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andrewp123 | 1 year ago

Correct - superposition doesn't fork the world - measurement does. And correct, you can't communicate with the other universe after the split has happened [1]. I'm glad you mentioned that quantum computers can't solve NP-complete problems - my next blog post was going to be about why. Here's an overview of what I plan on saying:

A typical quantum algorithm like Shor's works by sending every possible input through a gate, and so you get every possible output out in a superposition. If you were to just measure that, you'd get a random result - so instead, you need to somehow interfere the output to get the actual result. You do this by taking advantage of the fact that the superposition is a periodic function and the amplitude repeats. This is literally the core assumption of the algorithm.(a common way of doing this using the QFT).

Every quantum algorithm requires some kind of structure in the output like this. Deustch's algo, dumb ones like Simon's algo, etc. NP-Complete problems have no structure to them, so even if you build a gate that creates the superposition you want, it's not possible to destructively interfere it to get an answer (I don't know how to prove that there's no structure to NP-Complete outputs - it just feels trivial, since they're only solvable in exponential time, so there must be an exponential amount of "structure" there).

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[1] The only way to communicate with the other universe would be to try to use quantum mechanics with something like an entangled pair. But no information can be communicated through an entangled pair if all you just have 1 of the 2 particles! Measurement collapses a state nonlocally, and if you could somehow measure one particle and change the probability distribution of the other, you'd be communicating faster than light. The measurement genuinely changes the state and the amplitudes, but not in a way that the other person can detect. It's really interesting and leads to stuff like teleportation.

discuss

order

eynsham|1 year ago

Since quantum computers can stimulate classical computers, presumably they can solve NP-complete problems, since classical computers, by definition, can. Perhaps you mean that they can’t solve NP-complete problems in polynomial time, but we don’t even know that of classical computers, so you would presumably have shown that P≠NP, which would be fairly impressive.

Dylan16807|1 year ago

So you're nitpicking it for being too strong of a statement and too weak of a statement at the same time?

How about this, they can't do the thing NP stands for. They can't run a generic polynomial-time algorithm in a nondeterministically-branching way, and then pick the winner.

andrewp123|1 year ago

I can't imagine you believe P=NP.

timmattison|1 year ago

Where is the best place a layman can dig into this statement “You do this by taking advantage of the fact that the superposition is a periodic function and the amplitude repeats.”? I’ve seen articles hinting at this in an obtuse way but I’d love to see something more approachable to help wrap my head around it.

andrewp123|1 year ago

I just tried finding a good resource and I can’t. All of them are mile long page scrolls… I don’t know how they have so much stuff to spew. Qiskit had amazing lessons with cool illustrations (although they did spew at the end) but I can’t even find that anymore on their site.

Don’t worry though, even the professional researchers I’ve worked with think it’s a waste of time. The field is screwed.

Here’s a quick explanation from me- The state |x> means you have some qubits that represent the number x. Say you want to represent the number 13, that just means you have |1,0,1,1>, it just means you have 4 qubits in this configuration (quits can be 0 or 1). It’s also written |13>. If you want the state “13 AND 14 AND 15” in superposition where qubits are both 0 and 1, that’s represented by |1,0,1,1> + |1,1,0,0> + |1,1,0,1>. It’s in that superposition and can interact with itself until you choose to measure it. When you do go to measure it, you might measure any of the values (you dont get to choose which). Maybe you measure 15, that means the state is now |1,1,0,1>, you just deleted all the terms that aren’t 15.

This is a full pic of Shor’s algorithm https://images.app.goo.gl/ZE5rDxHScm4LUqms6

If you look at the pic, main idea is the first layer of H’s creates the state sum_x=0…2^n-1 |x, 0>, then gate U turns that state into sum_x |x, f(x)>, then the measurements measure which f(x) you have, deleting all the terms that don’t have that f(x) in them, so for example if you measure that f(x) is 13, the state is now |0, 13> + |15, 13> + |30, 13> + |45, 13> + … This is the periodic state. Now that we have it we can just apply a gate that takes the QFT (finds the frequency, which here turns the state into roughly |15, 13>), and then measures it, giving the answer period=15.