If you want to see this kind of thing done “really in real time” check out Terry Tao’s recent mathstodon posts where he learns Lean while formalizing a paper of his own. Fascinating stuff.
“In fact, it isn’t even clear that depositors were going to be wiped out, absent federal intervention. When SVB was shut down, it still had real assets that were worth money, which can be sold to pay back investors. ”
Would conservatives suggest investors really should be paid back first? Unlikely to be popular.
Argh - yes it can. Bell explicitly have a pretty trivial one in one of his early papers. QM cannot be explained by a non-contextual hidden variables model. (The simplest physical version of contextuality is locality - so as Bell showed you cannot explain quantum theory with a local hidden variables model).
Presumably talking to the wind here as this thread is old, but nonlocal hidden variables do not break causality! This can be seen explicitly in Bell's trivial model, where the state of a system is just the regular quantum state plus a uniform random number between 0 and 1 (hidden). Or put it this way, if this model "breaks causality" then so does regular non-relativistic quantum theory.
A few months ago a couple of physics postdocs set up a website (vanityindex dot com) and proposed a few "vanity metrics". Can be fun to check out your own vanity index and that of your colleagues.
A clear description but potentially very misleading and one that will certainly screw up your intuition about what to expect from quantum computers.
As pointed out below, a classical probability distribution of N bits is a 2^N dimensional vector of real numbers. There are many very good reasons to think of the quantum state as "more like" this classical vector than the physical state of the "N classical bits" themselves.
Here is one:
- If I send you the physical systems which encode N classical bits then sure enough, I can communicate to you only N classical bits of information despite it taking exponentially many real parameters to specify the distribution/state I prepared them in.
- If I send you the physical systems which encode N quantum bits then sure enough, I can communicate to you only N classical bits of information despite it taking exponentially many real parameters to specify the distribution/state I prepared them in.
There are many other reasons to think of quantum states as not "inherently real" and more like the classical probability distribution. A key one is that both "instantaneously collapse" when you get information about the outcome of an observation.
The issue of course is that while we know the "real states of the world are" of a conventional computer, nobody agrees on what (if any) they are for quantum systems (though many constraints on such purported real states are known, I write about some of them in _Q is for Quantum_)
Here is a way to see the fallacy of the OMG, its 10^300 variables, thats crazy style of “argument”.
Consider a probabilistic classical algorithm on 500 bits. Perhaps a Monte Carlo simulation of an Ising model for example.
Note first that the most general probability distribution over the 500 classical bits takes 2^500 real numbers to specify. (You have to specify P(000…0) and P(000…1) and… P(111…1)). [You should compare this to the 2^501 real parameters it takes to specify the quantum state of 500 qubits.]
To generate the most general such distribution perhaps you are restricted to using circuits where the gates act on at most n-bits at a time. Each gate can be described by a 2^n x 2^n bistochastic matrix comprised of (n-1)^2 real parameters. [You should compare this to the n^2 real parameters it takes to specify a 2^n x 2^n unitary matrix for a quantum gate acting on n qubits.]
Obviously its nuts to imagine you can generate all classical probability distributions over the 2^500 real parameters, particularly if you’re so mad as to think you are going to do it using a circuit comprised only of these n-bit gates!
Therefore useful classical monte carlo computing is obviously decades away.
(Oh and please trust me, I'm well know in stuff that isn't quantum computing.)
Someone comes to you with two formal models of computing. Both models involve representing the state of the computer as a vector of real numbers, they both involve finite dimensional subsystems combined with a tensor product, both involve gates defined over the reals also combined via the tensor product and so on. That is, both models are just about evolution of a vector in some (very high) dimensional real vector space according to gates acting on a small number of subsystems. In fact these two models are identical, except for the fact that in model A the readout procedure involves computing a property of the output vector with the 1-norm, while in model B the 2-norm is used.
This is not an analogy, these are valid mathematical formulations of classical and quantum computing, the correspondences (and differences!) are well understood and rigorous.
Now you read an IEEE article that vociferously objects to the feasibility of building a computer based on model B, but all the objections are to do with properties of model B that it fully shares with model A. And model A you know can be very well approximated already in the physical world, which means reality was somehow was not inhibited by those objections. To try and refute the physicality of model B with an argument based on premises already satisfied by model A is silly.
