So last time, we talked about how ordinary, non-quantum, computers are built out of many, many logic gates. These logic gates defined operations on pairs of bits, like the and gate that follows this rule
1 and 1 = 1
0 and 1 = 0
1 and 0 = 0
0 and 0 = 0
They’re called logic gates because if you think of “1” as “true” and “0” as “false” then they look like logical ideas like “a and b are true if both a and b are true”. Combine enough of these logic gates together and you’ve got all the operations on numbers that a computer needs to function.
Now we get to the punchline of this three part story: how quantum logic gates, things that can be used to build up all the operations you need for a quantum computer, are different than ordinary logic gates.
So recalling a couple of issues ago, when we covered the basic idea of quantum computers, we said that a quantum bit—qbit—held data that could be used to compute the probability the qbit returns 1 or 0 when read. Now it’s finally time to talk about what that data even is. Surprise: it’s a point on a circle. The next part is going to involve a little bit of mathematics: namely the Pythagorean theorem and the square root operation. If you haven’t seen these things before, that’s okay, you can just follow along to get the idea!
To explain how “a wave that can be used to calculate probability”, which is what we’ve called qbits before, could be thought of as a point on a circle we need to use Pythagoras’s theorem!
So the Pythagorean theorem says that, for a circle of radius 1, if you take any point on the circle (x,y) then
x^2 + y^2 =1
Okay, well here’s the other fun little bit: when we talk about probability we generally deal with numbers that range from 0 to 1, where 1 means it’s 100% going to happen and 0 means 0% chance of it happening. If something is 50% likely, you’d write that as 0.5. This means that
(probability the qbit is 0) + (probability the qbit is 1) = 1
Okay, if we compare this to the above we get something kinda funny!
x^2 = probability the qbit is 0
y^2 = probability the qbit is 1
This means we can relate the points x and y to the probabilities. That means that every single point on the circle corresponds to some possible qbit state.
Mathematical aside over!
If the data in a qbit is a point on a circle, rather than a 1 or a 0, then what is a qgate? Well, a quantum gate is just a rotation of the circle. So to follow this, imagine drawing a point on the top of the circle. In our understanding of points as qbit data, then we know that this point means “the qbit is definitely 1”. If we rotate it 45 degrees clockwise, then we get the new state “the qbit is equally likely to be 1 or 0”. If we rotate it 45 degrees further then we get “the qbit is definitely zero”.
Now, what starts to get impressive is when we add multiple qbits. If you have two qbits, then qgates become rotating spheres in three-dimensions! If you have three qbits then qgates become rotating…uhh…well we don’t have a good word for dimensions higher than three but they’re like spheres but with more coordinate axes. So while a single ordinary logic gate can just operate on a few ones and zeros at a time quantum gates can do these big complicated things on higher-dimensional shapes you can’t even picture!
The other cool thing about quantum gates is that this means that you can always reverse the operation, the same way you’d reverse rotating a circle or a ball. You just go back the other way! You can’t do that with ordinary logic gates like and.
Here you can build really simple gates out of pieces and then see how they change the probabilities. But, really, the main lesson from all this is that quantum gates are really fascinating!