I think our questioner is aware of the difficulties of switching fields and if not he or she soon will be, so let me be naive and try to be constructive.
For Quantum Computation, Isaac Chuang and Michael Nielsen's "Quantum Computation and Quantum Information" has become a standard introduction to the subject, suitable for a graduate student in either mathematics, physics or computer science.
Since I have no idea what background you have in PDEs (you could be a specialist in D-modules for all I know and find these suggestions childish), here are some texts I've become acquainted with:
-V.I. Arnold's "Lectures on Partial Differential Equations" gives a beautifully geometric and intuitive understanding of PDEs, introducing and weaving together contact and symplectic geometry. The table of contents looks quite basic, but it contains the depth you should expect from Arnold.
-Lawrence C. Evans "Partial Differential Equations" is nice and contains the basic notions from Functional Analysis, Sobolev Spaces, Weak Theory and Regularity Theory. It does a good job of being self-contained and trying to give physical interpretations of various PDEs.
-Gilbarg and Trudinger have the classic "Elliptic PDEs of Second Order", which is dense, but a classic nonetheless.
As a mathematician you don't need to learn how physicists think in the next year. Physicists have different ways of looking at problems and are constrained to their own paradigms just as mathematicians are. It is often quicker to pick up advanced physics if you know advanced mathematics, with many excellent bridge texts by world-class mathematicians. Examples that come to mind are Bott's "Morse Theory Indomitable" which includes an exposition of some of Witten's ideas for a mathematician. Atiyah's "Geometry and Physics of Knots" is also an excellent example of this. Feynman's Lectures are great, but won't advance you to research. It's more like a Caltech undergraduate degree bound in 3 volumes.
Finally, as a note of inspiration, I have heard of at least two new faculty who self-studied PDEs in their post-doctoral years. One was supplanting a thesis in deformation theory and integrable systems, the other in knot theory and Floer homology. It is definitely a hard path to follow, but is sometimes necessary for growth. Also, bear in mind that Ed Witten was a history major as an undergrad, dropped out of economics grad school before applying for Princeton applied math and then switching to physics. Raoul Bott switched from electrical engineering to mathematics after his PhD (a much harder path, one might argue). Finally, my personal hero, Douglas Hofstadter, after quitting his Berkeley math PhD and finishing a 7+ year physics PhD in Oregon, then lived at home for a few years re-tooling himself as an AI researcher. Now he has a Pulitzer and tenure at a university -- not too shabby.