Quantum braiding: It’s all in (and on) your head.

Morning sunlight illuminated John Preskill’s lecture notes about Caltech’s quantum-computation course, Ph 219. I’m TAing (the teaching assistant for) Ph 219. I previewed lecture material one sun-kissed Sunday.

Pasadena sunlight spilled through my window. So did the howling of a dog that’s deepened my appreciation for Billy Collins’s poem “Another reason why I don’t keep a gun in the house.” My desk space warmed up, and I unbuttoned my jacket. I underlined a phrase, braided my hair so my neck could cool, and flipped a page.

I flipped back. The phrase concerned a mathematical statement called “the Yang-Baxter relation.” A sunbeam had winked on in my mind: The Yang-Baxter relation described my hair.

The Yang-Baxter relation belongs to a branch of math called “topology.” Topology resembles geometry in its focus on shapes. Topologists study spheres, doughnuts, knots, and braids.

Topology describes some quantum physics. Scientists are harnessing this physics to build quantum computers. Alexei Kitaev largely dreamed up the harness. Alexei, a Caltech professor, is teaching Ph 219 this spring.1 His computational scheme works like this.

We can encode information in radio signals, in letters printed on a page, in the pursing of one’s lips as one passes a howling dog’s owner, and in quantum particles. Imagine three particles on a tabletop.

Peas 1

Consider pushing the particles around like peas on a dinner plate. You could push peas 1 and 2 until they swapped places. The swap represents a computation, in Alexei’s scheme.2

The diagram below shows how the peas move. Imagine slicing the figure into horizontal strips. Each strip would show one instant in time. Letting time run amounts to following the diagram from bottom to top.

Peas 2

Arrows copied from John Preskill’s lecture notes. Peas added by the author.

Imagine swapping peas 1 and 3.

Peas 3

Humor me with one more swap, an interchange of 2 and 3.

Peas 4

Congratulations! You’ve modeled a significant quantum computation. You’ve also braided particles.

2 braids

The author models a quantum computation.

Let’s recap: You began with peas 1, 2, and 3. You swapped 1 with 2, then 1 with 3, and then 2 with 3. The peas end up ordered oppositely the way they began—end up ordered as 3, 2, 1.

You could, instead, morph 1-2-3 into 3-2-1 via a different sequence of swaps. That sequence, or braid, appears below.

Peas 5

Congratulations! You’ve begun proving the Yang-Baxter relation. You’ve shown that  each braid turns 1-2-3 into 3-2-1.

The relation states also that 1-2-3 is topologically equivalent to 3-2-1: Imagine standing atop pea 2 during the 1-2-3 braiding. You’d see peas 1 and 3 circle around you counterclockwise. You’d see the same circling if you stood atop pea 2 during the 3-2-1 braiding.

That Sunday morning, I looked at John’s swap diagrams. I looked at the hair draped over my left shoulder. I looked at John’s swap diagrams.

“Yang-Baxter relation” might sound, to nonspecialists, like a mouthful of tweed. It might sound like a sneeze in a musty library. But an eight-year-old could grasp the half the relation. When I braid my hair, I pass my left hand over the back of my neck. Then, I pass my right hand over. But I could have passed the right hand first, then the left. The braid would have ended the same way. The braidings would look identical to a beetle hiding atop what had begun as the middle hunk of hair.


The Yang-Baxter relation.

I tried to keep reading John’s lecture notes, but the analogy mushroomed. Imagine spinning one pea atop the table.

Pea 6

A 360° rotation returns the pea to its initial orientation. You can’t distinguish the pea’s final state from its first. But a quantum particle’s state can change during a 360° rotation. Physicists illustrate such rotations with corkscrews.


Pachos corkscrew 2

A quantum corkscrew (“twisted worldribbon,” in technical jargon)

Like the corkscrews formed as I twirled my hair around a finger. I hadn’t realized that I was fidgeting till I found John’s analysis.

Version 2

I gave up on his lecture notes as the analogy sprouted legs.

I’ve never mastered the fishtail braid. What computation might it represent? What about the French braid? You begin French-braiding by selecting a clump of hair. You add strands to the clump while braiding. The addition brings to mind particles created (and annihilated) during a topological quantum computation.

Ancient Greek statues wear elaborate hairstyles, replete with braids and twists.  Could you decode a Greek hairdo? Might it represent the first 18 digits in pi? How long an algorithm could you run on Rapunzel’s hair?

Call me one bobby pin short of a bun. But shouldn’t a scientist find inspiration in every fiber of nature? The sunlight spilling through a window illuminates no less than the hair spilling over a shoulder. What grows on a quantum physicist’s head informs what grows in it.


1Alexei and John trade off on teaching Ph 219. Alexei recommends the notes that John wrote while teaching in previous years.

2When your mother ordered you to quit playing with your food, you could have objected, “I’m modeling computations!”

Free Feynman!

Last Friday the 13th was a lucky day for those who love physics — The online html version of Volume 1 of the Feynman Lectures on Physics (FLP) was released! Now anyone with Internet access and a web browser can enjoy these unique lectures for free. They look beautiful.

Mike Gottlieb at Caltech on 20 September 2013. He's the one on the right.

Mike Gottlieb at Caltech on 20 September 2013. He’s the one on the right.

On the day of release, over 86,000 visitors viewed the website, and the Amazon sales rank of the paperback version of FLP leapt over the weekend from 67,000 to 12,000. My tweet about the release was retweeted over 150 times (my most retweets ever).

