Quantum computing gets a lot of attention, but we don’t hear much about quantum mathematics. The very idea is an affront to Platonist thinkers everywhere—those of us who consider the elements of mathematics to be independent of the physical universe. Is the truth of the Pythagorean theorem subject to the same uncertainty as the fate of Schrodinger’s cat? Surely the counting numbers 1, 2, 3,… should be the same in any universe, whether quantum or classical; indeed, if you’re a full-gospel, whole-Bible Platonist, you might well argue that those numbers existed even before there was a universe. (At the opposite extreme are the constructivists, who warn that the only numbers you can count on are those you have fingers for.)

Questions like these are easily settled by experiment: Just annihilate the universe, abolish space and time, and try doing a few sums in the void. I would put the proposition to the test right now except that I don’t want to miss this week’s episode of “1 vs. 100.”

Paul Benioff of Argonne National Laboratory has been thinking and writing about these issues for some time. A few weeks ago he posted a preprint on the arXiv titled “Space of quantum theory representations of natural numbers, integers, and rational numbers.” I’ve been struggling to understand that paper, along with a couple of earlier ones on the same theme (here and here). I’m still a long way from mastering this material, but here’s what I’ve made of it so far.

Even if you go along with the Platonist view that the true home of all things mathematical is an ideal realm outside of time and space, we don’t live in that realm. Whenever we want to do something with numbers (or other mathematical entities), we have to represent them somehow in this universe. We chalk them on the blackboard; we twiddle the beads of an abacus; we load patterns of bits into a computer memory; even mental arithmetic requires a mind, which in turn seems to require a body. Thus no one does mathematics without atoms and photons and other toys and tools of the physicist, and so perhaps it’s not utterly crazy to suppose that what we can accomplish in mathematics may depend to some extent on the laws of physics. All the evidence suggests that those laws are inescapably quantum mechanical.

Benioff argues that numbers have to be represented and manipulated in ways that accord with quantum principles; for example, all operations should be reversible, and they should enforce “unitarity,” meaning that the probabilities of all possible outcomes should sum to exactly 1. At the same time, it’s important that numbers behave like numbers: They have to obey familiar axioms of arithmetic, or they’re of no use to mathematics. In the case of the counting numbers (a.k.a. the natural numbers) we have to be able to add and multiply. For the integers (the natural numbers augmented with zero and negative values), subtraction must also be allowed. The rationals introduce division. And for all these kinds of numbers we should be able to determine if two numbers are equal, and put them in order if they are not.

Quantum computing is usually discussed in terms of qubits, or the quantum equivalent of binary digits. Benioff extends the discussion to qukits, the quantum equivalent of base-k digits. Think of a qukit as a black box that holds one of k values, but until you open the lid, you can’t be sure which value. Once you look inside, the uncertainty is resolved, and the box is found to contain a specific value. The question Benioff addresses is: How do you do arithmetic with such unruly numbers? If you can’t even be sure of what numbers you’re working with, how can you add or subtract them, or test them for equality? The answer, ultimately, is that you have to put up with a degree of uncertainty. In the quantum world, the commutative law of addition, a+b = b+a, is not a bedrock principle but a statement whose truth is a matter of probabilities.

Benioff presents a specific implementation of quantum-theoretical numbers, based on a two-dimensional lattice of qukits. Each number occupies its own row of the lattice, with the digits arrayed from left to right. For integers, a designated qukit (actually a qubit, since it has just two states) serves as a sign bit. For rationals, this special qubit also marks the position of the decimal point (or “k-al” point), separating the integer part from the fractional part. On first glance, this lattice of qukits doesn’t seem too different from the hardware of an ordinary computer, but the quantum nature of the qukits brings some peculiarities. For example, the result of every operation has to go into a newly allocated row of the lattice; you can never erase or overwrite an existing value, because quantum operations have to be reversible and cannot destroy information.

Benioff introduces a set of parameters for his numbering system: m and h are the coordinates of a number within the lattice of qukits, k is the base of the qukits, and g is a “gauge-fixing function.” This last item sounds quite arcane, but I think it has a fairly simple explanation. You can imagine a qukit as a vector that can point in any of k directions; the gauge-fixing function defines a reference direction from which all the others are measured.

