O brave new quantum world!
From a “woman seeking man” ad:
OBSESSIVE COLLECTOR OF ANTIQUE MATH-TEXTBOOKS Pure math is a lot like art. Relatively soon I will be leaving Seattle to travel around the world. Make my aquaintance now, before I die of dysentary or malaria in the dark jungles of Swaziland. minitrue, 22 #115063
And no, I didn’t find this ad myself but was led to it by a friend who was using the personals as part of an art project.
Another interesting letter in “Perfectly Reasonable Deviations From The Beaten Track: The Letters Of Richard P. Feynman” by T. Ferris (forward), R.P. Feynman (of course!), and M. Feynman (editor) is the following:
Mr. Todd Pramberg
Stockholm, Sweden
Dear Sir:
The fact that I beat a drum has nothing to do with the fact that I do theoretical physics. Theoretical physics is a human endeavor, one of the higher developments of human beings-and this perpetual desire to prove that people who do it are human by showing that they do other things that a few other humans do (like playing bongo drums) is insulting to me.
I am human enough to tell you to go to hell.
Sincerely,
Richard P. Feynman
Why do I find this letter interesting? Well when I was senior at Caltech a movie about Feynman, “Infinity” staring Matthew Broderick, was released (I’ve never seen the movie, but I’ve heard it’s a stinker.) CNN was doing a spot about the movie and Feynman’s legacy and they needed a token undergraduate to blab about Feynman so myself and the smartest physicist in my class, Sebastian Maurer, were interviewed for the spot. Sebastian attempted to get a quote on T.V. about the Feynman lectures on physics, which, if you listened to it carefully could actually be interpretted as a statement about Mao’s little red book (Feynman’s lectures on physics used to come as a series of three red books.) Here is what I said about Feynman:
Mention his name to physics students at Cal Tech[sic] today and watch their eyes light up: “One of the reasons it was easier to become a physicist was because he was so exciting and he wasn’t the typical, you know, nerd who doesn’t say anything,” said Cal Tech[sic] senior Dave Bacon
So you see, the above letter makes me realize that what I said was exactly the sort of thing which would have driven Feynman crazy. So I kind of feel like I’ve been chewed out from beyond the grave.
Last night I finished reading “Perfectly Reasonable Deviations From The Beaten Track: The Letters Of Richard P. Feynman” by T. Ferris (forward), R.P. Feynman (of course!), and M. Feynman (editor). I wouldn’t recommend this collection of letters to everyone, but it is interesting for those who have read a lot of the other stuff about or by Feynman as the letters help flesh out Feynman’s character. There really aren’t that many “anecdotal” letters in the book, which is, of course, what everyone comes to expect from a Feynman collection (to be known as one gigantic anecdote generating machine…what a legacy). However, the following letter, in which information was requested about the infamous Lars Onsager, is rather amusing:
On one occasion when we were standing together, a young man came up to explain his ideas on superconductivity to us both. I didn’t understand what the fellow was saying-so I thought it must be nonsense (a bad habit I have.) I was surprised to hear Onsager say, “Yes, that seems to be the solution to the problem.” Did he mean the puzzle of superconductivity was solved-and I didn’t even know what the young man said? I guess so. I have never been sure-I think the young man could have been Cooper. Could you check?
This reminds me of a picture of Feynman’s blackboard at the time of his death. On this blackboard is a list of things “TO LEARN.” Included on the list are “the Bethe Ansatz Prob.”, “Kondo”, “2-D Hall”, “accel Temp.”, “Non-linear classical Hydro”. “Kondo” has been crossed out. This list is amazing, first in that it contains probably some of the most interesting problems in physics (“accel Temp.” refers to the Unruh effect where a uniformaly accelerating observer observes the vacuum as a thermal bath with a temperature proportional to the acceleration.) But even more amazing is that the great Richard Feynman, who legend has always portrayed as knowing everything there is to know about physics, still had “to learn” these famous problems. One wonders if Feynman’s “to learn” was a lot different than everyone elses “to learn?”
The pigeon hole principle states that if you put m pigeons into n<m holes, then there are at least two pigeons in one hole. Consider, for example, the following problem
Given a list [tex]$a_1,dots,a_n$[/tex] of residues mod [tex]$p$[/tex] and suppose [tex]$n>log_2 p$[/tex], find two distinct subsets [tex]$S_1,S_2 subset {1,2,dots,n}$[/tex] so that [tex]$prod_{i in S_1} a_i =prod_{i in S_2}a_j~{rm mod}~p$[/tex].
