I am not aware of many examples of exploring alternative histories with regards to science, and by that I mean in more detail than what steampunk has to offer, although William Gibson and Bruce Sterling do a pretty good job of imagining a world in which Charles Babbage succeeded in introducing a mechanical computer to the world in their book "The Difference Engine". The subject matter is certainly a worthwhile topic for another post , especially when contrasted with the challenges now to go beyond the Turing machine by getting Quantum Computing to the market. (*h/t vznvzn*)

The untimely death I am contemplating here is that of William Kingdon Clifford. If you are not immersed in physics and math, you have probably never heard his name, because we live in a world where he died young.

That meant it fell to Gibbs and Heaviside to clean up the Maxwell equations, which gave us the insufferable cross-product that confused leagues of students by requiring them to distinguish between polar and axial vectors. It also meant that complex function theory got stuck in two dimensions, and that group theory was developed without the obvious geometric connection. Which in turn, once this approach started to take over, provoked older physicists, such as Schrödinger, to coin the term "Gruppenpest" (group pestilence). It also created a false symmetry between the electric and magnetic fields, motivating the quest for the ever elusive magnetic monopol. Last but not least, it led to the confused notion that spin is an intrinsically quantum mechanical property, something that is still taught in universities across the globe to this day.

It's impossible to do Geometric Algebra (GA) justice in one short blog post, but David Hestenes managed to do so in a fairly concise and highly readable paper, the 2002 Oersted Medal Lecture.

It is hard to overstate the profound effect this paper had on me. The only thing it compares to is when I first learned of Euler's formula many years ago in my first physics semester. And the similarities are striking, not only due to the power of bringing together seemingly disparate areas of mathematics by putting them into a geometric context. In the latter case, the key is the imaginary unit, which was originally introduced to solve for negative square roots, and thus allows for the fundamental theorem of algebra. In fact, it turns out that complex numbers can be neatly embedded into geometric algebra and are isomorphic to the 2d GA case. Also, Quaternion are part of the 3d geometric algebra and have a similarly satisfying geometric interpretation.

All this is accomplished by introducing a higher level concept of vector. For instance, rather than using a cross product, an outer product is defined that creates a bivector that can be thought of as a directed plane segment.

Hestenes makes a convincing case that geometric algebra should be incorporated into every physics curriculum. He wrote some excellent textbooks on the subject, and thankfully, numerous other authors have picked up the mantle (outstanding is John W. Arthur's take on electrodynamics and Chris Doran's ambitious and extensive treatment).

The advantages of geometric algebra are so glaring and the concepts so natural that one has to wonder why it took a century to be rediscovered. John Snygg puts it best in the preface to his textbook on differential geometry:

]]>Although Clifford was recognized worldwide as one of England’s most distinguished mathematicians, he chose to have the first paper published in what must have been a very obscure journal at the time. Quite possibly it was a gesture of support for the efforts of James Joseph Sylvester to establish the first American graduate program in mathematics at Johns Hopkins University. As part of his endeavors, Sylvester founded the American Journal of Mathematics and Clifford’s first paper on what is now known as Clifford algebra appeared in the very first volume of that journal.

The second paper was published after his death in unfinished form as part of his collected papers. Both of these papers were ignored and soon forgotten. As late as 1923, math historian David Eugene Smith discussed Clifford’s achievements without mentioning “geometric algebra” (Smith, David Eugene 1923). In 1928, P.A.M. Dirac reinvented Clifford algebra to formulate his equation for the electron. This equation enabled him to predict the discovery of the positron in 1931. (...)

Had Clifford lived longer, “geometric algebra” would probably have become mainstream mathematics near the beginning of the twentieth century. In the decades following Clifford’s death, a battle broke out between those who wanted to use quaternions to do physics and geometry and those who wanted to use vectors. Quaternions were superior for dealing with rotations, but they are useless in dimensions higher than three or four without grafting on some extra structure.

