Aram Harrow asked: "Is there any place this is written up?"
Update Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in our Soldier Healing Seminar's notes for 2013; and various motivations for caring about these properties are surveyed in a post to Dick Lipton and Ken Regan's Gödel's Lost Letter weblog. For engineers, the practical question at-issue is whether thermodynamics "just works" on varietal state-spaces, not only at regular points, but at singular points.
Introduction In considering the question "Are quantum states exponentially long vectors?" (arXiv:quant-ph/0507242) Scott Aaronson provocatively argued that:
The real reason to study quantum computing is not to learn other people’s secrets, but to unravel the ultimate Secret of Secrets: Is our universe a polynomial or an exponential place? …
For me, the main weakness in the arguments of quantum computing skeptics has always been their failure to suggest an answer to the following question: What criterion separates the quantum states we’re sure we can prepare, from the states that arise in Shor’s factoring algorithm?
In a (wonderfully interesting!) subsequent debate, the thesis that our universe is (in Aaronson's phrase) "an exponential place" has been ably defended by Aram Harrow, whereas Gil Kalai has marshalled cogent reasons — for example, in a UW/PIMS colloquium (Friday November 23; details here) — "Why quantum computers cannot work."
The questions asked The questions asked concern the classification of secant varieties of Segre varieties of Veronese varieties (SSVs). Viewed as dynamical manifolds immersed in an embedding Hilbert state-space, SSVs find broad applications in quantum simulation problems.
The questions asked concretely concern a dynamical Hilbert space of 27 dimensions, which is viewed as a 26-dimensional projective space $\mathbb{P}^{26}$. Numerical studies indicate that the following five varieties are immersed submanifolds of dimension 25, that is, the following five varietal state-spaces are Hilbert-deficient by precisely one dimension:
$$\begin{alignedat}{2} \sigma_4\big(\,&\text{Seg}(\,\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^2\,)\big)\\ \sigma_5\big(\,&\text{Seg}(\,\mathbb{G}^2\times\mathbb{P}^2\times\mathbb{P}^2\,)\big)\\ \sigma_7\big(\,&\text{Seg}(\,\mathbb{G}^2\times\mathbb{G}^2\times\mathbb{G}^2\,)\big)\\ \sigma_9\big(\,&\text{Seg}(\,\mathbb{G}^8\times\mathbb{P}^2)\,\big)\\ \sigma_{13}\big(\,&\text{Seg}(\,\mathbb{G}^{26}\,)\big) \end{alignedat}$$
The Four Questions
Q1 In what mathematical respects (if any) do these five varietal spaces differ from one another, in regard to their topology, singularity structure, metric structure, and symplectic structure? Do they all pushforward/blowdown to the same varietal submanifold of Hilbert space? Or do they represent five essentially different "foamy" algebraic/metric/symplectic geometries?
Q2 By what physical measurement processes (if any) can dynamical trajectories on these manifolds be distinguished from one another, and from trajectories on the (flat, singularity-free) Hilbert space in which they are embedded?
Q3 In what quantum mechanical respects (if any) are trajectory unravelings on these manifolds sufficiently deficient in any and all forms of "quantum goodness" as to invalidate their claim to be "Sure-Shor separators" (in the sense of Aaronson)?
Q4 What classification theorems (if any) exist to help quantum systems engineers in exploiting these varietal spaces for purposes of large-scale quantum dynamical simulation?
Physical remarks The state-spaces are stipulated to inherit by pullback the standard metric, symplectic, Hamiltonian, and Lindbladian structures of the immersing Hilbert space, such that (from a Harrow-esque perspective) any quantum experiment envisioned to be performed in the Hilbert space, has an algebraically natural pullback/blowup/resolution of singularities on the varieties named, such that simulated quantum trajectories unraveled on the varietal state-space are a reasonable approximation to the "true" quantum state-space.
Alternatively (from a Kalai-esque perspective), any quantum experiment envisioned to be performed in the varietal state-space, has an algebraically natural pushforward/blowdown onto a fictitious Hilbert space of exponentially many dimensions, that is a maximally symmetric (but computationally intractable) approximation to the "true" varietal state-space.
In a nutshell These varietal spaces are candidate manifolds for Aaronson-style "Sure-Shor separators", and the difficulty of distinguishing geometric perturbations of dynamical trajectories on these "foamy" algebraic manifolds from noise perturbations helps us appreciate the depth and subtlety of Gil Kalai's reasons "why quantum computers cannot work."
Notational remarks The question adopts and slightly generalizes the notation of Landsberg's Tensors: Geometry and Applications as follows: $\sigma_r$ is the usual rank-$r$ secant join, $\text{Seg}$ is the usual Segre embedding, and $\mathbb{P}^k$ is the usual projective space
$$\mathbb{P}^k(\,\boldsymbol{\xi}\,) = [\xi_0,\xi_1,\dots,\xi_k]$$
The projective space $\mathbb{G}^k$ is (what mathematicians call) a Veronese embedding
$$ \mathbb{G}^k(a,b) = [\,{% \textstyle{{k}\choose{0}}^{1/2}a^k b^0,% {{k}\choose{1}}^{1/2}a^{k-1} b^1,% \dots,% {{k}\choose{k-1}}^{1/2}a^{1} b^{k-1},% {{k}\choose{k}}^{1/2}a^{0} b^{k}% }\,]\hookrightarrow \mathbb{P}^k.$$
Physicists regard this Veronese embedding as the space of coherent states; the "$\mathbb{G}$" of the above notation is a mnemonic for Roy Glauber, who pioneered the application of coherent states to broad-ranging practical problems, first in quantum optics and nowadays in many branches of science and engineering; the connexion between the physics viewpoint and the algebraic geometry viewpoint is surveyed by Dorje Brody and Eva-Maria Graefe's recent Coherent states and rational surfaces; the complex numbers $a,b$ are associated to the quaternionic rotation coordinates $\boldsymbol{q} = \{q_0,q_1,q_2,q_3\}$ that are favored by engineers by
$$ \begin{aligned} a(\boldsymbol{q}) &= q_0 + i q_3\\ b(\boldsymbol{q}) &= i q_1 - q_2\ ; \end{aligned} $$
and finally, the binomial weights ${{k}\choose{0}}^{1/2}$ etc. that appear in the definition of $\mathbb{G}^k$ commonly are omitted in the mathematical literature. The binomial weights ensure that the Veronese embedding is a metric and symplectic isometry, as is desirable to preserve various aspects of "quantum goodness" for simulation purposes; Katsumi Nomizu (1976) seems to have been among the first to appreciate their rationale.
Motivation This question is constructed as a tribute to Aaronson's fine "Sure-Shor Separator" question, and as a tribute to the outstanding Kalai/Harrow debate that Aaronson's question helped stimulate. For engineers, Gil Kalai's proposition "Why quantum computers CAN'T work" belongs to a broad class of strategically crucial propositions that include "How large-scale quantum simulation DOES work" and "How atomic-resolution microscopes CAN work", and include too unanswered fundamental physics questions like "Is Nature's quantum dynamical state-space EXPERIMENTALLY static and flat?"
Acknowledgements Appreciation and thanks are extended to Scott Aaronson, Gil Kalai, and Aram Harrow, for their sustained and admirably collegial commitment to asking good questions and seeking good answers.