Here’s a taste of what Benioff has to say about his quantum numbers:

Transformations (k, (m, h), g) → (k’, (m’, h’), g’) in the parameter set induce transformations in the representation space. These consist of unitary translations that move the qukit strings on the lattice, transformations that change states of strings of base-k qukits to states of strings of base-k’ qukits, and unitary gauge transformations for each k….

An interesting result is that the axioms and theorems for each of the three types of numbers [i.e., natural numbers, integers and rationals] are invariant under these transformations. They represent symmetries of the systems. This is the case even though the specific expressions of the axioms and theorems in terms of basic arithmetic relations and operations are different for different representations. This is like the situation in physics where the laws of physics are invariant under Lorentz transformations even though their specific expression in different reference frames may be different.

Another interesting result is that qukits q_k where k is a prime number function as elementary qukits. These are the “elementary particles” as far as quantum representations of numbers are concerned. Qukits where k is not prime can be considered as composites of the prime number q_k.

If you want to delve more deeply into these matters, I must send you to Benioff’s paper. Here I want to return to the broad question of whether this line of inquiry can really lead us to some kind of quantum mathematics, as opposed to an abstract version of quantum computing. Personally, I’m not quite persuaded, although I find the proposition intriguing.

In most of this work the focus is on the representation of numbers, rather than the numbers themselves—a preoccupation that seems more characteristic of computing than of mathematics. Interesting mathematical properties of numbers tend to be independent of their representation; for example, the number seven is a prime whether you write it in decimal notation as 7, in binary as 111 or as the Roman numeral vii. Of course you must choose some representation, and the choice can make a difference in what you can accomplish. Benioff points out that unary notation (in which seven becomes 1111111) is inherently less efficient than other schemes, because the amount of work expended in manipulating a number is proportional to the number itself rather than to the logarithm of the number. For similar reasons Benioff objects to building all of arithmetic on the successor function (n → n+1). But this fretting over efficiency again suggests a more computational than mathematical frame of mind. After all, unary numbers and the successor function are essential building blocks in theories of the foundations of mathematics, such as in the Principia Mathematica of Whitehead and Russell—who were blissfully unconcerned with efficiency.

If you accept that the physical representation of mathematical objects is a fundamental issue in mathematics, then it’s an easy step to the conclusion that any such representation has to be consistent with quantum principles. But is the mathematical imagination truly fettered by the bounds of the physical universe? Mathematicians routinely reason about objects and operations that have no explicit material representation—irrational numbers, for example, or Georg Cantor’s infinite sets. A century ago, in response to another challenge from those who wanted to fence in the scope of mathematics, David Hilbert defiantly proclaimed: “No one shall expel us from the paradise that Cantor has created for us.”

If I remain a tad doubtful about the mathematical status of quantum-theoretical numbers, I do think they offer an interesting perspective on quantum computing. So far, no one has succeeded in building a practical quantum computer with a large number of interacting qubits (or qukits). Technological skeptics contend it will not be done any time soon. Benioff’s work turns this argument on its head. Quantum computers are the only ones we can build, he says, because we live in a world where quantum physics is the law of the land. We may think we have classical computers, but that’s an illusion. We merely have quantum computers that are heavily biased toward specific classical outputs, but they always retain the possibility of delivering a quantum surprise. Usually we regard computing—whether it’s done with a machine or with pencil and paper—as an approximation to a mathematical ideal. If a calculation shows that a+b does not equal b+a, we don’t question the commutative law of addition; we look for a bug or an error. But maybe we need to consider the possibility that it’s the quantum computation that’s the ultimate reality, and the mathematical law is just our convenient and tidy approximation to it.

The story about young Gauss and his trick for summing an arithmetic series just won’t stop. My archive now includes 134 versions. Almost all the recent additions were discovered by Barry Cipra. (For a recap on what this is all about, see earlier bit-player postings here and here, or the American Scientist article here.)

I’ve also received an interesting e-mail from James Grant, developer of a Web site on Fibonacci:

… Fibonacci (better called Leonardo da Pisa) published the formula for adding a linear series in his pivotal book Liber Abaci in 1202, which persuaded Europe to use “Arabic numerals” (actually from India) instead of Roman numerals. He presents the method at the beginning of the 12th chapter. He also presents methods for calculating other series, including a series of squares.

Fibonacci does not claim to be the discoverer of these formulas. He credits Arab and Indian mathematicians.

