Quantum Spin Microscopy's
Emerging Methods, Roadmaps, and Enterprises

Our syllabus notes—now finally done!—are available as an arxiv server preprint,
and a substantial upgrade of the notes is nearing completion.

The main upgrade will be an explicit pseudo-code construction of certain projective structures that are central to our key theorems regarding state-space involutions, foliation isometries, and symplectomorphisms.

We're finding that folks who code simulations like to see abstract proofs and theorems reduced to concrete numerical algorithms, with the latter expressed in a Donald Knuth-style literate programming format  ... these concrete recipes are what we're adding to the preprint.

The (in-progress) upgrade reflects three lessons-learned in recent weeks:

• mathematicians want to review a one-page summary of proofs,
• physicists want to know how a simulation framework "really" works, and
• software engineers want concrete examples of working code.

The upgrade will attempt to keep all three groups (reasonably) happy! 

We're inspired by Feynman's 1961 textbook Quantum Electrodynamics, which (as we discuss later) develops quantum electrodynamics from a trajectory-oriented point-of-view, using as little as feasible of the operator-oriented spectral toolset that was the starting-point of contemporary quantum field theory textbooks.

Feynman's non-spectral framework proves to be congenial for quantum simulation, particularly if we combine it with Vladimir Arnold's geometric point-of-view that "Hamiltonian mechanics is geometry in phase space; phase space has the structure of a symplectic manifold."

Our QSE Group's dynamical point-of-view (about which we recently posted on Dick Lipton's blog) changes just one word of Arnold's description: "Quantum mechanics is geometry in phase space; phase space has the structure of a Kähler manifold."

We then specify Feynman-Arnold symplectic trajectories in terms of Lindblad's informatic description of quantum dynamics, particularly in its synoptic unravelling.

A major advantage of the resulting Feynman-Arnold-Lindblad symplectic/synoptic framework is that it teaches us quantum dynamics forwards-and-backwards.

We find that *both* directions respect traditional and emerging ideals of dynamical naturality ... and *both* directions encourage us to create efficient dynamical simulations.

If you care to read the fine print on any slide, just click on it to see a PDF version.

This slide refers to our (short) PNAS article Spin microscopy's heritage, achievements, and prospects, which derived Shannon channel capacities for the three main imaging technologies of the conference: magnetic resonance force microscopy (MRFM) cantilevers, NV (diamond nitrogen-vacancy) centers, and ferromagnetic resonance microscopy (FMRM) devices.

The good news is, everyone's still got plenty of quantum headroom relative to existing device performance, before we begin to approach quantum limits.

Quantum spin imaging and quantum simulation both are easier than we thought; imaging because it turns out that 0.8 nm is good enough for many practical purposes; quantum simulation because the noise and measurement processes that are ubiquitous in spin microscopy actually help us to simulate spin dynamics more efficiently.

Note: the face in the center (being lovingly embraced by Prof. Frederick Frankenstein) is that of Göran Lindblad, whose ideas are seminal to our quantum simulation framework (see the following slide).

This slide is the subject of a (light-hearted) post about "The New French Revolution" on Scott Aaronson's blog Shtetl Optimized. It is also mentioned on Dick Lipton's blog Gödel's Lost Letter in the discussion of Dick's essay Quantum Algorithms: a Different View.

The top two bullets can be technically stated as "unravel Lindblad maps as stochastic processes on symplectic state-spaces; the resulting dynamical compression reduces the state-space dimensionality and allows efficient simulation."

Although noise commonly is viewed as making classical simulations more difficult, the opposite is true in quantum simulations, in consequence of the dimension-reducing drift that is associated to Lindbladian quantum processes.

That noise makes quantum simulation generically feasible is is what we call "Göran Lindblad's Loophole."

The point of this slide is to remind ourselves that we 21st century scientists and engineers are poised to recapitulate—in the quantum domain—the 19th century evolution of geometry and dynamics.

As the slide reminds us, the history of modern geometry begins (in essence) with the non-Euclidean navigational framework of Nathaniel Bowditch (1807); it was Bowditch's practical framework that provided concrete (and startlingly modern) foundations for the subsequent intrinsic mathematical framework of Gauss (1827) and Riemann (1854).

Analogously, our framework seeks to apply to practical problems of quantum simulation, the non-unitary dynamical evolution of Göran Lindblad as pulled-back onto the non-Hilbert quantum state-spaces of Abhay Ashtekar and Troy Schilling.

The following slide was added mainly to create a concrete reason to thank Dick Lipton for an outstanding topic on his blog Gödel's Lost Letter and P=NP, in particular the topic Quantum Algorithms: A Different View

Let's pursue the idea that if there are different views of quantum algorithms (each view with its own merits), then there must be different paths to learn quantum algorithms (each path with its own merits).

