26 Concepts
BY CLARISSA LITTLER
Gateways to Computation In the last issue, we talked about the physics of quantum mechanics and the ways quantum computers calculate things. This time we're going to talk about how the hardware of a computer works and how this relates to building quantum computers. Logical thinking Let's start with the concept of "logic gates". These are the fundamental building blocks of how a processor is built. How fundamental? Well a typical modern CPU has millions—even billions—of logic gates. Each logic gate performs a specific operation on a few bits at a time. One of the first we'll introduce is the and-gate. The and-gate takes two bits and returns 1, if both of the bits it's given are 1, and 0 otherwise. In Python, you'd write it like: def andGate(b1,b2): if (b1 == 1) and (b2 == 1): return 1 else: return 0 So if you think of bits as "1" means true and "0" means false, then an and-gate is basically the same thing as the and you use in if-statements in Python. There's also an or-gate that gives back a 1
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if either—or both—of the bits you give it are 1. And the last important gates are the not-gate that sends 0 to 1 & 1 to 0, and the xor-gate [read this as "ex or" for "exclusive or"] that returns 1 if only one of the inputs is 1. From here, we can demonstrate something kind of cool: let's add two bits together using logic gates. As a reminder of how binary numbers work, a binary number is a series of 1 or 0 and you read them as powers of two. So 101 is 4 + 1 which is 5 or 0110 is 2 + 1 which is 3. With that we can add two bits together like this: 0 + 0 = 00 1 + 0 = 01 0 + 1 = 01 1 + 1 = 10 If we look at the two output bits, we can notice a pattern. The right output bit is 1 when only one of the input bits is 1. The left output bit is 1 only when both of the input bits are 1. Huh. Those sound kind of like logic gates. To write this as Python code, it's: def halfAdder(b1,b2): return (andGate(b1,b2),xorGate(b1,b2)) So you've got the start of a general addition function just from using two gates. We call this a "half adder", though, because it only operates on one pair of bits at a time and doesn't easily chain together to add multi-bit numbers together. If you can understand how the half-adder works, though, then you're well on the way to understanding how to do all arithmetic with logic gates. Something to try: implement the xorGate function yourself and
then test out this addition function on ones and zeros. Gateways that go both ways So now we come to quantum computers and qbits. Quantum bits, as we discussed in the last issue, aren't just on or off, 1 or 0, but instead they store a wave that describes the probability the bit is a 1 or 0. These waves can interfere with each other, like noise-cancelling headphones interfere with sound waves, and this allows for very different kinds of math than just adding bits together. These operations will be carried about by quantum gates. Quantum gates are inherently more complex than the simple gates we’ve seen so far: for example, all quantum gates must be reversible. What does that mean? Well, a reversible operation is one you can undo: opening a backpack is reversible because you can close it. Turning left is reversible because you can then turn right. Adding two is reversible because you can subtract two. Crushing a soda can isn't reversible because you can never get it completely back to its old shape. Multiplying by zero isn't reversible, because you can't recover the original number just by looking at zero. And here's the punchline: logic gates, like the and-gate and xor-gate, are not reversible, similarly to why multiplying by zero isn't reversible. If an and-gate returned a 0 can you tell me what the inputs were? If things like and and xor gates aren't reversible, how can we do math with a quantum computer? We'll cover that next time. b