(Note that even if it were the case that complex numbers were necessary for quantum computing, which they are not - see eg. my book Q is for Quantum - you can map the quantum density matrix on n qubits to a real vector over the basis of Hermitian matrices).
Hey I'm not familiar with this way of comparing classical and quantum computation. Can you point me to some more details? I have Nielson's book but don't remember seeing this analogy before!
I presume its explained in Scott Asronsons book, its implicitly there in Nielsen and Chuang. But the best way to understand it is by example - try to write out how you would describe classical probabilistic computation on two classical bits to mimic the quantum circuit type of picture, and if you succeed the generalization will be obvious.
Likewise, you don't ever explore the entire state space when you perform quantum computation. As it happens, most states are inaccessible. As shown by Poulin et al. in Physical Review Letters, 106, 170501 (2011), you can only ever explore a very small fraction of quantum states in polynomial time.
Edit: tezthenerd has said pretty much the same thing below. I just included the reference in case you'd like a formal proof.
Monte Carlo simulation doesn't generate all classical probability distributions though. Attempting to do that would be nuts. Monte Carlo simulation only samples from a single, fixed distribution (or maybe a small number of distribution).
I'm not super convinced by the argument based on number of parameters either, but your analogy doesn't refute it at all.
We also will not attempt to generate all quantum states on a quantum computer. As with classical monte carlo, we will only ever generate a tiny fraction of the possible quantum states/distributions, and will also sample from a fixed distribution (whatever the quantum circuit outputs, we measure it in a fixed basis and always draw samples from that single, fixed distribution).
We will also achieve robustness against the tyranny of the real numbers in our gate parameters in a very similar way that a classical computer does when it approximates some idealized Monte Carlo algorithm.
I might not be using the same definitions as you are or maybe I'm getting something mixed up, so I'd be glad if you could elaborate on what exactly you mean by "determinism", "realism" and especially "locality".
(In my book, the Copenhagen interpretation is a non-realistic(∆), non-deterministic and local theory, which would contradict your statement.)
(∆) Assuming, of course, that the wave function is not an object of reality, as I think is standard.
Many would argue that Copenhagen is not local, but its very hard to even define locality if you are genuinely non-realist about everything.
Regardless, if it was possible to be "local + non-deterministic" many of us would be fine with that. But its not - Bell rules out "locality + realism", regardless of whether the realistic theory is deterministic or non-deterministic.
Who? I believe the point of view that the Copenhagen interpretation is local is the standard one.[1]
> but its very hard to even define locality
How so? There are various definitions of locality—in terms of no faster-than-light transmission of information, in terms of commutators of field operators at spacelike distances vanishing as well as in terms of C* algebras—and the Copenhagen interpretation fulfills them all.
> if you are genuinely non-realist about everything
I'm being non-realist only about things whose existence we can't prove, e.g. the wave function.
> Regardless, if it was possible to be "local + non-deterministic" many of us would be fine with that. But its not - Bell rules out "locality + realism"
I don't see how this disproves anything of what I said. While I think you're right[2] about the fact that the violation of Bell's inequality rules out local realism—irrespective of determinism—, I already said that, in my book, Copenhagen is not a realistic theory because the wave function is not an object of reality.
[2]: "However, Fine's theorem shows that, this deterministic assignment of properties is not required to prove Bell's theorem. This is because the set of statistical distributions for measurements on two parties, once locality has been assumed, are independent of whether or not determinism is also assumed." (https://en.wikipedia.org/wiki/Principle_of_locality#Local_re...)
Where did I invoke mysticism? And where did I end the quest for answers?
I said "in my book" for a reason. You might of course argue that the wave function is real, which would make the Copenhagen interpretation a realistic, albeit non-local theory (as the collapse of the wave function would happen instantaneously everywhere). I'd be very happy to discuss this but please bear in mind that this was not what the discussion was about.
Try reading "Q is for Quantum", by the end of Part I you will understand one quantum algorithm thoroughly. All necessary gates are explained carefully and the linear algebra gets sneaked in without using bras and kets and so on...