If we order—as in the slide below—the various ideals of mathematical naturality that enter into quantum dynamics, then we find that the most mysterious and magical aspects are associated to spectral naturality ... this encompasses non-classical phenomena like "collapse to eigenstates", for example.

Since most students are in a hurry to learn about these spectral mysteries, most textbooks are in a hurry to teach them! And the fastest way to teach them is to lump every other aspect of quantum dynamics into a set of axioms ... `cuz hey ... then we don't have to explain them ... they're axioms!

Two terrific explanations of quantum mechanics from the spectral-first/axiomatic point-of-view are the on-line lectures by Scott Aaronson Quantum computing since Democritus (Lecture 9) and the textbook by Nielsen and Chuang Quantum computation and quantum information ... and there are hundreds more books and articles that follow a similar spectral-first/axiomatic path.

If we want to follow a spectral-last, non-axiomatic path to understanding quantum mechanics, then (regrettably) we have fewer options ... these slides and the accompanying arxiv preprint are an attempt to fill this lacuna.

But why would we even want to follow an alternative path? The answer is given (implicitly) in Dick's blog, namely, there are some aspects of quantum mechanics that seem to be unnatural in all of the ways that we know how to teach ourselves quantum mechanics.

So let's just take a lightning-fast tour of questions about quantum mechanics that we don't know (presently) know how to answer in any way at all, and that in particular, seem to have no natural answers from a spectral/axiomatic point-of-view. We give no answers to these questions ... except links to student-accessible articles or lectures that discuss them.

Some tough complexity questions that are associated to quantum naturality

Let's begin with a real tough question: Why is the quantum separability problem in the complexity class NP-hard? This doesn't seem to have an easy-to-follow, physically motivated, and geometrically natural explanation from any point-of-view.

Ed Witten has pointed out that field theory goes astray (minute 16:28) as soon as we write a quantum field as φ(x), because the coordinate x is not gauge-covariant. Have we similarly made another mistake (disguised as an axiom!) in assuming that φ is an unbounded operator on a linear Hilbert space? Here the point is that an operator-valued φ permits unbounded particle densities, which general relativity teaches us is physically impossible, via the Bekenstein bound.

At first sight, it seems that abandoning Hilbert space is infeasible ... until we notice that the arch-trickster Feynman's book Quantum Electrodynamics never uses the word "Hilbert", or speaks of density matrices at all. For whatever reason, Feynman's book develops the entirety of quantum electrodynamics using as few as feasible of the spectral elements of quantum theory.

Hmmm ... perhaps the entirety of the Standard Model could be developed from a similar same non-spectral perspective? This is mainly a question in quantum pedagogy: is it better for students to learn the elements of spectral naturality first as quantum axioms, or last as an approximation toolset?

Well ... common sense suggests that the best way to learn quantum mechanics most likely depends on the student, of course ... and conversely, that every student should be aware of these multiple learning paths ... and (eventually) learn to travel more than one of them ... and (with luck) even map-out new paths for others to follow. And this path-finding has happened over-and-over again, seemingly without end, not only in quantum dynamics, but in classical dynamics too.

Would this Feynman-style non-spectral idiom help us appreciate Ashtekar and Schilling's much-cited argument that "the linear structure which is at the forefront in text-book treatments of quantum mechanics is, primarily, only a technical convenience and the essential ingredients---the manifold of states, the symplectic structure and the Riemannian metric---do not share this linearity"?

In particular, what would modern articles like Scott Aaronson's Multilinear Formulas and Skepticism of Quantum Computing, and the associated (outstanding) lecture The computational complexity of linear optics (which heck, is pure Feynman-compatible QED) look like if they were rewritten from a Feynman-style non-spectral/non-Hilbert point-of-view?

And return to the beginning, why is quantum linearity so closely tied to the P=NP problem, according to Leonid Gurvits' celebrated proof?

Another way to ask these questions, is to transpose Scott Aaronson's linear optics lecture into the language of Russel Impagliazzo's review article A personal view of average-case complexity.

Are we living in a Impagliazzo-style heuristica universe, in which all experiments can be simulated in PTIME with classical resources, or are we living (perhaps) in a Impagliazzo-style cryptomania universe, in which quantum machines (golly, does this include our own brains?) are capable of behaviors that PTIME pen-and-pencil calculations can never predict?

No matter whether we humans live in the heuristica universe (in which simulation is easy) or in the cryptomania universe (in which simulation is hard), these are wonderfully challenging questions for students to ponder. The above links provide NO final answers ... but instead (and better) they provide lively hints and starting points ... so have fun!

Usually in simulation theory, we think of pullback as a method for reducing the dimensionality of the state-space. But the mathematical capabilities of pullback are much broader: pullback also serves as the mathematically natural means to augment state-space dimensionality.

In retrospect, perhaps the 20th century view of quantum dynamics—that quantum dynamics is embedded in a (linear) Hilbert space having exponentially many dimensions—was synthesized inadvertently, as the maximal extension of dimensional augmentation-by-pullback.

