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Table of contents
- The Traditional Ballad Index -- Supplemental Tradition
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- About This Blog
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- Confessions of the Lovelorn ()
Scott offered me free ad space if I wrote a guest post, so here we are. The position is in all areas of CS theory, including QC theory although the search is not limited to that. In this post, I wear the hat of a pure mathematician in a box provided by Archimedes. In my happy world, I like quantum supremacy as a demonstration of a beautiful coincidence in mathematics that has been known for more than years in a special case.
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The single-qubit case was discovered by Archimedes, duh. Much as I would like to clobber you with highly abstract mathematics, I will wait for some other time. I will call it the Born map, since it expresses the Born rule of quantum mechanics that amplitudes yield probabilities.
If you pretend to be a bad probability student, then you might not be surprised by this coincidence, because you might suppose that all probability distributions are uniform other than treacherous exceptions to your intuition. The theorem of Archimedes is that a natural projection from the unit sphere to a circumscribing vertical cylinder preserves area.
The projection is the second one that you might think of: Project radially from a vertical axis rather than radially in all three directions. This transformation from a quantum state to a classical randomized state is a linear projection to a vertical axis. This is the map that squishes Greenland along with the top of North America and Eurasia to give them proportionate area. There was no Internet back then to easily find out who did what first. The polar angle is the angle from vertical in spherical coordinates, as depicted in red in the Bloch sphere diagram.
This is a coincidence which is special to the 2-sphere in 3 dimensions. If you want to step through this in even more detail, Scott mentioned to me an action video which is vastly spiffier than anything that I could ever make. The projection of the Bloch sphere onto an interval also shows what goes wrong if you try this with a rebit. If you linearly project a circle onto an interval, then the length of the circle is clearly bunched up at the ends of the interval and the projected measure on the interval is not uniform.
That the higher-dimensional Born map also preserves measure is downright eerie. Scott challenged me to write an intuitive explanation. I remembered two different but similar proofs, neither of which is original to me.
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Scott and I disagree as to which proof is nicer. Because you can approximate any such region with a union of products of thin annuli. The third and final step is the paint principle for comparing surface areas. If you paint the hoods of two cars with the same thin layer of paint and you used the same volume of paint for each one, then you can conclude that the two car hoods have nearly same area.
The key is that the exponent 2 appears in two different ways: as the dimension of the complex numbers, and as the exponent used to set probabilities and define spheres. The higher-dimensional real Born map similarly distorts volumes, whether or not you restrict to unit-length states. Although Wikipedia says that no one knows why Legendre defined it this way, I wonder if his goal was to do what the Catholic church later did for itself in It put a Pole at the origin. Scott wanted to censor this joke. Well, the Czechs are cool too.
There are inevitably various proofs of this result, and I will give another one. This time we rescale the vector until its sum is 1. The gist of the proof is that the Born map takes the Gaussian algorithm to the exponential algorithm. We will need it, and it is another way to see that the theorem relies on the fact that the complex numbers are two-dimensional.
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Since the Born map takes the Gaussian algorithm to the exponential algorithm, and since each algorithm produces a uniformly random point, the Born map must preserve uniform measure. Scott likes this proof better because it is algorithmic, and because it is probabilistic.
Now about quantum supremacy. In a sense, the statistical distribution of the bit strings is almost the same almost every time, independent of which random quantum circuit you choose to generate them.
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The catch is that the position of any given bit string does depend on the circuit and is highly scrambled. I picture it in my mind like this:. It is thought to be computationally intractable to calculate where each bit string lands on this exponential curve, or even where just one of them does. That is one reason that random quantum circuits are supreme.
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I wish them the best of luck in fixing the problem. Around , one of the first things I ever did in quantum computing theory was to work on a problem that Fortnow and Rogers suggested in a paper: is it possible to separate P from BQP relative to a random oracle? Or to the contrary: suppose that a quantum algorithm Q makes T queries to a Boolean input string X. It would underscore that superpolynomial quantum speedups depend on structure. I never managed to solve this problem. Around , though, I noticed that a solution would follow from a perhaps-not-obviously-related conjecture, about influences in low-degree polynomials.
Why would this conjecture imply the statement about quantum algorithms?
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Basically, because of the seminal result of Beals et al. Whenever that happens, halt and output the constant c. The key is that by hypothesis, this algorithm will halt, with high probability over X, after only poly T steps. Andris and I decided to write up the two directions jointly. My plan had been as follows: 1 Read their paper in detail. Understand every step of their proof. Unfortunately, this plan did not sufficiently grapple with the fact that I now have two kids.
Confessions of the Lovelorn ()
It got snagged for a week at step 1. Unfortunately, the polynomial blowup is quite enormous: from a quantum algorithm making T queries, Keller and Klein apparently get a classical algorithm making O T 18 queries. But such things can almost always be massively improved.
Feel free to use the comments to ask any questions about this result or its broader context. Update Nov. Modulo some facts about noise operators, hypercontractivity, etc. First, you hit your bounded degree-d function f with a random restriction to attenuate its higher-degree Fourier coefficients reminiscent of Linial-Mansour-Nisan. Every time you find another influential coalition, that norm goes down by a little, but by approximation theory, it can only go down O d 2 times until it hits rock bottom and your function is nearly constant.
Finally, you get rid of the log n term by using the fact that f essentially depended on at most exp O d variables anyway.
But happy to discuss more in the comments, if anyone else is reading it. Meanwhile, Drew Potter , an expert on topological qubits, rejoined our physics faculty after a brief leave. My own group now has five postdocs and six PhD students—as well as some amazing undergrads striving to meet the bar set by Ewin Tang.
If you want to work with me or John Wright on quantum algorithms and complexity, apply to CS I can also supervise physics students in rare cases. If you want to work with Drew Potter on nonabelian anyons or suchlike, or with Allan MacDonald , Linda Reichl , Elaine Li , or others on many-body quantum theory, apply to physics. Admissions are extremely competitive, so I strongly encourage you to apply broadly to maximize your options.
To apply, just send me an email by January 1, with the following info: — Your CV — 2 or 3 of your best papers links or PDF attachments — The names of two recommenders who should email me their letters separately. Our CS, physics, and ECE departments are all open to considering additional candidates in quantum information, both junior and senior. This morning a humanities teacher named Richard Horan, having read my NYT op-ed on quantum supremacy , emailed me the following question about it:.
Is this pursuit [of scalable quantum computation] just an arms race? A race to see who can achieve it first? To what end? Will this achievement yield advances in medical science and human quality of life, or will it threaten us even more than we are threatened presently by our technologies? You seem rather sanguine about its possible development and uses.