
A first for physics: Fundamental quantum physics problem proved unsolvable - fluxic
http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html
======
fitzwatermellow
Companion article to original paper:

Paradox at the heart of mathematics makes physics problem unanswerable:
Gödel’s incompleteness theorems are connected to unsolvable calculations in
quantum physics.

[http://www.nature.com/news/paradox-at-the-heart-of-
mathemati...](http://www.nature.com/news/paradox-at-the-heart-of-mathematics-
makes-physics-problem-unanswerable-1.18983)

Toby Cubitt, a "quantum-information theorist" at UCL should win this years
relevant username contest ;)

~~~
pavel_lishin
A great example of nominative determinism.

~~~
ethbro
To be fair, his name only collapsed into a determinative state once we looked
at it in the context of this article. /pun

------
colanderman
_The proof combines Hamiltonian complexity techniques with aperiodic tilings,
to construct a Hamiltonian whose ground state encodes the evolution of a
quantum phase-estimation algorithm followed by a universal Turing machine._

Just when I thought linking quantum mechanics with Turing completeness was
cool, they went one step further and linked them using aperiodic tilings.

~~~
fallingfrog
So if I'm understanding this right, they proved that you can't calculate the
bandgap of some class of systems of arbitrary size in any predetermined finite
period of time, right? I guess when you really think about it, the opposite
finding would be much more surprising. (That you could calculate the bandgap
of any system of any size in some finite time period, however large).

~~~
sp332
It's not a constant period of time. The amount of time is allowed to increase
with the size of the system. In fact it's allowed to increase quadratically,
or exponentially, or factorially, or whatever - it will never be enough.
Whatever algorithm you come up with for calculating the rate of expansion, you
will always underestimate the amount of time (for a sufficiently large
system).

[https://en.wikipedia.org/wiki/Busy_beaver#Non-
computability_...](https://en.wikipedia.org/wiki/Busy_beaver#Non-
computability_of_.CE.A3)

------
mathgenius
Long version:
[http://arxiv.org/abs/1502.04573](http://arxiv.org/abs/1502.04573)

Short version:
[http://arxiv.org/abs/1502.04135](http://arxiv.org/abs/1502.04135)

------
mojoe
I can't determine what the postulates are that this proof is based upon. If
someone has a relatively succinct explanation, I'd appreciate it.

~~~
mathgenius
Mathematically, this is in the land of linear algebra (complex vector spaces.)
The particular question at hand concerns the spectrum of a (Hermitian) linear
operator, known as "the Hamiltonian". The "gap" refers to a separation in the
lowest two eigenvalues ("energy levels") of this operator.

It seems plausible that one could easily encode a turing incomplete problem
into linear algebra (think of tensor networks, constraint satisfaction, etc.)
but what these guys have done is place very strong restrictions on the form of
the linear operator (in jargon: two dimensional, nearest neighbour
interactions, translational invariance). This now comes near to a class of
physical systems that one may hope to build, or at least understand
theoretically, and the result of the paper puts a stark upper limit on what we
may hope to achieve in this vein.

~~~
hndl
Thank you for this.

------
littletimmy
This might be an ignorant question, but can one still force a decidable
solution by assuming a stronger set of axioms? And if so, does this
undecidability have any physical significance?

~~~
jordigh
Yes, you can add more axioms, but your new system with more axioms will have
other undecidable statements.

I don't know much physics, but my interpretation is this: we have a
mathematical model of physics in which we have found a "naturally-ocurring"
undecidable statement (scare quotes because most famous undecidable statements
were specifically constructed to be undecidable). This does not really say
anything about the natural world, merely that our mathematical modelling of it
is incomplete, but it always must be incomplete.

~~~
noobermin
Is it really impossible to have some system of axioms where most statements
you care about (ie. can make a physical experiment out of) can be decided? If
you can't measure something (ie., decide it someway from your theory or leave
it as a free parameter), it's not science, string-theory nonwithstanding.

~~~
jordigh
> Is it really impossible to have some system of axioms where most statements
> you care about (ie. can make a physical experiment out of) can be decided?

Of course it's possible. Most statements we care about are decidable.
Otherwise, mathematics would be pretty hopeless. :-) It's just that any system
has undecidable statements, as long as it's strong enough to do arithmetic.
That's Gödel's theorem.