I do not claim here that Gauss read Liber Abaci in time to impress his teacher. It seems quite fair to credit him with re-discovering this formula. I just think it’s important for us to be aware that the formula was known perhaps 1,000 years before Gauss sprang it on his teacher.

Here is part of the passage from Liber Abaci, transcribed from the translation of Lawrence E. Sigler (Springer-Verlag, 2002):

When you wish to sum a given series of numbers which increases by some given number, as increasing by ones, or twos, or threes, or any other numbers, then you multiply half the number of numbers in the series times the sum of the first and last numbers in the series, or you multiply half the sum of the first and last numbers in the series by the number of numbers in the series, and you will have the proposition. For example, I wish to sum 7 numbers that increase by threes from seven up to 31, namely 7, 10, 13, and so forth up to 31. The number of the aforesaid numbers is indeed 9, that is there are nine numbers in the aforesaid series, of which the first is the seven…. Therefore the sum of the extremes, namely the 7 and the 31 is 38; therefore if you multiply half the … 9 by the 38, or half the 38 by the 9, then the result is 171 for the sum of the posed series of nine numbers….

As Grant notes, Fibonacci is not the inventor of this method; it appears three centuries earlier in a manuscript attributed to Alcuin of York; furthermore, Don Knuth has pointed out that it can also be found in Archimedes. Among the early sources, however, Fibonacci makes the best effort at explaining the algorithm; other accounts merely state the problem and solution.

I watched the spelling bee on TV a couple of weeks ago and was stumped by word after word: aniseikonia, oberek, randkluft, cachalot, schuhplattler, cilice. It’s all enough to send you reeling back to Andrew Jackson or Mark Twain or Winston Churchill or whoever the hell it was who said “I don’t give a damn for a man that can only spell a word one way!” As it happens, I’ve been writing lately about words that get spelled and misspelled in lots and lots of ways. My Computing Science column in the July–August issue of American Scientist asks the question: “How many ways can you spell V1@gra?”

Disclaimer: I ask the question but I can’t answer it—or at least I can’t give a definite number or a close approximation.

Another question also goes unanswered. The curious and creative spellings prevalent in spam are (presumably) intended to evade the filters that most of us have installed on our e-mail. Because the word “Viagra” is uncommonly common in spam, most e-mail that mentions it gets dumped in the junk bin. So what do you do if you frequently need to discuss Viagra in your correspondence? In particular, what about Pfizer, the company that makes and markets the stuff? Surely their corporate mail servers can’t be running filters that block all references to their own product.

I tried to find out how Pfizer deals with this problem. I sent an e-mail query to their public relations department. I got no response—which could of course be taken as an answer to my question. I tried following up by telephone, but no one I spoke with was able to shed any light on the issue. So, if anyone from Pfizer should read this, please get in touch; I’d still like to know the answer.

By the way, has anyone noticed that “Pfizer” looks a spelling invented by a spammer?

Among the 250 million Rubik’s cubes manufactured since 1980, how many lie abandoned in a scrambled state, having never regained their original configuration since being taken out of the box? Most of them, I would guess. Now comes word that those cubes might be restored to pristinity with a little less effort. The upper bound on the number of moves required to solve any state of Rubik’s cube has been lowered from 27 to 26. The result is reported by Daniel Kunkle and Gene Cooperman of Northeastern University in a preprint available here.

It’s well-known that Rubik’s cube isn’t really a toy or a puzzle but rather a group-theory machine in disguise. Each possible move, or twiddle—rotating a face by some multiple of 90 degrees—is a transformation that permutes the positions and orientations of the 26 “cubies.” From any position there are just 18 distinct nontrivial moves, but in various combinations they generate a total of 43,252,003,274,489,856,000 permutations. What is the maximum number of moves needed to transform any one state of the cube into any other? That’s the question Kunkle and Cooperman are addressing. In other words, they are looking for the longest shortest path.

In principle we could get the answer by exhaustive search. It would be done by constructing the “Cayley graph” for the Rubik group: a graph in which each possible configuration of the cube is represented by a node, and two nodes x and y are connected by an edge if some single move allows configuration x to be transformed into y. The most efficient solution for any state is the shortest path (i.e., sequence of edges) leading from that state to the node representing the solved state. The worst-case solution length for the entire puzzle is the maximum of the shortest solution paths for all the nodes. The Cayley graph approach was used by Cooperman and two colleagues to find the worst-case path for the 2 × 2 × 2 version of Rubik’s cube. The Cayley graph for this device has 88,179,840 nodes, and it turns out the longest paths within it have 11 steps. For the 3 × 3 × 3 cube, however, exhaustive search is just too exhausting.

