4 ms·

I don't know what a bunch of words like "event" and "amplitude" (of... what?) mean in context, to the point that I was able to nod along but didn't actually und

I don't know what a bunch of words like "event" and "amplitude" (of... what?) mean in context, to the point that I was able to nod along but didn't actually understand any of it.

jandrese 7y ago
My takeaway is that amplitude is used because we're thinking about wave patterns and in particular the interference from those waves.
monktastic1 7y ago
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.
ramshorns 7y ago
An amplitude is kind of like a probability. I guess it's called an amplitude because it's used in complex quantum wave equations. But the rule to get a probability from an amplitude is basically that you square its absolute value, which gives you a positive real number.
- p1esk 7y ago
  This doesn’t really help. How is this amplitude different from traditional probability? How can an event have negative amplitude? What does it even mean?
  - empath75 7y ago
    Quantum mechanics makes a lot more sense if you don’t care about what it means and just learn the math.
  - aeternum 7y ago
    The traditional probability = amplitude squared. An event can have negative amplitude, but due to the square the probability cannot be negative. I'm not sure why they don't just say that, it is the most straightforward part of QM.
    - wtallis 7y ago
      Amplitudes can be non-real, so simply squaring them doesn't ensure that you have a (non-negative) probability; you have to also take the absolute value. It's still a pretty straightforward process to derive a probability from an amplitude, but fully explaining that process is a bit of a distraction from making the point that amplitudes really aren't probabilities and shouldn't be thought of as equivalent to or even intuitively similar to probabilities.
      - a1369209993 7y ago
        To be more pedantic, it's actually the product of [amplitude] times [complex conjugate of amplitude][0]. Confusingly, this is sometimes called the "square" of a value - usually in contexts where complex-element vectors and matrices are involved, since most[1] real-element formulas calling for matrix transpose (dot products[2] are a special case of this) actually want conjugate transpose[3] but didn't notice because conjugate is identity on reals. 0: zz* = (z)·(z*) = (a+bi)·(a-bi) 1: Possibly all - I have yet to happen across a counterexample. 2: https://en.wikipedia.org/wiki/Dot_product#Algebraic_definition https://en.wikipedia.org/wiki/Dot_product#Algebraic_definiti... 3: https://en.wikipedia.org/wiki/Conjugate_transpose https://en.wikipedia.org/wiki/Conjugate_transpose
NoodleIncident 7y ago
I don't think it's important to know how "amplitude" relates to the underlying mathematics. In the context of the comic, it is constrained by the following statements: * A qubit is special because it's a superposition of "0" and "1" * A superposition isn't described by being "both" of them, or "maybe one or the other in weird ways"; it's "a complex combination" that's also a "unit vector" * Quantum mechanics is a generalization of probability * (Not in comic, but common knowledge) Any "event" has a certain "probability" that it will occur * Classical probability can only range from 0 to 1, along a straight 1D line * Quantum events are determined by the 2D probability "amplitude", where this word describes the superposition of "0" and "1" of the qubit above * (Common knowledge) Taking a measurement of a qubit or any other quantum system collapses the superposition * When you measure a qubit, there is a rule that converts the "amplitude" into a classical probability. In the case of a qubit, the "event" is whether it will be a "0" or a "1" * Because amplitudes are actually 2D (while the qubit is in superposition), the rules for adding two amplitudes are different than for adding two 1D probabilities. In particular, they can "interfere" and cancel out. Think of two vectors that point in opposite directions, or (1 + 0i) + (-1 + 0i) = (0 + 0i) Some of the above is probably cheating, since I read the other replies to this comment before rereading the comic. However, I think if you move even further away from the detailed math, the usage of "amplitude" is pretty clear. Non-quantum bits would have a "probability" of being measured as "0" or "1". A qubit in superposition has a certain "amplitude" that is analogous to probability, determining whether it will be measured as "0" or "1". Quantum computing exploits the special probabilities of this "amplitude" to do calculations you can't normally do, for very specific problems. It's very hard to keep even a few small qubits in the state of "superposition", which is required to exploit these properties.