Free html versions of Volumes 2 and 3 are in preparation. Soon pdf versions of all three volumes will be offered for sale, each available in both desktop and tablet versions at a price comparable to the cost of the paperback editions. All these happy developments resulted from a lot of effort by many people. You can learn about some of the history and the people involved from Kip Thorne’s 2010 preface to the print edition.

A hero of the story is Mike Gottlieb, who spends most of his time in Costa Rica, but passed through Caltech yesterday for a brief visit. Mike entered the University of Maryland to study mathematics at age 15 and at age 16 began a career as a self-employed computer software consultant. In 1999, when Mike was 39,  a chance meeting with Feynman’s friend and co-author Ralph Leighton changed Mike’s life.

At Ralph’s suggestion, Mike read Feynman’s Lectures on Computation. Impressed by Feynman’s insights and engaging presentation style, Mike became eager to learn more about physics; again following Ralph’s suggestion, he decided to master the Feynman Lectures on Physics. Holed up at a rented farm in Costa Rica without a computer, he pored over the lectures for six months, painstakingly compiling a handwritten list of about 200 errata.

Kip’s preface picks up the story at that stage. I won’t repeat all that, except to note two pivotal developments. Rudi Pfeiffer was a postdoc at the University of Vienna in 2006 when, frustrated by the publisher’s resistance to correcting errata that he and others had found, he (later joined by Gottlieb) began converting FLP to LaTeX, the modern computer system for typesetting mathematics. Eventually, all the figures were redrawn in electronic form as scalable vector graphics, paving the way for a “New Millenium Edition” of FLP (published in 2011), as well as other electronically enhanced editions planned for the future. Except that, before all that could happen, Caltech’s Intellectual Property Counsel Adam Cochran had to untangle a thicket of conflicting publishing rights, which I have never been able to understand in detail and therefore will not attempt to explain.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

The proposal to offer an html version for free has been enthusiastically pursued by Caltech and has received essential financial support from Carver Mead. The task of converting Volume 1 from LaTeX to html was carried out for a fee by Caltech alum Michael Hartl; Gottlieb is doing the conversion himself for the other volumes, which are already far along.

Aside from the pending html editions of Volumes 2 and 3, and the pdf editions of all three volumes, there is another very exciting longer-term project in the works — the html will provide the basis for a Multimedia Edition of FLP. Audio for every one of Feynman’s lectures was recorded, and has been digitally enhanced by Ralph Leighton. In addition, the blackboards were photographed for almost all of the lectures. The audio and photos will be embedded in the Multimedia Edition, possibly accompanied by some additional animations and “Ken Burns style” movies. The audio in particular is great fun, bringing to life Feynman the consummate performer. For the impatient, a multimedia version of six of the lectures is already available as an iBook. To see a quick preview, watch Adam’s TEDxCaltech talk.

Mike Gottlieb has now devoted 13 years of his life to enhancing FLP and bringing the lectures to a broader audience, receiving little monetary compensation. I asked him yesterday about his motivation, and his answer surprised me somewhat. Mike wants to be able to look back at his life feeling that he has made a bigger contribution to the world than merely writing code and making money. He would love to have a role in solving the great open problems in physics, in particular the problem of reconciling general relativity with quantum mechanics, but feels it is beyond his ability to solve those problems himself. Instead, Mike feels he can best facilitate progress in physics by inspiring other very talented young people to become physicists and work on the most important problems. In Mike’s view, there is no better way of inspiring students to pursue physics than broadening access to the Feynman Lectures on Physics!

Stephen Hawking wins $3M Milner Prize

The official announcement won’t come until tomorrow, but The New York Times is reporting that Stephen Hawking will receive a “special” $3M Prize from Yuri Milner’s Fundamental Physics Prize Foundation.

This is fantastic news! I assume the Prize recognizes Stephen’s great discovery that black holes radiate, one of the most transformative developments in theoretical physics during my lifetime. That’s just one of Stephen’s many important contributions. And of course his supreme skill as a popularizer and the unparalleled courage he displays in response to his disability have made him the most famous living scientist in the world. Congratulations, Stephen!

Stephen has a long-standing relationship with Caltech. He spent a sabbatical year here during 1974-75, when he wrote his famous paper formulating the black hole information paradox, and he has made more or less annual extended visits to Caltech since the 1990s. Stephen and I had many memorable discussions about black holes over the years, culminating when he conceded a bet, for which I received far more attention than I deserved. I’ve been proud to be Stephen’s friend for the past 30 years, and we’ve shared a lot of laughter.

With Kip Thorne and Stephen Hawking, 2005.

With Kip Thorne and Stephen Hawking, 2005.

Continue reading

Accelerometer: Part I

This blog has made the terrible decision to ask me to do more regular posts.  Well, before trial and error catches up with me, let’s have some fun together…

As young single Caltech graduate students, we have become accustomed to making hearts race with our science.  We turn measurements and derivations into heart palpitations.  While this has been manifestly obvious for quite some time, we in the Oskar Painter group have recently been interested in quantitatively measuring this effect.  Because, as any good Caltech physics graduate student believes, anything (even sex appeal) is uninteresting unless fully quantified in a dataset.  We set out to make an accelerometer with enough resolution to sense the irregular and skipped heartbeats of our fawning admirers.

What follows is a multi-part treatise on the optomechanical accelerometers we developed.

Continue reading