Nevertheless, Planck did not understand well enough the requirements for a good system of fundamental units of measurement (because he was a theoretician, not an experimentalist; he had computed his constants by a better mathematical treatment of the experimental data provided by Paschen), so he did not find any good way to integrate Planck's constant in a system of fundamental units and he has made the same mistake made by Stoney 25 years before him (after computing the value of the elementary electric charge) and he has chosen the wrong method for defining the unit of mass among two variants previously proposed by Maxwell (the 2 variants were deriving the unit of mass from the mass of some atom or molecule and deriving the unit of mass from the Newtonian constant of gravitation).
All dimensionless systems of fundamental units are worthless in practice (because they cause huge uncertainties in all values of absolute measurements) and they do not have any special theoretical significance (for now; such a significance would appear only if it became possible to compute exactly from theory the values of the 2 constants of the electromagnetic interaction and gravitational interaction, instead of measuring them through experiments; until now nobody had any useful idea for a theory that could do such things).
For the number of independently chosen fundamental units of measurement there exists an optimum value and the systems with either more or fewer fundamental units lead to greater uncertainties in the values of the physical quantities and to superfluous computations in the mathematical models.
The dimensionless systems of units are not simpler, but more complicated, so attempting to eliminate the independently chosen fundamental units is the wrong goal when searching for the best system of units of measurement.
My point is that the values of the so-called "Planck units" have absolutely no physical significance, therefore it is extremely wrong to use them in any reasoning about what is possible or impossible or about anything else.
The "Planck units" are not unique, there also exists a very similar system of "Stoney units", proposed a quarter of century before the "Planck units", where the values of the units are different, and there are also other variants of dimensionless systems of units proposed later. None of them is better than the others and all are bad, the worst defect being that the huge experimental uncertainties from measuring the value of the Newtonian constant of gravitation are moved from that single value into the values of all unrelated physical quantities, so that no absolute value can be known precisely, but only the ratios between quantities of the same kind.
In a useful system of fundamental units, for all units there are "natural" choices, except for one, which is the scale factor of the spatio-temporal units. For this scale factor of space-time, in the current state of knowledge there is no special value that can be distinguished from other arbitrary choices, so it is chosen solely based on the practical ease of building standards of frequency and wave-number that have adequate reproducibility and stability.
The only historical value of the "Planck units" is that they provide a good example of how one should NOT choose a system of units of measurement. The fact that they are still frequently mentioned by some people in any other context than criticizing such a system just demonstrates the very sad state of physics education, where no physics textbook includes an adequate presentation of the foundation of physics, which is the theory of the measurement of physical quantities. One century and a half ago, Maxwell began his treatise on electricity and magnetism with a very good exposition of the state of metrology at that time, but later physics textbooks have become less and less rigorous, instead of improving.
The theoretical entropy for a Schwartzchild black hole is nicely expressed using the Planck area.
So…
No. Your assertion that they have no value in theory, is wrong.
(Also, like, angular momentum is quantized in multiples of hbar or hbar/2 or something like that.)
It may be true that they aren’t a good system of units for actual measurements, on account of the high uncertainty (especially for G).
But, there is a reason why it is common to use units where G=c=hbar=1 : it is quite convenient.
More realistically it breaking down would hopefully give us a new physics frontier.
2^256 states are comfortably distinct in that many dimensions with amplitude ~1. Their distinctness is entirely direction.
The obvious parallels to vector embeddings and high-dimensional tensor properties have some groups working out how to combine them in "quantum AI", and because that doesn't require the same precision (like trained neurel nets still work usefully after heavy quantization and noise), quantum AI might arrive before regular quantum computation, and might be feasible even if the latter is not.
Forget the talk about amplitudes. What I find hard to believe is that nature will let us compute reliably with hundreds of entangled qubits.
Shor's algorithm starts with the qubits in a superposition of all possible bitstrings. That is the only place we have exponentially small amplitudes at the start (in a particular choice of a basis), and there is no entanglement in that state to begin with.
We do get interesting entangled states after the oracle step, that is true. And it is fair to have a vague sense that entanglement is weird. I just want to be clear that your last point (forgetting about amplitudes, and focusing on the weirdness of entangled qubits) is a gut feeling, not something based in the mathematics that has proven to be a correct description of nature over many orders of magnitude.
Of course, it would be great if it turns out that quantum mechanics is wrong in some parameter regime -- that would be the most exciting thing in Physics in a century. There is just not much hope it is wrong in this particular way.
Yes, that is exactly the point. The example statevector you guys are talking about can (tautologically) be written in a basis in which only one of its amplitudes is nonzero.
Let's call |ψ⟩ the initial state of the Shor algorithm, i.e. the superposition of all classical bitstrings.
|ψ⟩ = |00..00⟩ + |00..01⟩ + |00..10⟩ + .. + |11..11⟩
That state is factorizable, i.e. it is *completely* unentangled. In the X basis (a.k.a. the Hadamard basis) it can be written as
|ψ⟩ = |00..00⟩ + |00..01⟩ + |00..10⟩ + .. + |11..11⟩ = |++..++⟩
You can see that even from the preparation circuit of the Shor algorithm. It is just single-qubit Hadamard gates -- there are no entangling gates. Preparing this state is a triviality and in optical systems we have been able to prepare it for decades. Shining a wide laser pulse on a CD basically prepares exactly that state.
> Changing basis does not affect the number of basis functions.
I do not know what "number of basis functions" means. If you are referring to "non zero entries in the column-vector representation of the state in a given basis", then of course it changes. Here is a trivial example: take the x-y plane and take the unit vector along x. It has one non-zero coefficient. Now express the same vector in a basis rotated at 45deg. It has two non-zero coefficients in that basis.
---
Generally speaking, any physical argument that is valid only in a single basis is automatically a weak argument, because physics is not basis dependent. It is just that some bases make deriving results easier.
Preparing a state that is a superposition of all possible states of the "computational basis" is something we have been able to do since before people started talking seriously about quantum computers.
Even preparing the initial state that accurately is only trivial on paper.
- I am not saying that you have to find a basis in which your amplitudes are not small, I am saying that such a basis always exists. So any argument about "small amplitudes would potentially cause problems" probably does not hold, because there is no physical reality to "an amplitude" or "a basis" -- these are all arbitrary choices and the laws of physics do not change if you pick a different basis.
- In classical probability we are not worried about vanishingly small probabilities in probability distributions that we achieve all the time. Take a one-time pad of n bits. Its stochastic state vector in the natural basis is filled with exponentially small entries 1/2^n. We create one-time pads all the time and nature does not seem to mind.
- Most textbooks that include Shor's algorithm also include proof that you do not need precise gates. Shor's algorithm (or the quantum Fourier transform more specifically) converges even if you have finite absolute precision of the various gates.
- Preparing the initial state to extremely high precision in an optical quantum computer is trivial and it has been trivial for decades. There isn't really much "quantum" to it.
- It is fair to be worried about the numerical stability of a quantum algorithm. Shor's algorithm happens to be stable as mentioned above. But the original point by OP was that physics itself might "break" -- I am arguing against that original point. Physics, of course, might break, and that would be very exciting, but that particular way of it breaking is very improbable (because of the rest of the points posted above).
But when N>2 this gets tougher rapidly.
If we add 10^12 complex amplitudes and each one is off by one part in 10^{-6}, we could easily have serious problems with the accuracy of the sum. And 10^12 amplitudes is "only" around 40 qubits.