~~~
noobermin
I know due to Godel in principle that a logical system can't be complete, but
science is a little more specified beyond just logical consistency, it has to
be experimentally verifiable, which is a constraint that the mathematics used
by science doesn't have. What I'm hinting at is there is a subset of logical
statements that the _mathematics_ of science contains defined by the set of
experimentally testable statements, and this subset is decidable given
independent axioms and free parameters.

EDIT: to expand upon this further, things like the continuum hypothesis are
interesting from a mathematical perspective, let's say I'm Godel and I assert
that CH is false and there is some set A "in between" the reals and N in
cardinality. However, if my friend Enrico Fermi wants to do experiments and he
wants me to do a calculation, I won't use elements from A because it wouldn't
be useful for his experiments in with particle physics. Worse, I can't even
use all the real numbers, because he'll need numbers he can compare to his
measurements, so I'm restricted to a countable subset of even the reals, and
of course, I'm restricted to using statements from the decidable subset of
mathematics.

There's a big difference between what mathematics is capable of saying and
what we observe in the real world. Even if the real world had a set of
undecidable facts, the fact that the only way science can understand it is by
experiments restricts _falsifiable scientific statements_ to a subset of
mathematical statements which is incomplete (because of the need of axioms and
free parameters like the mass of particles or their charge, fundamental
constants, etc), but _decidable_ within it's axioms, because otherwise, it's
not testable and it's not science.

~~~
jordigh
As Einstein said, "As far as the laws of mathematics refer to reality, they
are not certain; and as far as they are certain, they do not refer to
reality."[1]

Forgive me for not addressing directly what you said, as you seem to be using
"countable" to mean something other than "in bijection with the natural
numbers". I don't really understand exactly what you mean. But it appears to
me that you're trying to perfectly reconcile mathematics with experimental
reality. This is impossible. Also, a lot of mathematical physics is very
distant from experimental reality.

\--

[1] [http://www-groups.dcs.st-
and.ac.uk/history/Extras/Einstein_g...](http://www-groups.dcs.st-
and.ac.uk/history/Extras/Einstein_geometry.html)

~~~
noobermin
I might be confusing this thread with another. The posters there claimed that
when considering the infinite number of possible systems that you end up with
undecidable statements. Probably countability has nothing to do with this.

Also, the popular article linked elsewhere seems to imply that this has very
real implications on experiment, which is upsetting.

------
swehner
I can't tell how much of this is "quantum physics" and how much is
"mathematical theory that also arises in quantum physics".

As in, I doubt their construction allows "building something" which then
allows solving a conventionally unsolvable problem. With new physics you may
be able to build machines that can compute more; however I don't think they
are talking about such physics here.

But, they do quote: Marian B. Pour-El and Jonathan I. Richards. Computability
in Analysis and Physics. Springer, 1989.

M. B. Pour-El and J. Richards have earlier results:

[http://www.sciencedirect.com/science/article/pii/00018708819...](http://www.sciencedirect.com/science/article/pii/0001870881900013):
The wave equation with computable initial data such that its unique solution
is not computable (1981)

[http://philpapers.org/rec/POUTWE](http://philpapers.org/rec/POUTWE): The wave
equation with computable initial data whose unique solution is nowhere
computable (1997)

The Spectral Gap authors seem to call these results "easier" (page 8 at
[http://arxiv.org/pdf/1502.04573v2.pdf](http://arxiv.org/pdf/1502.04573v2.pdf))

------
j2kun
It's nice that they showed this for the problem of finding the ground state of
a Hamiltonian, but we already knew the axioms of quantum computing encode
undecidable problems because it includes all of classical computing.

------
Confusion
Reminds me of an adage I once read and have taken to heart (but I can't recall
the source and I'm paraphrasing): what is impossible in theory is often not
nearly as interesting as what is possible in practice.

------
drdeca
Wait,

Does that imply that it could be possible to construct a system which gives
uncomputable results?

Does it assume an infinite plane or something? (Assume might be the wrong
word)

Because, if there's something uncomputable, due to halting problem, how does
that fit with the bekenstein bound?

~~~
misja111
But spectral gaps do occur in nature. Does that mean that even though it is
impossible to compute if a spectral gap will occur or not, we could just
observe such a system in nature and that way we could build a computer which
would compute uncomputable results?

