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It's hard to answer your question without giving a course in QM, but this is a good place to share a pet peeve about this. For some deeply unfortunate reason, "

It's hard to answer your question without giving a course in QM, but this is a good place to share a pet peeve about this. For some deeply unfortunate reason, "amplitude" can refer to: (1) The (complex) coefficient of a component of a state vector. (2) If we treat that number c as re^(iθ), then r is sometimes called c's "amplitude" (though it's more often called the "norm" or the "modulus"). (3) Some authors even use it to refer to θ ("The argument is sometimes also known as the phase or, more rarely and more confusingly, the amplitude" [1]) Luckily, in quantum mechanics, they almost always mean (1). That's the usage here. [1] http://mathworld.wolfram.com/ComplexArgument.html http://mathworld.wolfram.com/ComplexArgument.html

boblivion 7y ago
Wait, why do they call it amplitude when it's the rotational component? What is the reasoning to call this amplitude?
- monktastic1 7y ago
  I don't know, because they hate us?
mrkstu 7y ago
My knowing-nothing coming in takeaway is that the 'wave' state of QM is more like an ocean wave than the two dimensional wave of a sine wave on a oscilloscope and 'interferes' similar to how wave encountering each other on the ocean might. Is that at all part of what is trying to be communicated?
- monktastic1 7y ago
  This is a bit tricky to explain without more math. The sine wave is 1-dimensional and the ocean wave is 2-dimensional, but both have the property that the amplitude at a point (say, f(x) or f(x, y)) is a real number. In QM, to describe position, we have a 3-d wave whose amplitude is complex. This is obviously hard to visualize. But consider for a moment a particle confined to one dimension. Its wave function is a function f(x). This function can also be thought of as a vector in an infinite-dimensional Hilbert space called the L^2 function space[1]. In quantum computing, we generally deal with two-dimensional Hilbert spaces (and their tensor products), which you can sort of think of as discrete waves with only two points. This is because, while a particle can take on an infinite number of positions, there are only two possible spins (for example). If you follow Prof. Aaronson's lectures, these are the sorts of "waves" that matter. Leaving that aside, you could imagine two 1-D sine waves (with different speeds and/or phases) interfering, just as you could two ocean waves. The difference in QM is that you're summing complex numbers. [1] https://en.wikipedia.org/wiki/Square-integrable_function https://en.wikipedia.org/wiki/Square-integrable_function
kazinator 7y ago
If someone refers to phase as amplitude, there is justification in ceasing to read their paper and switching to something else.

boblivion 7y ago

Wait, why do they call it amplitude when it's the rotational component? What is the reasoning to call this amplitude?

monktastic1 7y ago
I don't know, because they hate us?

mrkstu 7y ago

My knowing-nothing coming in takeaway is that the 'wave' state of QM is more like an ocean wave than the two dimensional wave of a sine wave on a oscilloscope and 'interferes' similar to how wave encountering each other on the ocean might. Is that at all part of what is trying to be communicated?

monktastic1 7y ago
This is a bit tricky to explain without more math. The sine wave is 1-dimensional and the ocean wave is 2-dimensional, but both have the property that the amplitude at a point (say, f(x) or f(x, y)) is a real number. In QM, to describe position, we have a 3-d wave whose amplitude is complex. This is obviously hard to visualize. But consider for a moment a particle confined to one dimension. Its wave function is a function f(x). This function can also be thought of as a vector in an infinite-dimensional Hilbert space called the L^2 function space[1]. In quantum computing, we generally deal with two-dimensional Hilbert spaces (and their tensor products), which you can sort of think of as discrete waves with only two points. This is because, while a particle can take on an infinite number of positions, there are only two possible spins (for example). If you follow Prof. Aaronson's lectures, these are the sorts of "waves" that matter. Leaving that aside, you could imagine two 1-D sine waves (with different speeds and/or phases) interfering, just as you could two ocean waves. The difference in QM is that you're summing complex numbers. [1] https://en.wikipedia.org/wiki/Square-integrable_function https://en.wikipedia.org/wiki/Square-integrable_function

kazinator 7y ago

If someone refers to phase as amplitude, there is justification in ceasing to read their paper and switching to something else.