Here we have said “find”, but how do we even know that two subsets which solve this problem exist? The pigeon hole principle! The number of possible subsets of [tex] {1,2,dots,n}$[/tex] is [tex]$2^n$[/tex] (because each element either is or is not in the subset, and “is” and “is not” is binary, and so we have a binary string of length [tex]$n$[/tex].) But [tex]2^n>p[/tex], so we are putting more objects into our “holes” which run from [tex]$0$[/tex] to [tex]$p-1$[/tex]. Thus by the pigeon hole principle there must be two distinct subsets (two pigeons) whose product is the same.
In computational complexity the above problem belongs to a class of search problems called PPP, which, according to the complexity zoo, stands for “Polynomial Pigeonhole Principle.” These are problems where we are searching for a solution to an instance of a problem, this solution can be verified to be a solution in polynomial time, and the existence of a solution is guaranteed by a pigeon hole argument. The second of these should be familiar from the class NP, and the first of these is what you’re doing when you go beyond decision problems and actually want to find a solution.
PPP is interesting for various reasons, but one of the reasons I know of it is because it is at least as hard as the discrete log problem. In the discrete log problem is given two elements [tex]$a,b in {mathbb Z}_p$[/tex] find the smallest [tex]$s$[/tex] such that [tex]$a^s=b~{rm mod}~p$[/tex]. One of the neat things about a quantum computer is that it can efficiently solve the discrete log problem, whereas there is no known efficient classical algorithm.
So what do we know about the power of quantum computers to solve problems in PPP? Well we know a little. We know that a naive approach to solving these problems will fail. Suppose that we try to solve the above PPP problem by querying the function [tex]$f_{a_1,dots,a_n}(b_1,dots,b_n)=a_1^{b_1} a_2^{b_2} dots a_n^{b_n}$[/tex], where [tex]$b_i in {mathbb Z}_2[/tex]. Then a result by Scott Aaronson says that using a quantum algorithm requires [tex]$Omega((2^n)^{1/5})$[/tex] queries of this function (later improved to [tex]$Omega((2^n)^{1/3})$[/tex] by Yaoyun Shi) to solve this problem. So this naive approach to attacking problems in PPP fails. An interesting open question remains, however, whether there is a more sophisticated approach to efficiently solving problems in PPP using a quantum computer.
Interestingly, the problem Scott and Yaoyun work on is also relavent to a physicist in the lab. The problem they consider is called the collision problem. Suppose you have a function [tex]$f$[/tex] from [tex]${mathbb Z}_N$[/tex] to [tex]${mathbb Z}_N$[/tex] which is promised to be either 1-to-1 or 2-to-1 and the problem is to distinguish between these two cases. Scott and Yaoyun’s result says that if you do this you need to query this function [tex]$Omega((2^n)^{1/3})$[/tex] times (and, it turns out that there is a quantum algorithm which uses this number of queries) to distinguish between these two cases. Now how does this have relevance to a physicist? Suppose we follow the standard query method and query the function to produce the state [tex]$N^{-1/2} sum_{x in {mathbb Z}_N} |x>|f(x)>$[/tex]. Discarding the second register produces a mixed state of rank [tex]$N$[/tex] if [tex]$f$[/tex] is 1-to-1 but of rank [tex]$N/2$[/tex] if [tex]$f$[/tex] is 2-to-1. The entropy of these states is thus either [tex]$log_2 N$[/tex] or [tex]$log_2 N-1$[/tex] respectivally. Now suppose you are a physicist in the lab and you want to measure the entropy of a quantum state which you can prepare multiple copies of. You might be interested in how many copies you will need to measure this entropy. Scott and Yaoyun’s result then imply that you will need to perform a number of experiments which is at least [tex]$Omega(d^{1/3})$[/tex] such experiments to measure the entropy, where [tex]$d$[/tex] is the dimension of your quantum states. This is bad news if you really want to measure the entropy of a many-qubit system! In a related manner one can think of the period finding function of Shor’s algorithm as a method for determining the entropy for special class of states which have a particular promise (that they are periodic).
