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Frankly I am so tired of this whole branch of research where people try to be foundational about "quantum theory" but at the same time boil it down to qubits,

Frankly I am so tired of this whole branch of research where people try to be foundational about "quantum theory" but at the same time boil it down to qubits, gates, bell tests and, well, two-by-two matrices. Here is my viewpoint, which somehow some people find controversial: quantum theory is first and foremost a description of individual particles. To describe their time evolution, we use the Schrodinger equation: i d_t Psi = H Psi What is that "i" there? Oh right, the imaginary unit. So... quantum theory uses complex numbers. Now you are free to search for another theory without the "i", and perhaps even find something that is somehow mathematically consistent. But that theory either describes experiments just as well as ordinary quantum theory, in which case it is physically equivalent and of no advantage (except to those with strong allergies to complex numbers), or it does not, and then it is wrong. Of course the last logical possibility is that your theory might do better than quantum theory... but that is the dream only of those who do not known quantum field theory. /rant, with apologies

jjk166 1y ago
The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness. Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way. I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.
- Retric 1y ago
  The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data. Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.
  - IcyWindows 1y ago
    More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were. If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.
  - jjk166 1y ago
    And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity. Quantum mechanics likewise is just an approximation of quantum field theories.
    - Retric 1y ago
      It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.
      - jjk166 1y ago
        Epicycles wasn't a theory, it was a model. It did not try to explain why the planets moved in the sky as they did, it only predicted where they'd be. Neither, for that matter, were copernican or keplerian mechanics theories. They too required unending tweaking because they also were only approximations of what was actually happening. For the first few centuries after heliocentrism was proposed, it gave worse results, and demanded more tweaking. What really won people over was that the phases of the moons of jupiter were accurately predicted by the model as well. The only way to achieve that result with epicycles was to rearrange everything to be mathematically equivalent to a heliocentric model. You can reconstruct our modern understanding of the motion of the planets in the reference frame of a static earth and produce a mathematically equivalent path that draws out epicycles which predict the positions of planets with exactly the same accuracy as our regular formulations. You can rework the representation of the laws of gravity such that they spit out positions in this reference frame. It is an equally valid model of the cosmos, with exactly the same number of starting assumptions, it's just remarkably more complex.
nathan_compton 1y ago
There is really nothing to the appearance of complex numbers in QM. In QM we must design wave functions which do the double duty of representing the probability of measurement outcomes AND capture the symmetries implicit in the system related to the fact that there are degrees of freedom between preparation of a state and measurement (for example, we may rotate our detector any way we wish before we make a measurement of a particle in a given prepared spin state). To accomplish this we need some number-like objects to denote our wave function in that square to real numbers but have enough structure to represent (in this case) the rotations. As you venture further into the universe of QFT you find that you need even more exotic number like objects like spinors with their own peculiar structures, but the essence is the same: they must serve the purpose of representing probabilities and symmetries. The complex numbers in QM mean nothing at all except in that they serve these purposes. If we wish to speak informally and wave our hands a bit we can say that it isn't so surprising that we find the complex numbers and related number like objects because the complex numbers are a promise to square something at a later date and recover a real number, which is what we need to satisfy the requirement to represent probabilities. In fact, we can formulate classical probabilistic mechanics with complex numbers (the Koopman von Neuman operator theory) and again, they appear because we want to operate on objects living in a nice Hilbert space which also square to probabilities. In only took me 20 years to understand this, so I can sympathize with confusion.
- omnicognate 1y ago
  It's a long time since I read it, but there's a book called "The Structure and Interpretation of Quantum Mechanics" [1] by R. I. G. Hughes. The "Structure" part of it begins by building up most of the mathematical framework (including use of complex numbers, Hilbert spaces, operators, etc), motivated only by the desire to build a physical theory that is probabilistic in nature. It then shows how you can add one extra ingredient that turns the framework into that used for quantum mechanics [2]. I assume that everything discussed up to that point applies equally to Koopman-von Neumann. It's a really nice book, very self-contained. I think anyone with a basic mathematical education (A-Level or equivalent) could get through it without having to read other things to acquire prerequisites, though they should be prepared to think quite hard. 1. The resemblance to the titles of Gerald Jay Sussman's "Structure and Interpretation" books appears to be coincidental. The title is meant literally: the book is split into two sections, one on the (mathematical) structure of QM and one on its (philosophical) interpretation. There are no similarities in style, pedagogy or subject matter to Sussmann's books and no use of, or reference to, programming. The author was a professor of philosophy at the University of South Carolina. 2. He actually lists a collection of alternatives for that extra ingredient, any one of which has the same effect when added.
Strilanc 1y ago
Yes, the post is focusing on the overall effect of operations (unitaries) rather than their continuous trajectories (hamiltonians acting on system via Schrodinger equation) (analogous to working with impulses rather than forces). To make the continuous case interesting as a compilation problem, you'd need some alternate formulation of the Schrodinger equation, e.g. based on the limit of small powers of unitaries rather than on the matrix exponential, so that deleting i didn't delete literally all processes. Or you could arbitrarily declare real-only hamiltonians are permitted, despite the Schrodinger equation saying "i". But that'd be kinda lame, imo. (Note: am author of post)
- wasabi991011 1y ago
  Gidney, that's you? Huge fan of your work! I just started my PhD in distributed quantum computing, and my Masters was applying that framework to the QFT. I came across a number of papers you authored in the process, as well as your blog. In particular, big fan of Kahanamoku-Meyer et al.'s optimistic QFT circuit. Anyway, keep up the great work!
- prof-dr-ir 1y ago
  I made a related comment about two years ago: https://news.ycombinator.com/item?id=38255476 https://news.ycombinator.com/item?id=38255476