To make the search feasible, Cubists have had to scale back their ambitions and accept an upper bound rather than a true maximum. First it was shown that the worst-case solution path cannot require more than 52 moves; then the bound was lowered to 30, and then 29; a year ago Silviu Radu got the number down to 27.

Kunkle and Cooperman’s strategy is to factor the problem into two pieces by segregating half-turn (180-degree) and quarter-turn (±90-degree) moves. The subgroup of states reachable by half-turn moves is quite small by Rubik group standards, with just 663,552 elements, and the number is further reduced to only 15,752 after various symmetries are eliminated. Calculating the best solution for each of these states took only about a day on an ordinary PC. The longest such paths have 13 steps.

The second phase of the analysis tackles the “cosets” of the half-turn subgroup. For each of the 663,552 positions reachable by making half-turns only, there are 65,182,537,728,000 additional positions reachable by making quarter-turn moves. Kunkle and Cooperman had to search for the longest shortest path connecting any two elements of this set. They exploited various symmetries to reduce the scope of the problem and devised fast algorithms for some of the basic steps in the calculation. Even so, they wound up running their computer program for 63 hours on 128 processors at the San Diego Supercomputer Center. At one point the data store for intermediate results ballooned to seven terabytes. At the end of the run, they found that the longest shortest path consisted of 16 moves.

Combining the two phases of this solution yields 13 moves for the half-turn part of the path and 16 moves for the coset part, for a total of 29 moves. This is not an improvement over the best existing result of 27. However, Kunkle and Cooperman found that the two-stage algorithm reports path lengths greater than 26 only for a comparative handful of states—about 14,000. By a brute-force search they were able to find solutions of length 26 or less for each of these states, and thereby established the 26-move bound overall. It’s possible that further brute-force searching will lower the limit further to 25 moves.

Other experiments done by Herbert Kociemba, Richard Korf and others have shown that most randomly selected cube states can be solved in 18 moves or less. Korf has conjectured that all configurations are solvable in about 20 steps. We’ll see whether this is true—but probably not soon.

Anthony G. Pakes of the University of Western Australia shines further light on the “factoidal” function, discussed earlier here on bit-player and in the May-June issue of American Scientist:

I found your article about factoid(n) very interesting, and I offer you some analytically derived facts about it.

I will denote factoid(n) by f(n). You probably realize that 2f(2) is the payout in the St Petersberg game, i.e. the payout if tossing a fair coin repeatedly shows the first head on toss number n. Its probability law is exactly known, so there is not much more to say about it.

For n=2, 3, …, there is a critical number t_n such that 1 = t₂ < t₃ < t₄ < …, and the moment of order t of f(n) is finite iff t < t_n. I have an explicit formula for this moment. I believe that, for large n, t_n is asymptotically proportional to 1/n log(n) (natural logs), but my ‘proof’ of this is not entirely rigorous.

The distribution tail of f(n) is indeed fat; it decays in proportion to t_n^–x. I have values of the constants of proportionality. In this respect the distribution of f(n) is similar to members of the stable domain of attraction with index t_n. Also, it is nothing like a lognormal law, which has finite moments of all orders.

There is a limit law too: log(f(n))/n log(n) converges in distribution to the standard exponential law. This confirms your remarks about the probability of f(n) being bigger than n!. It does indeed converge to 1/e. In fact the probability that f(n) > (n!)^x converges to exp(–x).

A couple of general remarks. There is a book devoted to Products of Random Variables: by J Galambos & I Simonelli, Marcel Dekker Inc., (2004). The stopping rule defining f(n) is quite different to the model considered by Reed and Hughes; in their case it is independent of their summand random variables. For f(n) it is defined by its random factors, although it is a stopping time. Finally, f(n) is a function. For each n, it is a function defined on some sample space which could be made explicit, but almost never is. Besides being a random variable, it differs from n! in that f(n+1) cannot be computed from f(n) simply by multiplying by another random variable; changing n changes the probability law of the factor random variables. The situation is very similar to triangular arrays studied in the general summation theory of independent random variables, infinite divisibility and related stuff.