Note this answer is superseded by a more detailed answer to which the reader is referred.
I would very much like to write the book (or much easier, read someone else's book) Geometric Dynamics with Practical Applications in Classical and Quantum Systems Engineering.
The envisioned book would encompass, via geometric methods, many of the dynamical and informatic themes that Michael Nielsen and Isaac Chuang so excellently cover via algebraic methods, in their Quantum Computation and Quantum Information (2000).
It would be the book that—in an alternative history of 20th century quantum physics—would have grown from Saunders Mac Lane's 1970 Chauvenet Lecture Hamiltonian mechanics and geometry, had Mac Lane first read Nielsen and Chuang's now-classic text, and then read Abhay Ashtekar and Troy Schilling's Geometrical formulation of quantum mechanics (1999), and finally read Carlton Caves' (on-line) note Completely positive maps, positive maps, and the Lindblad form (2000, revised 2002 and 2008).
As for notation, this book would embrace the notation of Jack Lee's admirably clear Introduction to Smooth Manifolds (2003) ... no need for "bras" and "kets"!
Our UW Quantum Systems Engineering (QSE) Group first explored these ideas in an article Practical recipes for the model order reduction, dynamical simulation and compressive sampling of large-scale open quantum systems. This article is sufficiently lengthy (at 96 pages) as to provide much of the material for a short textbook. However, an all-recipe textbook on quantum simulation would lack an overall ordering perspective.
Thus, a void in the existing quantum dynamics literature is an integrative synthesis of the above references by Mac Lane, Nielsen and Chuang, Ashtekar and Schilling, Caves; moreover many other authors—Arnold, Carmichael, Abraham and Marsden, Thurston, etc.—could be added to this list. It takes quite a bit of reading to appreciate that these authors' ideas and formalisms are naturally congruent.
The arxiv preprint Elements of naturality in dynamical simulation frameworks for Hamiltonian, thermostatic, and Lindbladian flows on classical and quantum state-spaces (arXiv:1007.1958) is a first-draft summary of those results needed to establish a coherent set of geometric ideas associated to practical large-scale quantum simulation (fortunately, not very many new results are needed).
Current research interests focus on verification, validation, and runtime estimation (VVR), with a view toward establishing consonance between practical VVR and various "no go" results that complexity theory provides. These investigations are still in an early stage; they largely motivate our TCS StackExchange questions "Are runtime bounds in P decidable? (answer: no)" and "Do runtimes for P require EXP resources to upper-bound? … are concrete examples known? (answer: yes and yes)".
We are working on a narrative that integrates these ideas, our MathOverflow answer to the question "What a Geometer Should Know" is our working outline for that narrative.
The PNAS article "Spin Microscopy's Heritage, Achievements, and Prospects" describes our technical objectives, which originate in roadmaps set forth by von Neumann, Wiener, and Feynman in the era 1946-59; most of my questions and answers here on MathOverflow, and also on TCS StackExchange, are conditioned upon these concrete (and predominantly medical) objectives.
And finally, this first book Geometric Dynamics with Practical Applications in Classical and Quantum Systems Engineering is envisioned as the first of two books ... the second book will be Applications of Quantum Spin Microscopy in Regenerative Medicine. That second book will someday be a topic for a second post.