I don't know what a bunch of words like "event" and "amplitude" (of... what?) mean in context, to the point that I was able to nod along but didn't actually understand any of it.

jandrese 7y ago

My takeaway is that amplitude is used because we're thinking about wave patterns and in particular the interference from those waves.

monktastic1 7y ago

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.

ramshorns 7y ago

An amplitude is kind of like a probability. I guess it's called an amplitude because it's used in complex quantum wave equations. But the rule to get a probability from an amplitude is basically that you square its absolute value, which gives you a positive real number.

p1esk 7y ago
This doesn’t really help. How is this amplitude different from traditional probability? How can an event have negative amplitude? What does it even mean?
- empath75 7y ago
  Quantum mechanics makes a lot more sense if you don’t care about what it means and just learn the math.
- aeternum 7y ago
  The traditional probability = amplitude squared. An event can have negative amplitude, but due to the square the probability cannot be negative. I'm not sure why they don't just say that, it is the most straightforward part of QM.
  - wtallis 7y ago
    Amplitudes can be non-real, so simply squaring them doesn't ensure that you have a (non-negative) probability; you have to also take the absolute value. It's still a pretty straightforward process to derive a probability from an amplitude, but fully explaining that process is a bit of a distraction from making the point that amplitudes really aren't probabilities and shouldn't be thought of as equivalent to or even intuitively similar to probabilities.
    - a1369209993 7y ago
      To be more pedantic, it's actually the product of [amplitude] times [complex conjugate of amplitude][0]. Confusingly, this is sometimes called the "square" of a value - usually in contexts where complex-element vectors and matrices are involved, since most[1] real-element formulas calling for matrix transpose (dot products[2] are a special case of this) actually want conjugate transpose[3] but didn't notice because conjugate is identity on reals. 0: zz* = (z)·(z*) = (a+bi)·(a-bi) 1: Possibly all - I have yet to happen across a counterexample. 2: https://en.wikipedia.org/wiki/Dot_product#Algebraic_definition https://en.wikipedia.org/wiki/Dot_product#Algebraic_definiti... 3: https://en.wikipedia.org/wiki/Conjugate_transpose https://en.wikipedia.org/wiki/Conjugate_transpose

NoodleIncident 7y ago

I don't think it's important to know how "amplitude" relates to the underlying mathematics. In the context of the comic, it is constrained by the following statements: * A qubit is special because it's a superposition of "0" and "1" * A superposition isn't described by being "both" of them, or "maybe one or the other in weird ways"; it's "a complex combination" that's also a "unit vector" * Quantum mechanics is a generalization of probability * (Not in comic, but common knowledge) Any "event" has a certain "probability" that it will occur * Classical probability can only range from 0 to 1, along a straight 1D line * Quantum events are determined by the 2D probability "amplitude", where this word describes the superposition of "0" and "1" of the qubit above * (Common knowledge) Taking a measurement of a qubit or any other quantum system collapses the superposition * When you measure a qubit, there is a rule that converts the "amplitude" into a classical probability. In the case of a qubit, the "event" is whether it will be a "0" or a "1" * Because amplitudes are actually 2D (while the qubit is in superposition), the rules for adding two amplitudes are different than for adding two 1D probabilities. In particular, they can "interfere" and cancel out. Think of two vectors that point in opposite directions, or (1 + 0i) + (-1 + 0i) = (0 + 0i) Some of the above is probably cheating, since I read the other replies to this comment before rereading the comic. However, I think if you move even further away from the detailed math, the usage of "amplitude" is pretty clear. Non-quantum bits would have a "probability" of being measured as "0" or "1". A qubit in superposition has a certain "amplitude" that is analogous to probability, determining whether it will be measured as "0" or "1". Quantum computing exploits the special probabilities of this "amplitude" to do calculations you can't normally do, for very specific problems. It's very hard to keep even a few small qubits in the state of "superposition", which is required to exploit these properties.