Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3


Just about everybody calls it weird—the bizarre ability of atomic and subatomic particles to be in two states at the same time, rotating this way and that, being here and there, simultaneously. But it’s getting less weird—or at least a bit more comprehensible—as physicists and other researchers push farther into the quantum world. It’s a world of the atom and its parts, a vanishingly small world where the physical laws defy our common sense. Indiana University Bloomington mathematics professors Zhenghan Wang and Michael Larsen are among this world’s pioneers.

photo of Zhenghan Wang
Zhenghan Wang is assistant professor of mathematics at Indiana University Bloomington. photo ©2001 Tyagan Miller

A force of nature

One of the most enticing results of understanding the quantum world and being able to manipulate it could be the quantum computer. Right now, it’s an inconceivable machine to most of us, but if—or when—scientists manage to deploy quantum weirdness to power a computer, it will be, as Discover magazine put it, “the ultimate computer, less a machine than a force of nature.”

Why? Because in a quantum computer, computational speed and power could take an extreme vertical leap. Such a computer could, for example, crack the RSA encryption scheme, which is the foundation of most methods that currently preserve data security on the Internet (including credit card transactions). The RSA scheme is based on finding the prime factors of very large numbers—for example, the prime numbers that can be multiplied together to equal a number such as 57,322,061. Today’s fastest computers might take hundreds of thousands of years to factor a number that is, say, 250 digits long—billions of years, perhaps, to factor a number that is 1,000 digits. The ability to break the RSA scheme in some reasonable amount of time would have a profound effect on all secure electronic data transmission.

That possibility alone has generated the widest interest in quantum computers, but there are many other problems that might be solvable only with quantum computers and some scientific applications that would be much better done with a quantum machine.

Two bits, four bits, six bits, a qubit

The computers we’re familiar with—called “classical” because they operate according to the laws of classical mechanics set forth by Sir Isaac Newton—use electronic bits to encode and manipulate information. It takes eight bits to store a single character such as the “I” at the beginning of this sentence. These bits are either “on” or “off.” A quantum computer could, instead, use atomic or subatomic particles as qubits, or quantum bits, to encode and manipulate information. Because the quantum world is weird, qubits exist in some mixture of on and off at the same time (called a superposition of states). This property would allow the computer to perform multiple computations—and get multiple answers—all at the same time. In other words, quantum computers could use a bunch of qubits to solve problems along different paths simultaneously, a mind-boggling example of parallel processing.

Building such a computer on a usable scale involves formidable challenges, though, not the least of which are error control and correction. Quantum computation must contend with errors introduced by various sources—including spontaneous, uncontrollable changes in the subatomic particles and random disturbances from our outside “classical” world, such as the radiation from light.

photo of Michael Larsen
Michael Larsen is professor of mathematics at IU Bloomington. photo©2001 Tyagan Miller

Wang and Larsen are studying what it would take to create a feasible fault-tolerant quantum computing system. They are among only a handful of people in the world delving into the mathematical structures that underlie the physics of quantum computing.

Knots, braids, and doughnuts

To devise a fault-tolerant system, Wang and Larsen rely on an area of mathematics called topology. Topology is the mathematical study of what happens to objects (circles, spheres, curves, and the like) as they are stretched, twisted, or folded. A quantum computer based in topological theory adds another layer of weirdness to the quantum machine. For one thing, it would involve neither silicon chips nor circuit boards. Its exotic physical system requires an electron liquid on a two-dimensional surface chilled to an extremely low temperature and subjected to an extremely strong magnetic field.

“When it’s that cold,” says Wang, “and with such a strong magnetic field, you’d expect some strange things to happen. And they do.”

When the electron liquid is under extreme cold conditions, strange composite particles called anyons form. The anyons move on the two-dimensional surface. “They’re dancing around,” says Wang. “We describe this as a braid.”

illustration of braid and anyons
The entangled lines above, called a "braid," describe the trajectories of four composite particles, called anyons, on a two-dimensional plane. The bottom illustration is a cross-section of the braid, showing the anyons at a certain point in time. The arrows indicate how the anyons "dance" from place to place. The dancing pattern corresponds to the braid above. Courtesy Zhenghan Wang

The idea behind a topological quantum computer is to make use of the braid-like movement of anyons. In mathematics, braids relate to the topological study and classification of knots. Mathematical knots are closed curves in three-dimensional space (think of the string a child uses to make string figures, only with no loose ends). These mathematical knots are composed of braids.

Some years ago, Vaughan Jones, a professor of mathematics at University of California, Berkeley, discovered a method of mapping a braid to a matrix, or a grid of numbers. Each matrix has an invariant property, a property that remains constant regardless of the shape or form the braids and knots take. The braids created by the trajectories of anyons share this property, which can be used to encode information and protect it from errors. Think of a shape like a doughnut, suggests Wang. If your information is encoded along a line following the curve around the top surface of the doughnut, then even if a piece is missing or damaged, you still know enough of the doughnut shape to fill in the missing piece.

According to Wang, the operation of a topological quantum computer would not depend on “the exact details of a path, only on the way the paths (or braids) of the anyons are linked.” Since the braids and their matrices are used to perform quantum computations, the result would be a robust and stable system. To have a usable computer, you don’t need to come up with matrices for all possible computations, only for a certain set of them from which other computational functions can be built. (Think of how we use the right combination of simple arithmetic, like adding, to do more complicated arithmetic, like multiplying or squaring numbers.)

Having proven mathematically that such a system could work, Wang and Larsen are exploring how we could actually use it to do computations, in preparation for the day when the physical system exists. The next step, Wang says, is for physicists “to find a material in nature which, when subjected to extremely low temperatures and high magnetic fields, yields the behavior that gives us an appropriate set of matrices, complicated enough to perform quantum computation.”

If you build it . . . ?

Wang and Larsen recently collaborated with Michael Freedman and Alexei Kitaev of Microsoft Research to simulate a quantum computer using topological quantum computation.

“Now,” says Larsen, “we are in a similar position to that of computer science in the 1930s, when people were proving that the various models of computation were computationally equivalent. The question now is: Which type of computer can we build?”

Quantum computers have been produced, but the largest so far is only seven qubits, which corresponds to 128 bits. Building a useful quantum computer would require, among other things, a colossal scaling up to much larger groups of qubits without letting the outside world interfere. Topological quantum computation could offer a way to scale up qubits without having to grapple with errors introduced by the outside world, so it may prove one of the more promising approaches to building a quantum computer.

Wang has received a grant from the National Science Foundation that will fund the work of a Topological Quantum Computing Numerical Laboratory at IUB. This laboratory will use some of IU’s supercomputing resources plus the university’s automated virtual environments, CAVE and the Immersadesk, for 3-D visualization. Wang’s first step will be to simulate a single anyon, study the boundary conditions that control its motion, and simulate manipulation of those conditions. The laboratory will also be used to develop new quantum algorithms for solving various computational problems. Wang and others will use part of the NSF grant to fund a Web site that will explain the work and describe its progress.

The payoff

The topological quantum computation approach is speculative, but it may well turn out to help researchers past major obstacles to quantum computing. Although we know some of the tantalizing benefits of quantum-based technology, we can’t know them all until we have a working quantum computer of a useful size.

“We don’t really know what the payoff would be,” says Larsen. “It would at least be substantial, and it might be enormous.”

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