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

Morning sunlight illuminated John Preskill’s lecture notes. The notes concern 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.

Yang-Baxter

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!”

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

  1. Another use of braiding diagrams is in change ringing of church bells. However here there will typically be 6 or more peas (bells) and more than one adjacent bells will be swapping places at the same time. Some resources are here http://www.ringing.info/

    • Thanks for drawing our attention to change ringing. The more parallels amongst related ideas, the merrier! My experience with change ringing consists mostly of reading Neal Stephenson’s Anathem (and hearing about the role played by group theory piqued my interest). I imagine that readers of this blog will appreciate the resources you recommended.

  2. Hi Nicole Yunger Halpern, “Quantum Braiding” was/is creative, brilliant, clear, insightful and engaging – and that’s coming from a computer scientist/physicist innovation specialist who reads an encyclopedia’s worth of stuff a week, talks/interacts/builds with a tech/creative spectrum of folks around the world and have been moved to write a comment like this only twice in the last decade.

    The way you see and write is breathtakingly refreshing, deeply insightful and, dare I say, fun! Congratulations – I’m now going to read all your previous posts – no matter what they’re about!

    Thanks for the ‘headshot’! My mind is refreshed and invigorated just from experiencing how you saw and wrote that, a too rare example of a mind dancing at play in a wholeness I’m sure all recall on reading – and, yes, I learned a lot too!

    K. David

    • Many thanks for your comment, Ken. I’m thrilled if you enjoyed the post (and I could do with waking up to such day-brightening feedback more often!). One aspect of physics that I appreciate most is the opportunity—the need—to draw surprising parallels. Pendulums look nothing like springs, prima facie. But if both can obey the same differential equation, perhaps there’s hope for us crazies who see anyons on the nape of a neck.

  3. It was a typical night in Ramadan. I had been waiting for my Kabab to be prepared and looking for a useful way to kill time until I stumbled upon this eloquent article.
    Thank you for writing this article

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