Determining whether the quantum state-space of Nature is really a Hilbert space is the hardest of this talk's challenges for the 21st century.

By the way, the long PDF version of this talk has a cool animation of a tumbling water molecule (provided you run it it Adobe Illustrator, or alternatively, run the animation here).

Nowadays, a pretty fair fraction of all the computer cycles in the world are devoted to tumbling water molecules, so this classical problem retains appreciable economic interest.

For us this problem is mainly a test-bed for proving dynamical theorems about involutions, foliation isometries, and symplectomorphisms; we use these theorems to validate and verify our classical and quantum simulations.

Here are the quantum-specific elements of simulation naturality ...

What did Feynman really say about quantum simulation?

Well, Feynman is a very tricky lecturer ... he never says it is infeasible! Arnold, Mac Lane, Ashtekar, and Schilling all have ideas to contribute too.

Here is a long-term challenge whose experimental aspect was inspired by Scott Aaronson's article "Multilinear formulas and skepticism of quantum computing" and by Scott's (wonderfully interesting, we think) ongoing research with Alex Arkhipov (as recently updated in Scott's presentation at the August 2010 Barriers II conference).

The slide below distinguishes between "proved results" and "verified demonstrations" and ...

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"Proved results" versus "verified demonstrations":
    what's the difference (if any)?

Our working definition of a "proved result" refers to a world that (1) assumes Hilbert-space quantum mechanics, and (2) assumes also the polynomial hierarchy of complexity theory.

Thus the "proved results" world is the idealized mathematical world of complexity theory.

Our working definition of "verifiable demonstration" refers to a world in which (1) the assumption of Hilbert-space dynamics is dropped (as discussed on Dick Lipton's blog), yet (2) the polynomial hierarchy is still respected.

Thus the "verified demonstrations" world is a common-sense engineering world in which verification and validation procedures are explicitly required to be computable with classical resources.

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The Platonic 21st century:
    "proved results" in complexity theory

Broadly speaking, focussing upon "proved results" leads to us to envision a late 21st century world that is Platonically utopian for complexity theorists:

• P≠NP is true, and moreover,
• it has proved that P≠NP is true, and moreover,
• quantum oracle devices have been demonstrated
  (possibly with post-selection), and moreover
• it has been concretely verified
  (uhhh ... will the verification be via protocols in P?  ...
  or perhaps a quantum 'web of trust' will be required?)
  that their capabilities are not in P, and so
• we have solid evidence (at last) that quantum state-space
  does have linear Hilbert geometry and exponential dimension, and
• all this has been demonstrated via elegant mathematics
  (with the help of some elegant experiments).

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The utopian 21st century:
    "verified demonstrations" in physical experiments

Broadly speaking, focussing upon "verified demonstrations" leads us to envision a late 21st century world that is an experimental utopia for physicists:

• quantum state-space has a non-Hilbert geometry
  (see again Lipton's blog), and moreover,
• this geometry is dynamical, and moreover,
• the dynamics is thermodynamical, and therefore,
• all quantum systems (perhaps even the universe itself?)
  can be simulated with computational resources in P, and
• so that Riemann/Einstein and Hilbert/Kähler state-spaces
  are unified dynamically and geometrically, and
• all this has been demonstrated via elegant experiments
  (with the help of some elegant mathematics).

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The pragmatic 21st century:
    practical quantum systems engineering (QSE)

We quantum systems engineers are (of course) pragmatically agnostic regarding these two utopian worlds ... they both seem pretty great to us ... because they both have elegant mathematics and elegant experiments ... whose insights and proof technologies help us engineers to create elegant technologies and new global-scale enterprises.

If the 21st century is very lucky, then perhaps these worlds may both come true: the first world strictly and mathematically; the second world effectively and technologically.

From a pragmatic point-of-view, such a vigorous 21st century would be utopian indeed, for all of humanity's STEM enterprises!

Here is an intermediate-term challenge, namely, to derive the Lindbladian stochastic potentials naturally, and to augment our understanding of their relation to causality.

Here is an near-term challenge, namely, to extend our understanding of independent-spin dynamics to fermionic dynamics. This is important for bringing the power of quantum simulation to a broad span of practical problems in materials science, condensed matter physics, and quantum chemistry.

Atomic-resolution microscopy is the near-term challenge that was a primary focus of the 3rd Nano-MRI Conference, as it has been a primary focus of humanity's STEM enterprise for about the past 345 years.

Hmmm ... maybe this isn't such an easy challenge ... see our article Spin microscopy's heritage, achievements, and prospects for further discussion of the reasons why.

This conference has been wonderful ... how can we make future conferences even better?

Making a revolution is never easy ... but the French have always been good at it.

Our sincere thanks go the organizers of this magnificent conference!