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 qk 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 qk.
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.