In the beginning God created a compositional universe of pictures, and then quantum computing
(this talk does not require any background in quantum physics nor category theory!)
It is now exactly 75 years ago that John von Neumann denounced his own Hilbert space formalism: ``I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.''
(sic)  His reason was that Hilbert space does not elucidate in any direct manner the key quantum behaviors. So what are these key quantum behaviors then?
For Schrodinger this is the behavior of compound quantum systems, described by the tensor product [2, again 75 years ago]. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0's and 1's in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by stuff like continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct. In other words, we are looking for a `high-level quantum language'!
Over the past couple of years we have played the following game: how much quantum phenomena can be derived from `composition + epsilon'. It turned out that epsilon can be taken to be `very little', surely not involving anything like continuum, fields, vector spaces, but merely a `two-dimensional space' of temporal composition (cf `and then') and compoundness (cf `while'), together with some very natural purely operational assertion. In a very short time, this radically different approach has produced a universal graphical language for quantum theory which helped to resolve some open problems. It also paved the way to automate quantum reasoning  and has even helped to solve problems outside physics, most notably in modeling meaning for natural languages .
The `categorical quantum mechanics' research program started with . A recent survey paper on the approach is .
 M Redei (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Stud Hist Phil Mod Phys 27, 493-510.
 E Schroedinger, (1935) Discussion of probability relations between separated systems. Proc Camb Phil Soc 31, 555-563; (1936) 32, 446-451.
 L Dixon, R Duncan, A Kissinger and A Merry. dream.inf.ed.ac.uk/projects/quantomatic/
 B Coecke, M Sadrzadeh & S Clark (2010) Ling Anal 36. Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394
 S Abramsky & B Coecke (2004) A categorical semantics of quantum protocols. LiCS '04.
 B Coecke (2010) Quantum picturalism. Contemporary Physics 51, 59-83. arXiv:0908.1787