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.
Death, Taxes, and the 2nd Law
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.
It's Quantum So Should We Be Surprised?
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?
Saros Type 130
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.
LaTeX
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].
Why So Many Songs?
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:
Spore
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.
A Supporter
It’s always good to see the “quantum computing friendly” gain entry to high places:
Tony Hey to join Microsoft as Corporate Vice President
Professor Tony Hey is to join Microsoft Corporation as a corporate vice president to co-ordinate their Technical Computing Initiative. He will work across the company to co-ordinate Microsoft’s efforts to collaborate with the scientific community worldwide.
Currently Director of the UK’s e-Science Programme, Tony Hey has been a member of staff of the School of Electronics and Computer Science (ECS) since 1986, and was Head of School from 1994 to 1999.
He played a critical role in building the School into one of the UK’s leading academic departments, and has retained his Chair in the School throughout his period of secondment to UK e-Science.
‘This is wonderful news,’ said Professor Wendy Hall, Head of ECS, ‘and I am delighted to send our warmest congratulations to Tony on behalf of the School. His energy, vision, and sheer ability to make things happen will be of huge benefit to Microsoft and to future collaboration with researchers worldwide. At Southampton we are very glad that Tony will be retaining his Chair in the School of Electronics and Computer Science, and his strong links with the School and the University.’
Professor Hey is a Fellow of the Royal Academy of Engineering, the British Computer Society, the Institution of Electrical Engineers and a member of the Institution of Electrical and Electronic Engineers. In the New Year Honours List (2005) he received the CBE (Commander of the British Empire) for his services to science.
‘Today computation plays a critical role in advancing research across almost every field of science, yet far too often scientists must build all their own programming infrastructures and research their own algorithms to help advance their research effort,’ said Professor Hey. ‘By collaborating with the scientific community, I am confident that some of Microsoft’s advanced technologies can be used to accelerate their rate of discover.’
He has worked in the field of parallel and distributed computing since the early 1980s and was instrumental in the development of the MPI message-passing standard and in the Genesis Distributed Memory Parallel Benchmark suite. In 1991, he founded the Southampton Parallel Applications Centre (now the IT Innovation Centre), which has played a leading technology transfer role in Europe and the UK in collaborative industrial projects. His personal research interests are concerned with performance engineering for Grid applications but he also retains an interest in experimental explorations of quantum computing and quantum information theory.
For those who don’t remember, Tony Hey was the editor for “Feynman Lectures on Computation” and the companion volume entitled “Feynman and Computation.”
Quantum Documentary?
One question I’ve had for a long time is why there hasn’t been a good documentary produced about the field of quantum information science. There is certainly a feeling that quantum computing has leaked into the mainstream of geeks through various sci-fi stories, articles in Scientific American, and postings on Slashdot, and yet there hasn’t been a NOVA style documentary produced on the field in spite of the (I think) fascinating results which have emerged from the field in the last few decades (we’ve got to count quantum cryptography!)
Journal of Well Written Scientific Papers
Via Michael Nielsen I’ve found that Ben Schumacher, inventor of the word “qubit” and quantum information theorist extraordinaire, has a blog. Michael points to the following from quote from Ben:
But you know, so much of academic writing is bad. It is banal, orotund, unmusical, and stuffed with wads of unnecessary jargon. It is the sort of writing that does more to obscure meaning than to convey it. I see this stuff almost every day. I swim in it. OK, maybe I do exaggerate, a little. After all, I teach at a liberal arts college that is moderately well-known for teaching people how to write. Our faculty is full of novelists and poets and whatnot. But let me tell you, it’s here, too. It’s everywhere. It is like a fungus growing over all things, blurring their shapes — the verbal equivalent, maybe, of the ivy on academic buildings. And like the ivy, I guess, its main purpose is to conceal the shabby edifices beneath
Which made me think that it would be fun to create a scientific journal in which good writing was a requirment. No silly page limits either. You either write a killer article, which is comprehensible and well written, or your articled doesn’t get accepted. Even if you result is correct and even if your result is groundbreaking. Sure, not all journals could be this way, but it would if such a journal existed, I would certainly be a regular reader. I certainly know that there are people who are up to the task: every once in a while you stumble upon a piece of scientific writing which is so well written it just makes you cry the next time you look through Physical Review Letters.