jjk166 1y ago

The "i" is there because it is a convenient way in our system of mathematics to write out such an equation, but that really comes from the fact that complex numbers have two dimensionality. Our best understanding of the universe demands that higher dimensionality, not necessarily the imaginary-ness. Yes a different mathematical formulation may be rewritten into this imaginary form, and thus is mathematically equivalent. But by the same logic a heliocentric system of elliptical orbits is mathematically equivalent to a geocentric system of epicycles. From one perspective there is a certain deeper meaning there - the universe has no absolute reference frame; but if you view your cosmos in terms of epicycles its very difficult to develop an understanding of what drives those epicycles, namely gravity. Likewise thinking about quantum mechanics in terms of of imaginary numbers may allow for accurate calculations, but nevertheless be an intellectual stumbling block for understanding why the universe is this way. I personally have no issue with "imaginary" numbers having real physical meaning. Our inability to process the square root of negative 1 seems more like a limitation of our ape brains than the universe, and likewise for the majority of quantum weirdness. But in throwing up my hands saying the question can not be answered, I have guaranteed that I will never find the answer even if it does indeed exist.

Retric 1y ago
The issue with epicycles is you need an infinite number of them to produce the actual orbits and with an infinite number of epicycles you can describe any shape. Thus it is as complex as the underlying data. Quantum Mechanics on the other hand is incredibly constrained and therefore actually says something.
- IcyWindows 1y ago
  More complete astronomy data from telescopes showed that epicycles needed to be even more complicated then they were. If we manage to find better tools for QM where we don't need to perform as much post-selection of experimental data, perhaps we'll also find a simpler model.
- jjk166 1y ago
  And pure ellipses as predicted by newtonian gravitation also don't line up with actual orbits perfectly. In both cases they are just models approximating reality, one of which happens to be more elegant. I don't know how anyone would be able to jump straight from epicycles to general relativity. Quantum mechanics likewise is just an approximation of quantum field theories.
  - Retric 1y ago
    It’s not about elegance for the sake of it. The number of constants in a theory provides a meaningful point of comparison, especially if you need to increase them after an experiment.
    - jjk166 1y ago
      Epicycles wasn't a theory, it was a model. It did not try to explain why the planets moved in the sky as they did, it only predicted where they'd be. Neither, for that matter, were copernican or keplerian mechanics theories. They too required unending tweaking because they also were only approximations of what was actually happening. For the first few centuries after heliocentrism was proposed, it gave worse results, and demanded more tweaking. What really won people over was that the phases of the moons of jupiter were accurately predicted by the model as well. The only way to achieve that result with epicycles was to rearrange everything to be mathematically equivalent to a heliocentric model. You can reconstruct our modern understanding of the motion of the planets in the reference frame of a static earth and produce a mathematically equivalent path that draws out epicycles which predict the positions of planets with exactly the same accuracy as our regular formulations. You can rework the representation of the laws of gravity such that they spit out positions in this reference frame. It is an equally valid model of the cosmos, with exactly the same number of starting assumptions, it's just remarkably more complex.