Discrete log is a pigeon problem, naive quantum algorithms for pigeon problems fail, and pigeon problems put limits on how efficiently we can measure the entropy of a system. It’s topics like these, which spread from computer science all the way to relevance for a experimental physics, which make me happy to be working in quantum information science.
Seth Lloyd, in Nature 430, 971 (2004):
Nothing in life is certain except death, taxes and the second law of thermodynamics. All three are processes in which useful or accessible forms of some quantity, such as energy or money, are transformed into useless, inaccessible forms of the same quantity. That is not to say that these three processes don’t have fringe benefits: taxes pay for roads and schools; the second law of thermodynamics drives cars, computers and metabolism; and death, at the very least, opens up tenured faculty positions.
Over the past few years there have been quite a few results where methods of quantum computing have been used to prove or reprove results about classical objects. Among these are lower bounds on locally decodable codes (de Wolf, Kerenidis, Wehner), limitations on classical algorithms for local search problems (Aaronson), a proof that PP is closed under intersection (Aaronson), and ways to use quantum communication complexity to prove lower bounds on classical circuit depths (Kerenidis). Now we have more to add to this mix, with quant-ph/0505188, “Lower Bounds on Matrix Rigidity via a Quantum Argument” by Ronald de Wolf (CWI Amsterdam). The rigidity of a matrix is the number of entries in a matrix which need to be changed in order to bring the rank of the matrix down to a certain value. What Ronald does in this paper is show how to use a quantum argument to prove some lower bounds on the rigidity of the Hadamard matrix. Very nice!
Whenever I see one of these quantum proofs of classical problems, I think, well: why are these quantum proofs significantly or slightly easier? But perhaps this thinking is backwards. Because the universe is quantum mechanical, should we really be surprised that the language of this theory is so powerful? One often hears the statement that it is amazing that mathematics is so useful for describing the physical world. But really this disregards the fact that different parts of mathematics are and are not useful for describing the physical world. So in a realm like computer science, where you are explicitly reasoning about physical systems, should it really be a surprise that the mathematics of these physical systems (quantum theory) is the best way to attack the problems?
Today is my birthday. And if there has ever been a doubt that I am a lunatic, my birthday should set to rest that debate because I was born on a lunar eclipse. (“All work and no play make Homer something somthing.” “Go crazy?” “Don’t mind if I do!”) My mother describes being in the hospital and looking up at the moon and thinking “that’s a strange looking moon.” I guess if I was religious or superstitious, I might take this as some sort of an omen and maybe even develop a nice messianic complex. Bah! I can develop such a complex perfectly fine without the aid of superstition.
This is a test of the render LaTeX plugin.
My first physics TA was an astrophysicist named David Hogg. He told me the following story. He said a friend of his was walking down the street, and, well this friend was really geeky looking so when a truck full of macho guys passed by they yelled out “Hey Geek! [tex]$E=mc^2$[/tex]”. The friend thought for a second, and then shouted back: “Only in the rest frame!”
Because of course everyone knows [tex]$E=gamma mc^2$[/tex]. But of course, having to explain [tex]$gamma$[/tex] is a lot harder than explaining [tex]$E$[/tex], [tex]$m$[/tex] and [tex]$c$[/tex].
My apartment in Seattle has a very nice view from the skyline of Seattle, all the way up Lake Union, and then to the outskirts of Fremont (which is the self-proclaimed center of the universe, and well, if you have a statue of Lenin, a thirty foot rocket, and a troll under a bridge, well, I guess you do get to do a little braging.) On clear days, you can see the cascade in the distance as well as Mt. Rainier to the left of the skyline. And the sea planes take off and land right in the middle of the view. What is kind of nice is that every room has a nice view. The master bedroom a view of Lake Union, the main living room with nearly solid windows and deck nearly the entire view, and the guest bedroom a view of downtown.
But to the point. Here is a picture of a rainbow over Lake Union as seen from my porch which I took with my cell phone:
Part of me thinks this game sounds like a lot of fun, and part of me thinks that it will create a bunch of intelligent design worshiping children (oh…wait)
Next year, Electronic Arts will release Wright’s next attempted masterpiece, Spore, a game some are calling “Sim Everything.”
Spore will give players the chance to control life — from the ground up.
Starting with single-cell organisms, players work on designing life with ever more complexity. As the game progresses, players must figure out how to take creatures from individual animals to small tribes and then to cities, whole planets, solar systems and galaxies.