Eventually vectors won out. Since the structure of both quaternions and vectors are contained in the formalism of Clifford algebra, the debate would have taken a different direction had Clifford lived longer. While alive, Clifford was an articulate spokesman and his writing for popular consumption still gets published from time to time. Had Clifford

participated in the quaternion–vector debate, “geometric algebra” would have received more serious consideration.

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The ‘god particle’, aka the Higgs boson, received a lot of attention, not that this wasn’t warranted, but I can’t help but suspect that the justification of the CERN budget is partly to blame for the media frenzy. The gay particle, on the other hand, is no less spectacular - especially since its theoretical prediction by far pre-dates the Higgs boson. Of course, what has been discovered is, yet again, not a real particle but ‘only’ a pseudo particle similar to the magnetic monopol that has been touted recently. And as usual, most pop-science write-ups fail entirely to remark on this rather fundamental aspect (apparently the journalists don’t want to bother their audience with these boring details). In case you want to get a more complete picture this colloquium paper gives you an in-depth overview.

On the other hand, a pseudo particle quantum excitation in a 2d superconductor is exactly what the doctor ordered for topological quantum computing, a field that has seen tremendous theoretical progress as it has been generously sponsored by Microsoft. This research entirely hinges on employing these anyon pseudoparticles as a hardware resource, because they have the fantastic property of allowing for inherently decoherence-resistant qubits. This is as if theoretical computer science would have started writing the first operating system in the roaring twenties of the last century, long before there was a computer or even a transistor, theorizing that a band gap in doped semiconductors should make it possible to build one. If this analogy was to hold, we’d now be at the stage where a band gap has been demonstrated for the first time. So here's to hoping this means we may see the first anyon-based qubit within the decade.

In the here and now of quantum computing, D-Wave merrily stays the course despite the recent Google bombshell news. It has been reported that they now have 12 machines operational, used in a hosted manner by their strategic partners (such as 1Qbit). They also continue to add staff from other superconducting outfits i.e. recently Bill Blake left Cray to join the company as VP of R&D.

Last but not least, if you are interested in physics you would have to live under a rock not to have heard about the sensational news that numerical calculations presumably proofed that black holes cannot form and hence do not exist. Sabine Hossenfelder nicely deconstructs this. The long and short of it is that this argument has been going on for a long time, that the equations employed in this research has some counter-intuitive properties, and that the mass integral employed is not all that well-motivated.

Einstein would have been happy if this pans out, after all this research claims to succeed where he failed, but the critical reception of this numerical model has just begun. It may very well be torn apart like an unlucky astronaut in a strongly in-homogeneous gravitational field.

This concludes another quick round-up post. I am traveling this week and couldn't make the time for a longer article, but I should find my way back to a more regular posting schedule next week.

]]>Recently a friend of mine observed in an email discussion *"I must admit I find it a little difficult to keep up with the various definitions of quantum computing." *

A healthy sign for an enlightened confusion, because this already sets him apart from most people who still have yet to learn about this field, and at best think that all quantum computers are more or less equivalent.

As computers became an integral part of peoples everyday lives, they essentially learn the truth of Turing completeness - even if they have never heard the term. Now, even a child exposed to various computing devices will quickly develop a sense that whatever one computer can do, another should be able to perform as well, with some allowance for the performance specs of the machine. Older, more limited machines may not be able to run a current software for compatibility or memory scaling reasons, but there is no difference in principle that would prevent any computer from executing whatever has already been proven to work on another machine.

In the quantum computing domain, things are less clear cut. In my earlier post where I tried my hand at a quantum computing taxonomy, I focused on maturity of the technology, less so on the underlying theoretical model. However, it is the dichotomy in the latter that has been driving the heated controversy of D-Wave's quantumness.

When David Deutsch wrote his seminal paper, he followed in Turing's footsteps, thinking through the consequences of putting a Turing machine into quantum superposition. This line of inquiry eventually gave rise to the popular gate model of quantum computing.

