all of them, or like 4 / hate 4. ” you now know that either she likes all numbers, or likes half / hates half.
Closed-ness is comforting to have, because it means that when you’re doing basic math, no matter how you jump you’ll always have somewhere to land. To “solve” this problem Euler decided to make up a new “number” called ““, with the property that , and complex numbers were born. may patch the problem with , but does it just give rise to a new problem when you try to figure out what is? You can check this by squaring it: Weirdly enough, there is absolutely no combination of roots/exponentiations or multiplications/divisions or additions/subtractions that can break out of complex numbers.
Where the closed-ness of real numbers fail, complex numbers hold strong.
In addition, ij=k, jk=i, ki=j, and if you flip the order you flip the sign, so ji=-k.
Quaternions don’t “patch holes” that the complex numbers have, but they do help with some very complicated problems that other number-systems can’t handle easily.
So, while a normal computer can take exactly one input with N bits, a quantum computer can take inputs, simultaneously, with N qbits.
Like a Turing machine, the exact construction of a quantum computer (and there are a lot of different constructions) isn’t particularly important for the philosophy behind its functioning.
The internal mechanisms of a quantum computer are, similarly, in many states at the same time.