The QSE Roadmap and Journal is undergoing a major upgrade this week (Dec. 18-25) to a new graphical interface. Some of the Roadmap entries may be temporarily truncated: they will return after they have been converted. Happy Holidays to all!

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To apply for the UW ME Department's tenure-track faculty position(s) in quantum system engineering, please see this advertisement.

The ME Department encourages any and all qualified applicants, from both theoretical and experimental backgrounds, who seek to create and teach new technologies that push against the bounds that quantum mechanics imposes on the speed, accuracy, sensitivity, size, and power consumption of modern mechatronic devices.

This position provides a wonderful opportunity to participate in creating and teaching the new, exciting, strategically important, and rapidly growing engineering discipline of quantum system engineering (QSE).

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Entry #4: Monday, November 27, 2006

Welcome to the UW QSE Group's Roadmap and Journal. Each entry first comments upon our (working) QSE Roadmap, and then describes our recent activities.

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QSE Roadmap Comments

Dirac's “Ordinary People Can Make Extraordinary Contributions”

Today's roadmap commentary focuses on a comment attributed to Paul Dirac: "Golden eras occur when ordinary people can make extraordinary contributions."

High-resolution PDF versions: (top level), (lower level), (entire roadmap)

Note: physicist Valentine Telegdi frequently quoted this "Golden Era" maxim to graduate students, attributing it to the physicist Paul Dirac. A literature search suggests that it plausibly derives from the following article by Dirac:

@inProceedings{Dirac:75,
   author =    {P. A. M. Dirac},
   title =     {The Development of Quantum Mechanics},
   booktitle = {Directions in Physics},
   editor =    {H. Hora and J. R. Shepanski},
   chapter =   1,
   pages =     {6},
   publisher = {Wiley-Interscience, New York},
   year =      1978,
 mynote = {Lectures delivered during a 1975 visit to Australia
           and New Zealand. "[In the early days of quantum
           mechanics] it was a good description to say that it
           was a game, a very interesting game one could play.
           Whenever one solved one of the little problems, one
           could write a paper about it. It was very easy in
           those days for any second-rate physicist to do
           first-rate work. There has not been such a glorious
           time since. It is very difficult now for a first-rate
           physicist to do second-rate work."},
}

A central theme of the QSE Roadmap is that Dirac's principle especially applies to our 21st Century world, because:

• On the one hand, our planet stands in urgent need of new resources, as the present population of 6 billion surges toward 10 billion.
• On the other hand—fortunately—vast new resources are newly becoming available to humanity, due to the confluence of advances in biology, nanotechnology, quantum science, and information theory.

The QSE Roadmap seeks to apply these new scientific and technological advances in creating abundant new resources, in service of Dirac's Golden Era principle.

The Scale and Pace of the QSE Roadmap

Job creation is arguably the single most urgent global strategic priority, to forestall the world's new billions of people from joining the two billion global citizens who already live in abject poverty.

But creating (say) a billion new jobs is a dauntingly great challenge. Suppose we boldly envision the creation of a gigantic new corporation, the largest in the world, employing as many people as IBM, Siemans, Merck, and Hitachi put together. In order to create a billion jobs, eight hundred new such companies would have to be created.

These considerations encourage quantum system engineers to "think big," on scales much larger than even the boldest venture capitalists, politicians, and generals.

Biospace is seemingly the largest new frontier on the horizon for humanity. To the extent that we are committed to creating the jobs necessary for global peace and prosperity, we are led to contemplate exploration of the biospace frontier on the largest feasible scales and by the fastest feasible paths: this QSE Roadmap seeks to achieve this.

The Federative Challenge of the QSE Roadmap

Biospace today is like outer space in the 1950s and 60s: we realize that a new frontier has been discovered, we have begun to explore it, and we increasingly appreciate that this frontier holds vast new resources for humanity.

Now as then, our main challenges are federative: the challenges of marshalling and federating the science, the technology, the political will, and the entrepenurial vigor, that are needed to explore this new frontier.

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Journal entry for Monday, November 27, 2006

The Challenge of Quantum Model Order Reduction (QMOR)

In exploring new technology frontiers, engineers appreciate the central importance of detailed design analyses.

A good example is the central role that trajectory calculations played in the development of the space program. These calculations were carried out on vacuum-tube computers by a (then considerably smaller) IBM Company (links here).

Although trajectory calculations—or more broadly, system-level emulation calculationss—are not particularly glamorous, they are both essential and central to any modern technology development effort. This is why quantum system emulation techniques play a central organizing role in the QSE Roadmap.

The Dynamical Arena of Hilbert Space

Just as the design of Apollo's Saturn V booster was largely conditioned by IBM's trajectory calculations in outer space, the design of MRFM devices is largely conditioned by trajectory calculations in a mathematical space called "Hilbert Space", and we now turn our attention to this topic.

Hilbert space is the dynamical arena within which all quantum phenomena take place. For purposes of quantum spin microscopy by MRFM, the salient mathematical feature of quantum Hilbert space is that it has very many dimensions. In fact, the quantum Hilbert spaces associated with MRFM have so many dimensions that until very recently it was widely believed that it is generically infeasible to compute quantum trajectories in these spaces.

Engineering Approaches to Quantum Model Order Reduction (QMOR).