D-Wave, on the other hand, gambled on adiabatic quantum computing, and more specifically, an implementation of quantum annealing. In preparation for this post I sought to look up these terms in my copy of Nielsen and Chuang's 'Quantum Computation and Quantum Information' textbook. To my surprise, neither term can be found in the index, and this is the 2010 anniversary edition. Now, this is not meant to knock the book, and if you want to learn about the gate model I think you won't find a better one. It just goes to show that neither the adiabatic nor annealing approach was on the academic radar when the book was originally written - the first paper on adiabatic quantum computation (Farhi et al.) was published the same year as the first edition of this standard QIS textbook.

At the time it was not clear how the computational powers of the adiabatic approach compared to the quantum gate model. Within a year, Vazirani et al. published a paper that showed that Grover Search can be implemented on this architecture with quantum speed-up. And although the notoriety of Shore's algorithm overshadows Grover's, the latter has arguably much more widespread technological potential. The Vazirani et al. paper also demonstrated that there will be problem instances that this QC model will not be able to solve efficiently, even though they can be tackled classically.

In 2004 a paper was submitted with a title that neatly sums it up: "Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation" (Lloyd et al.)

If D-Wave had aimed for universal adiabatic quantum computation, maybe it would not have experienced quite as much academic push-back, but they pragmatically went after some lower hanging fruit i.e, quantum annealing. (Notwithstanding, this doesn't stop MIT's Seth Lloyd from claiming that the company uses his ideas when pitching his own QC venture).

An adiabatic quantum computing algorithm encodes a problem into a cost, or in this case energy function, that is then explored for its absolute minimum. For instance, if you try to solve the traveling salesman problem your cost function would simply be distance traveled for each itinerary. A simple classical gradient descent algorithm over this energy 'landscape' will quickly get stuck in a local minimum (for an analog think of balls rolling down the hilly landscape collecting at some bottom close to were they started and you get the idea). A truly quantum algorithm, on the other hand, can exploit the 'spooky' quantum properties, such as entanglement and the tunnel effect . In essence, it is as if our rolling balls could somehow sense that there is a deeper valley adjacent to their resting place and "tunnel through" the barrier (hence the name). This gives these algorithms some spread-out look-ahead capabilities. But depending on your energy function, this may still not be enough.

The graph bellow illustrates this with a completely made-up cost function, that while entirely oversimplified, hopefully still somewhat captures the nature of the problem. To the extent that the look-ahead capabilities of an adiabatic algorithm are still locally limited, long flat stretches with a relative minimum (a 'plain' in the energy landscape) can still defeat it. I threw in some arbitrary Bell curves as a stand in for this local quantum 'fuzziness' (the latter incidentally the correct translation for what Heisenberg called his famous relation).

To the left, this fuzzy width doesn't stretch outside the bounds of the flat stretch (or rather, it is negligibly small outside any meaningful neighborhood of this local minimum).

On the other hand, further to the right there is some good overlap between the local minimum closest to the absolute one (overlayed with the bell curve in green). This is where the algorithm will perform well.D-Wave essentially performs such an algorithm with the caveat that it does not allow completely arbitrary energy functions, but only those that can be shoe-horned into the Ising model.

This was a smart pragmatic decision on their part because this model was originally created to describe solid state magnets that were imagined as little coupled elementary magnetic dipoles, and the latter map perfectly to the superconducting magnetic fluxes that are implemented on the chip.

In terms of complexity, even in a simple classical 2-d toy model, the amount of possible combinations is pretty staggering as the video below nicely demonstrates. The corresponding energy function (Hamiltonian in QM) is surprisingly versatile an can encode a large variety of problems.

