I have two sets of explanations for you. One is super saucy but straightforward; the other is simple and intriguing:
The saucy version comes from Scott Aaronson, an expert in the subject and you can read it here:
Here is the "simple" explanation of quantum computers. I hope it is useful. Although oversimplified, it provides a basic understanding and power of quantum computer vs. even fastest, most powerful classical computer, even faster than super computers and massive data centers, while consuming fraction of electricity power. How does that work? Let's look at the simple explanation that make sense to me.
Semiconductor computing (classical computers) operates deterministically. It means when we run a program on our computers, everything's either Yes or No, On or Off with no in-between. The results are deterministic. We ask a question, we get an answer. We can ask the same question and the answer is always the same. And they (our PCs at home and Office) operate serially; they only do one thing at a time and can do it fast, but not fast enough to solve enormous problems, the seemingly intractable problems. In some cases, they can solve huge problems, but it takes time. In some cases, it takes weeks, months, years, and hundreds or even millions of years.
By contrast, quantum computers operate probabilistically and simultaneously. They (quantum computers) don't have a monitor and keyboard attached, but there is always a classical computer that control the input and out from the classical computer to quantum computer, going back and forth until we get answer.
Now let's illustrate how this works.
Imagine a computer trying to solve a maze. Th program can go Down - Right - Up - Left until it finds the exit. It means that the classical computer would exhaust every potential path and see if it came across a roadblock on the first path that tried. Then it would rule that path out, revert to its original position, try the next logical path, and so on -until it found a solution. This takes time as the maze become exponentially more complex and bigger.
To have a chance in solving it, you must:
Create a Solve() routine and recursively call itself:
if 1st, 2nd, 3rd, ... are true Solve has succeeded in finding a solution
if 1st, 2nd, 3rd, ... contains a false, it has to backtrack and find another way
You need to build a buffer of places you've been to avoid infinite loops
as you make moves it needs to keep tabs on it
when we hit a dead end, we need to erase bad moves
we can implement the above by burning in a guess and removing it if it's wrong.
This works nice with small size maze. Imagine running this on a giant maze. It takes a long time to solve this maze, using this classical computing method. A quantum computer could test every single path in one attempt (checking all possibilities in the parallel) at the maze with only a single trial. That is the power of quantum computer.
Again, as I said, this won't matter much - it'll be hardly perceptible with a small maze vs. a giant maze characterized by a complex potential set of road combinations. The difference in calculation time can quickly go from years with a classical computer to minutes with a quantum computer. In modern business world, a lot of complex and intractable problems are characterized by this maze-like quality.
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