Quantum computation and categories

[John Baez <john.c.baez@gmail.com>]

Dear Dusko -

You wrote:

bob coecke proposed to add quantum computing to andre joyal's list of
> important directions of categorical research, but andre rejected it.


most results in quantum computing are theorems about hilbert spaces. quantum
> computing is a *tensor calculus*. but it is a tensor calculus of a special
> kind: it attempts to describe a wildly unintuitive world. even the greatest
> contributors, like von neumann and feynman, deplored the gap between the
> quantum world, imposed on us in the lab, and the intuitions imposed on us in
> everyday life. now category theory often helps where the common intuitions
> fail. many of its applications demonstrate this. so
> quantum computation might be an opportunity for an effective application of
> *geometry of tensor calculus*.
>

Exactly!  Samson Abramsky, Bob Coecke, Peter Selinger and others have been
doing great work along these lines.

I think this line of research will eventually be the key to understanding
quantum gravity, because string diagrams reveal the common features of the
tensor category of Hilbert spaces (Hilb, fundamental to quantum theory) and
the tensor category of cobordisms (nCob, fundamental to our traditional
notion of spacetime).  I argued this case here, in a nontechnical way:

http://math.ucr.edu/home/baez/quantum/

And I think that regardless of whether quantum computers or quantum gravity
ever work, this line of research is very interesting.


> is it really wise to reject an attempt to develop this, as objectionable as
> it might be in any details?


Andre didn't precisely "reject an attempt to develop" these ideas.  He said
"I am not convinced that quantum computing can contribute significantly to
category theory".  And that's fine. The bold researchers listed above will
now redouble their efforts to convince Andre by proving lots of wonderful
theorems.

Here's one point where work on quantum computing, quantum gravity, and TQFT
could have a radical effect on category theory.  Researchers in these
subjects have been forced by the nature of the material to embrace
"dagger-categories".  I explain why in my article above, but I called them
"*-categories" instead of dagger-categories.

A dagger-category is a category C with a functor

F: C -> C^{op}

which is the identity on objects and has F^2 = 1.

Category theorists will note that the above definition is "evil", in the
technical sense of that term:

http://ncatlab.org/nlab/show/evil

Namely, it imposes equations between objects, so we cannot transport a
dagger-category structure along an equivalence of categories.

Often evil concepts (like the concept of "strict monoidal category") have
non-evil counterparts (like the concept of "monoidal category").  But in
this particular case I know no way to express the idea without equations
between objects.  Both Hilb and nCob are dagger-categories.  This fact is
important.  Try saying it in a non-evil way!

Once Andre told me some ideas about this, relating to the case of Hilb, but
unfortunately I don't see that how they could apply  to nCob.

I was very interested at Mark Weber's reaction to this problem.  He said,
roughly, "So dagger-categories aren't really categories with extra
structure.  Okay: they're something else!  And that's fine."  (I'd be happy
for him to correct my rough summary and make his point more precisely.)

I like this bold attitude, especially coming from someone like Mark, who
knows enough category theory to carry it off.  This could lead to really new
developments.

Best,
jb


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]