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But examining the D-Wave results led to the Google partnership. D-Wave uses a process called quantum annealing. Annealing translates the problem into a set of peaks and valleys, and uses a property called quantum tunneling to drill though the hills to find the lowest valley. The approach limits the device to solving certain kinds of optimization problems rather than being a generalized computer, but it could also speed up progress toward a commercial machine. Martinis was intrigued by what might be possible if the group combined some of the annealing in the D-Wave machine with his own group's advances in error correction and coherence time.

"There are some indications they're not going to get a quantum speed up, and there are some indications they are. It's still kind of an open question, but it's definitely an interesting question," Martinis said. "Looking at that, we decided it would be really interesting to start another hardware approach, looking at the quantum annealer but basing it on our fabrication technology, where we have qubits with very long memory times."

This leads to the next question: Will this Google chip be indeed similarly restricted to implementing the Ising model like D-Wave, or strive for more universal adiabatic quantum computation? The later has theoretically been shown to be computationally equivalent to gate based QC. It seems odd to just aim for a marginal improvement of the existing architecture as this article implicates.

At any rate, D-Wave may retain the lead in qubit numbers for the foreseeable future if it sticks to no, or less costly, error correction schemes (leaving it to the coders to create their own). It will be interesting to eventually compare which approach will offer more practical benefits.

]]>It is an hour long talk but very informative. John Martinis does an excellent job at explaining, in very simple terms, how hardware-based surface code error correction works.

Throughout the talk he uses the Gate model formalism. Hence it is quite natural to assume that this is what the Google chip will aim for. This is certainly reinforced by the fact that other publications, such as from the IEEE, have also drawn a stark contrast between the Martinis approach, and D-Wave's quantum annealing architecture. This is certainly how I interpreted the news as well.

But on second thought, and careful parsing of the press releases, the case is not as clear cut. For instance, Technology Review quotes Martinis in this fashion:

“We would like to rethink the design and make the qubits in a different way,” says Martinis of his effort to improve on D-Wave’s hardware. “We think there’s an opportunity in the way we build our qubits to improve the machine.”

This sounds more like Martinis wants to build a quantum annealing chip based on his logical, error corrected qubits. From an engineering stand-point this would make sense, as this should be easier to achieve than a fully universal gate-based architecture, and it will address the key complaint that I heard from developers programming the D-Wave chip i.e. that they really would like to see error correction implemented on the chip.

On the other hand, in light of Martinis presentation, I presume that he will regard such an architecture simply as another stepping stone towards universal quantum computation.

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As school starts, I should find my way back to a regular blogging schedule. I usually drive my kids to German Saturday school and then pass the time at a nearby Starbucks updating this blog.

Job and family demanded too much of my time this summer. The former has gotten very interesting, as I am documenting a bank stress testing system, but the learning curve is steep. And while I just had a pleasant one week vacation at a pristine Northern lake, it very much lacked in Wifi connectivity and was not conducive to blogging. Yet, I had plenty of time to read up on material that will make for future posts.

Back home, my kids incidentally watched the Nova episode that features D-Wave and Geordie Rose, which prompted my mother-in-law to exclaim that she wants stock in this company. Her chance to act on this may come in the not too distant future. Recently, D-Wave's CEO hinted for the first time that there may be an IPO in the offing (*h/t Rolf D*).

Readers who follow the QC blogs have undoubtedly already learned about an interesting paper that supports D-Wave’s approach, since Geordie highlighted it on the company’s blog. The fact that Robert R. Tucci is looking for an experienced business partner to start a QC algorithm venture with may also already qualify as old news - Bob is mostly focused on the Gate model, but is agnostic about the adiabatic approach, and certainly displays an impressive grit and track record in consistently turning out patents and papers.

When it comes to love and business, timing is everything. The US allows for software patent protection of up to 20 years. This is a sufficiently long time frame to bet on Gate QC becoming a reality. But there is still a bit of a chicken and egg problem associated with this technology. After all, it is much more difficult (Geordie Rose would argue unrealistically so) then what D-Wave is doing. Shor’s algorithm alone cannot justify the necessary R&D expense to develop and scale up the required hardware, but other commercially more interesting algorithms very well may. Yet you only invest in developing those if there is a chance that you’ll eventually (within 20 years) have hardware to run them on. Currently, it still falls to academia to breach the gap, e.g. such as these Troyer et al. papers that make hope that quantum chemistry could see tangible speed-up from even modestly sized gate based quantum computers.