The challenge of computing dynamical trajectories in spaces of large dimensionality is familiar to engineers, and the techniques for dealing with this problem are generically called Model Order Reduction (MOR) methods (see, e.g., Rewienski's MIT thesis for a review of classical MOR methods).

A major technical goal of our UW WSE Group is to create tools for quantum model order reduction (QMOR) that are similarly effective to those presently available for classical model order reduction (C-MOR).

Compressed Representations of Quantum Trajectories

For engineering purposes, a quantum trajectory can be regarded as a column of complex numbers ψ(t). The central challenge of QMOR is that ψ(t) has so many dimensions that its components cannot be stored within the memory of even the largest conceivable computer—a computer as big as the entire universe.

Quantum MOR finesses this too-large issue by writing ψ(t) always in the following compressed representation:

This QMOR representation encodes ψ(t) as a small set of c-numbers, which suffice for us to compute any desired component of ψ(t). To anticipate, we will find that in quantum spin microscopy by MRFM, each additional spin requires the storage of the order of 100 additional c-numbers. This means that QMOR MRFM simulations of up to several thousand spins can be readily stored in the memory of existing (large) computer facilities. This algorithmic compression of trajectories is the first step in making QMOR practical.

Parallels between Quantum Trajectory Compression and Image Compression

Broadly speaking, it is helpful to think of the all-components storage of ψ(t) as akin to a graphics file stored in TIFF format: a format that can describe any image with perfect fidelity, but at the cost of an inconveniently large file size.

In contrast, the compressed QMOR representation of ψ(t) is like a JPEG graphics file: the fidelity is less-than-perfect, but accurate enough for practical purposes, and the file size is conveniently small.

The reason that compressed JPEG graphics files cannot store arbitrary images is that their format is optimized for storing the kinds of images that humans care about: images that have lots of algorithmically compressible spatial structure.

Similarly, the above QMOR representation of ψ(t) cannot store arbitrary Hilbert states, but (as we will see) it is very good at storing the kinds of states that arise in practical quantum system engineering, namely, states whose high-order quantum correlations have been washed out by noise.

The Algebraic View of Quantum Model Order Reduction (QMOR)

From an algebraic point of view, the above QMOR representation can be regarded as a polynomial in the c-numbers. Such algebraic polynomials are very convenient for numerical purposes, because define a quantum state-space is naturally equipped with geodesics (straight lines).

A web of QMOR geodesics can be readily constructed as follows: starting from any QMOR state, if we hold all the c-number coordinates of that state fixed except one, and then vary that one coordinate, we obtain a path (called an integral curve) that is a geodesic in the embedding Hilbert space.

Therefore, from an algebraic point of view, and especially for purposes of numerical computation, the QMOR state-space can be envisioned a a web of geodesics, such that these geodesics are straight lines (rays) in the embedding Hilbert space.

Numerical trajectories can then be efficiently and rapidly computed by traversing this web of geodesic paths; we therefore call this network a shinkansen, after Japan's famous Shinkansen network of high-speed trains.

The Geometric View of Quantum Model Order Reduction (QMOR)

From a geometric point of view, the interesting question is, what is the shape of the QMOR state space between its shinkansen poths? The figure below illustrates that this is a nontrivial question even in spaces of quite low dimensionality: two-dimensional surfaces embedded in three-dimensional space.

It is easily seen that a surface composed of a mesh of shinkansen paths can nonetheless have quite strong curvature, just as Japan's Shinkansen network is composed of level tracks, even though the topography of Japan itself is quite mountainous.

This same phenomenon—a strongly curved surface composed of a mesh of intersecting rays—is even more pronounced in high-dimension QMOR state spaces. Each c-number defines a ray, and therefore each point in the QMOR state space has a large number of rays—typically hundreds or even thousands of them—arriving and departing from it.

QMOR State Space Viewed as a Kähler Manifold

QMOR state space belongs to a class of mathematical objects called Kähler manifolds, which have been extensively studied by a vigorous community of mathematicians that includes Shiing-Shen Chern and Shing-Tung Yau (see Yau's review here).

For quantum system engineering purposes, it is most convenient to view QMOR state space as comprised of a strongly curved Kähler manifold that is naturally endowed with an embedded shinkansen ray structure.

QMOR State Space Viewed as a Gabion: Algebra and Geometry United

Needing a short name for the conjoined geometric and algebraic structures that comprise QMOR state space, and not finding such a name in the mathematical literature, our QSE Group calls this intimately united algebraic and geometric structure a gabion.

Structural engineers will recognize the word "gabion" as a reference to familiar structures (link here) whose design is directly analogous to QMOR state space:

• a gabion's enclosing wickerwork cage is analogous to the shinkansen network that "holds" QMOR state space,
• a gabion's wickerwork cage encloses intricately curved shapes (stream-smoothed rocks, for example) that are analogous to QMOR state space's Kähler geometry.

Just as an engineering gabion is jointly characterized by its wickerwork and its contents, a QMOR gabion is jointly characterized by the algebra of its shinkansen and the geometry of its Kähler manifold.

The Origin of QMOR's Efficiency and Fidelity

This combined algebraic-geometric point of view provides a useful basis for understanding both the algorithmic efficiency of QMOR (which derives from its shinkansen structure) and the fidelity of QMOR (which derives from its strongly-curved geometry).

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