~~~
sharpneli
In nature it does not matter if the spectral gap is really tiny. It's de facto
without spectral gap if it's just small enough.

Whereas for this particular decision problem the mere existence of spectral
gap matters.

~~~
selimthegrim
Who says? Graphene gaps, (it must gap) even though its so small it's
impossible to observe in practice. This had very important consequences for
topological materials theory...

~~~
sharpneli
This is exactly what I meant.

Even if it has theoretical reasons to have a tiny gap in practice it works as
zero gap.

~~~
tobycubitt
Except that in our result, the spectrum is either continuous, or the spectral
gap is guaranteed to be >= 1 (in natural units). It does not become
arbitrarily small in the gapped case.

The reason you can't use this to compute the uncomputable is that real systems
are finite, and the spectral gap is always computable in principle (maybe with
a lot of effort) for any finite system. A real (possibly very large but still
finite) system will either have a gap or not, and you'll be able to measure
it. This definitely doesn't solve an undecidable problem.

However, the undecidability in the idealised infinite lattice limit "shows
through" to the experimentally accessible finite-size case, in the form of
some rather unusual finite-size physics. This is discussed in more detail in
the paper itself (the relevant section is quoted verbatim here
[http://mathoverflow.net/a/225905](http://mathoverflow.net/a/225905)) and in
the comments on Scott Aaronson's blog.

------
stillsut
I've wondered if this can go the other way as well:

e.g. "If Golbach's conjecture is true, then FTL signalling is possible. Since
physics says FTL is impossible, then GC must be false"

[If you could reduce GC to a scheme for moving information via quantum
entaglement. I use GC as an example because it is famous and an unproved, but
maybe a better example is the Erdos Discrepancy [0] which makes statements
about aritrarily long sequences in {1,-1} just like sampling quantum states!]

[0]: [http://arxiv.org/abs/1509.05363](http://arxiv.org/abs/1509.05363)

~~~
NickM
I think most mathematicians would not accept something like that as a proof.
Our understanding of physics is based on real-world observations and
experiments in a way that math is not.

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hasenj
Since physical states exist in reality, it makes no sense to me that a
physical problem is not mathematically decidable.

The only way a problem is mathematically undecidable is when it's malformed
(i.e. non-sensical).

~~~
nsns
But surely, any physical state would be available to you only as a
_representation_ , and not directly. Meaning, that in order to make it
decidable, you'll probably have to look at it differently.

------
incepted
Uncomputable or unsolvable?

~~~
pygy_
Undecidable.

[https://en.wikipedia.org/wiki/Undecidable_problem](https://en.wikipedia.org/wiki/Undecidable_problem)

See here for a concrete example of a similar problem (props to adrianN for the
link in the first place):

[https://en.wikipedia.org/wiki/Wang_tile#Domino_problem](https://en.wikipedia.org/wiki/Wang_tile#Domino_problem)

------
clebio
The post and linked article titles don't match at all (??).

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jmount
Such a statement just means they can implement computation in this formalism.
Not a huge surprise as you can implement computation with many other
structures.

~~~
tgb
This is being downvoted, but I'd like an explanation of what the alternative
is. Can someone to explain how this result differs from a claim "I can build a
computer which turns a light on when a given program terminates, so therefore
'light on-ness' is undecidable".

~~~
lmm
Whether a light is on or off is contingent. This is more like proving the
light would always be on as a fundamental fact of basic physics. Even if it
were done by showing that the existence of a computer connected to the light,
proving that from first physical principles would be a big achievement.

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JoachimS
But, but, but - what if I use a D-Wave machine to solve it? ;-)

~~~
kefka
I know you jest, but is there a proof (or good hypothesis) that allows one to
convert a formula to an appropriately sped-up version quantum optimized
algorithm?

Compare the Sieve of Atkin to Shor's algorithm, for what I am referring to.

~~~
tgb
My understanding from reading Scott Aaronson, is that his belief is that most
algorithms will not have significant speedups as a quantum algorithm, and that
the currently known algorithms with large (exponential) speedups aren't very
useful in the sense that they're just breaking cryptographic problems which
were used solely because they were hard to break. In other words, don't expect
everything to have a Shor's algorithm analogue.