While quantum computing will remain a main theme of this blog, I intend to also get back to some more biographical posts that reflect on how the history of physics has evolved. Just as any human history, it is full of the oddest turns and twists that are more often than not edited out of the mainstream narrative. And just to be clear, this is not to suggest some grand conspiracy, but just another expression of the over-simplification that afflicts most popular science writing. Writing for the least common denominator makes often for rather poor results, but just as Sabine

observesIn the same vein, the intricacy of how scientific discovery progresses deserves some limelight as it illuminates the roads less traveled. It also makes for interesting thought experiments, imagining how physics may have developed if certain experiments or math had been discovered earlier, or one scientist's life hadn't been cut too short.

My next post will deal in some such idle speculation.

**Update**: This just in, Google sets out on its own (*h/t bettinman*), ~~planning to put 3B to reach the post silicon area. Not clear yet what this will mean for their relationship with D-Wave.~~

The hype comes before the fall. In the context of information technology, Gartner copyrighted the aptly named term “hype cycle”. Every technology starts out in obscurity, but some take off stellarly, promising the moon but, with the notable exception of the Apollo program, falling a bit short of that. Subsequently, disillusionment sets in, sometimes going as far as vilification of the technology/product. Eventually sentiments hit rock bottom, and a more balanced and realistic view is adopted as the technology is mainstreamed.

Even Web technology followed this pattern to some extent, and this was clearly mirrored by the dot com stock bubble. At the height of the exuberance, the web was credited with ushering in a New Economy that would unleash unprecedented productivity growth. By now it has, of course, vastly improved our lives and economies, it just didn’t happen quite as rapidly and profitably as the market originally anticipated.

D‑Wave’s technology will inevitably be subjected to the same roller coaster ride. When you make it to the cover of Time magazine, the spring for the tear down reaction has been set, waiting for the trigger. The latter came in the form of the testing performed by Mathias Troyer et al. While all of this is as expected in the field of journalism, I was a bit surprised to see that one of the sites I link to in my blogroll followed this pattern as well. When D‑Wave was widely shunned by academia, Robert R.Tucci wrote quite positively about them, but now seems to have given up all hope in their architecture in reaction to this one single paper. He makes the typical case against investing into D-Wave that I've seen many times argued by academics vested in the gate model of quantum computing.

The field came to prominence due to the theoretically clearly established potential of gate based quantum computers to outperform classical machines. And it was Shor’s algorithm that captured the public’s imaginations (and the NSA’s attention). Widely considered to be a NP-intermediate problem Shor’s algorithm could clearly crack our encryption schemes if we had gate based QC with thousands of qubits. Unfortunately, this is still sci-fi, and so the best that has been accomplished so far was the factorization of 21 based on this architecture. The quantum speed-up would be there if we had the hardware, but alas at this point it is the embodiment of something that is purely academic with no practical relevance whatsoever.

There is little doubt in my mind that a useful gate based quantum computer will be eventually built, just like, for instance, a space elevator. In both cases it is not a matter of ‘if’ but just a matter of ‘when’.

I’d wager we won’t see either within the next ten years.

Incidentally, it has been reported that a space elevator was considered as a Google Lab’s project, but subsequently thrown out as it requires too many fundamental technological breakthroughs in order to make it a reality. On the other hand, Google snapped up a D-Wave machine.

So is this just a case of acquiring trophy hardware, as some critics on Scott’s blog contended? I.e. nothing more than a marketing gimmick? Or have they been snookered? Maybe they, too, have a gambling addiction problem, like D-Wave investors as imagined on the qbnets blog?

