I have fleshed-out this answer with an extended answer on MathOverflow to Gil Kalai's community wiki question "[What is] A Book You Would Like to Write."
The extended answer seeks to link fundamental issues in TCS and QIT to practical issues in healing and regenerative medicine.
This answer extends Peter Shor's answer
, which discusses the roles of matrix product states in TCS and physics. Two recent surveys in the Bulletin of the AMS
are relevant to matrix product states, and both surveys are well-written, free of pay-wall restrictions, and reasonably accessible to non-specialists:
The mathematical arena for Landsberg's survey is secant varieties of Segre varieties, while the arena for Pelayo's and Ngoc's survey is four-dimensional symplectic manifolds … it takes awhile to appreciate that these two arenas both are matrix product states, as viewed respectively from a computational perspective (Landsburg) and a geometric perspective (Palayo and Ngoc). Moreover, Palayo and Ngoc include in their survey a discussion of Babelon, Cantini, and Douçot's A semi-classical study of the Jaynes–Cummings model (noting that the Jaynes–Cummings model is often encountered in the literature of condensed matter physics and quantum computing).
Each of these references goes far to illuminate the others. In particular, it has been helpful in our own (very practical) spin dynamical calculations to appreciate that the quantum state-spaces that are described variously in the literature as tensor network states, matrix product states, and secant varieties of Segre varieties are richly endowed with singularities whose algebraic, symplectic, and Riemannian structure is at present very incompletely understood (as Pelayo and Ngoc review).
For our engineering purposes, the Landsburg/algebraic geometry approach, in which the state-space of quantum dynamics is viewed as an algebraic variety rather than a vector space, is emerging as the most mathematically natural. This is surprising to us, but in common with many researchers, we find that the toolset of algebraic geometric is gratifyingly effective in validating and speeding practical quantum simulations.
Quantum simulationists presently enjoy the puzzling circumstance that large numerical quantum simulations very often perform much better than we have any known reason to expect. As mathematicians and physicists arrive at a shared understanding, this puzzlement surely will diminish and the enjoyment surely will remain. Good! :)