nathan_compton 1y ago

There is really nothing to the appearance of complex numbers in QM. In QM we must design wave functions which do the double duty of representing the probability of measurement outcomes AND capture the symmetries implicit in the system related to the fact that there are degrees of freedom between preparation of a state and measurement (for example, we may rotate our detector any way we wish before we make a measurement of a particle in a given prepared spin state). To accomplish this we need some number-like objects to denote our wave function in that square to real numbers but have enough structure to represent (in this case) the rotations. As you venture further into the universe of QFT you find that you need even more exotic number like objects like spinors with their own peculiar structures, but the essence is the same: they must serve the purpose of representing probabilities and symmetries. The complex numbers in QM mean nothing at all except in that they serve these purposes. If we wish to speak informally and wave our hands a bit we can say that it isn't so surprising that we find the complex numbers and related number like objects because the complex numbers are a promise to square something at a later date and recover a real number, which is what we need to satisfy the requirement to represent probabilities. In fact, we can formulate classical probabilistic mechanics with complex numbers (the Koopman von Neuman operator theory) and again, they appear because we want to operate on objects living in a nice Hilbert space which also square to probabilities. In only took me 20 years to understand this, so I can sympathize with confusion.

omnicognate 1y ago
It's a long time since I read it, but there's a book called "The Structure and Interpretation of Quantum Mechanics" [1] by R. I. G. Hughes. The "Structure" part of it begins by building up most of the mathematical framework (including use of complex numbers, Hilbert spaces, operators, etc), motivated only by the desire to build a physical theory that is probabilistic in nature. It then shows how you can add one extra ingredient that turns the framework into that used for quantum mechanics [2]. I assume that everything discussed up to that point applies equally to Koopman-von Neumann. It's a really nice book, very self-contained. I think anyone with a basic mathematical education (A-Level or equivalent) could get through it without having to read other things to acquire prerequisites, though they should be prepared to think quite hard. 1. The resemblance to the titles of Gerald Jay Sussman's "Structure and Interpretation" books appears to be coincidental. The title is meant literally: the book is split into two sections, one on the (mathematical) structure of QM and one on its (philosophical) interpretation. There are no similarities in style, pedagogy or subject matter to Sussmann's books and no use of, or reference to, programming. The author was a professor of philosophy at the University of South Carolina. 2. He actually lists a collection of alternatives for that extra ingredient, any one of which has the same effect when added.

Strilanc 1y ago

Yes, the post is focusing on the overall effect of operations (unitaries) rather than their continuous trajectories (hamiltonians acting on system via Schrodinger equation) (analogous to working with impulses rather than forces). To make the continuous case interesting as a compilation problem, you'd need some alternate formulation of the Schrodinger equation, e.g. based on the limit of small powers of unitaries rather than on the matrix exponential, so that deleting i didn't delete literally all processes. Or you could arbitrarily declare real-only hamiltonians are permitted, despite the Schrodinger equation saying "i". But that'd be kinda lame, imo. (Note: am author of post)

wasabi991011 1y ago
Gidney, that's you? Huge fan of your work! I just started my PhD in distributed quantum computing, and my Masters was applying that framework to the QFT. I came across a number of papers you authored in the process, as well as your blog. In particular, big fan of Kahanamoku-Meyer et al.'s optimistic QFT circuit. Anyway, keep up the great work!
prof-dr-ir 1y ago
I made a related comment about two years ago: https://news.ycombinator.com/item?id=38255476 https://news.ycombinator.com/item?id=38255476