Of course none of this is the case. It just makes business sense. And this is readily apparent as soon as you let go of the obsession over quantum speed-up.

Let’s just imagine for a second that there was a company with a computing technology that was neither transistor nor semiconductor based. Let’s further assume that within ten years they managed to rapidly mature this technology so that it caught up to current CPUs in terms of raw performance, and that this was all done with chip structures that are magnitudes larger than what current conventional hardware needs to deploy. Also this new technology does not suffer from loss currents introduced via accidental quantum tunneling, but is actually designed around this effect and utilizes it. Imagine that they did all this with a fraction of the R&D sums spend on conventional chip architectures, and since the power consumption scaling is radically different from current computers, putting another chip into the box will hardly double the energy consumed by the system.

A technology like this would almost be like the kind that IBM just announced to focus their research on, trying to find a way to the post-silicon future.

So our 'hypothetical company' sounds pretty impressive, doesn't it? You’d think that a company like Google that has enormous computational needs would be very interested in test driving an early prototype of such a technology. And since all of the above applies to D‑Wave this is indeed exactly what Google is doing.

Quantum speed-up is simply an added bonus. To thrive, D‑Wave only needs to provide a practical performance advantage per KWh. The $10M up-front cost, on the other hand, is a non-issue. The machines are currently assembled like cars before the advent of the Ford Model T. Most of the effort goes into the cooling apparatus and interface with the chip, and there clearly will be plenty of opportunity to bring down manufacturing cost once production is revved up.

The chip itself can be mass-produced using adapted and refined Lithographic processes borrowed from the semi-conductor industry; hence the cost basis for a wafer of D‑Wave chips will not be that different from the chip in your Laptop.

Just recently, D‑Wave’s CEO mentioned an IPO for the first time in a public talk (*h/t Rolf D*). Chances are, the early D-Wave investors will be laughing at the naysayers all the way to the bank long before a gate based quantum computer factors 42.

Fortunately, there is another excellent blog out there which is solely focused on fusion technology and the various concepts in the field. The Polywell is covered in depth, but General Fusion also gets is due, for its innovative technology.

Another focus of mine, the trouble with contemporary theoretical physics also keeps falling through the cracks. From my past posts one may get the impression that I am just yet another String apostate, but I don't really have any trouble with String Theory as such, but rather with uncritical confirmation bias. Unfortunately, the latter cuts across all fields as in this recent post of hers.

]]> nicely demonstratesThe advantage of conventional open source code is that everybody can play with it. To the extent that the code faithfully emulates quantum annealing of an Ising spin model, this will be a net benefit to D-Wave as it allows programmers to perform early testing of algorithmic concepts for their machine (similar to the emulator that is part of D-Wave's evolving coding environment).

Matthias Troyer attached the source code to this earlier paper, and for easier download I put it into this zip file. As I am hard pressed for time these days, I didn't get to work with it yet, but I confirmed that it compiles on my OS X machine (*"make an_ss_ge_fi" *produced a proper executable)*.*

Meanwhile D-Wave continues to receive good press (*h/t* web 238) as the evidence for substantial quantum effects (entanglement and tunneling) keep mounting, indicating that they can successfully keep decoherence at bay. (This older post on their corporate blog gives an excellent introduction to decoherence, illustrating how thermal noise gets in the way of quantum annealing).

On the other hand, despite enormous theoretical efforts, it is yet not decisively shown what quantum resources are required for the elusive quantum speed-up, but this recent paper in Nature (arxiv link) claims to have made some substantial headway in isolating some of the necessary ingredients (*h/t Theo*).

While this is all quite fascinating in the academic sense, at this time I nevertheless regard quantum speed-up as overrated from a practical point of view. Once I manage to set some time aside I will try to explain in my next blog post why I don't think that this has any bearing on D-Wave's value proposition. There's no place in the multiverse for the snarky, dark movie script that Robert Tucci offered up the other day

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