Computing with Quantum Knots

56 SCIENTIFIC AMERICAN A PRIL 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. BY BRAIDING WORLD LINES (trajectories) of special particles, one can perform a quantum computation that is impossible for any ordinary (classical) computer. The particles live in a fluid known as a two-dimensional electron gas. Computing with Quantum Knots A machine based on bizarre particles called anyons that represents a calculation as a set of braids in spacetime might be a shortcut to practical quantum computation uantum computers promise to perform calculations be- lieved to be impossible for ordinary computers. Some of those calculations are of Qgreat real-world importance. For example, certain widely used encryption methods could be cracked given a computer capable of breaking a large number into its component factors within a reasonable length of time. Virtually all encryption methods used for highly sensitive data are vulnerable to one quantum algorithm or another. The extra power of a quantum computer comes about because it operates on information represented as qubits, or quantum bits, instead of bits. An ordinary classical bit can be either a 0 or a 1, and standard microchip architectures enforce that dichotomy rigorously. A qubit, in contrast, can be in a so-called superposition state, which entails proportions of 0 and 1 coexisting together. One can think of the possible qubit states as points on a sphere. The north pole is a classical 1, the south pole a 0, and all the points in between are all the possible superpositions of 0 and 1 [see “Rules for a Complex Quantum World,” by Michael A. Nielsen; Scien- tific American, November 2002]. The freedom that qubits have to roam across the entire sphere helps to give quantum computers their unique capabilities. Unfortunately, quantum computers seem to be extremely difficult to build. The qubits are typically expressed as certain quantum properties of trapped particles, GEORGE RETSECK By Graham P. Collins such as individual atomic ions or electrons. But their superposition states are ex- www.sciam.com SCIENTIFIC AMERICAN 57 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. ceedingly fragile and can be spoiled by squashing and bending but not by cut- are instead quasiparticles—excitations the tiniest stray interactions with the ting or joining. It embraces such subjects in a two-dimensional electronic system ambient environment, which includes as knot theory. Small perturbations do that behave a lot like the particles and all the material making up the computer not change a topological property. For antiparticles of high-energy physics. itself. If qubits are not carefully isolated example, a closed loop of string with a And as a further complication, the qua- from their surroundings, such distur- knot tied in it is topologically different siparticles are of a special type called bances will introduce errors into the from a closed loop with no knot [see anyons, which have the desired mathe- computation. box on opposite page]. The only way to matical properties. Most schemes to design a quantum change the closed loop into a closed Here is a what a computation might computer therefore focus on finding loop plus knot is to cut the string, tie the look like: first, create pairs of anyons ways to minimize the interactions of the knot and then reseal the ends of the and place them along a line [see box on qubits with the environment. Research- string together. Similarly, the only way page 60]. Each anyon pair is rather like ers know that if the error rate can be to convert a topological qubit to a dif- a particle and its corresponding antipar- reduced to around one error in every ferent state is to subject it to some such ticle, created out of pure energy. 10,000 steps, then error-correction pro- violence. Small nudges from the envi- Next, move pairs of adjacent anyons cedures can be implemented to compen- ronment will not do the trick. around one another in a carefully deter- At first sight, a topological quantum computer does not seem much like a computer at all. sate for decay of individual qubits. Con- At first sight, a topological quantum mined sequence. Each anyon’s world structing a functional machine that has computer does not seem much like a line forms a thread, and the movements a large number of qubits isolated well computer at all. It works its calculations of the anyons as they are swapped this enough to have such a low error rate is a on braided strings—but not physical way and that produce a braiding of all daunting task that physicists are far strings in the conventional sense. Rath- the threads. The quantum computation from achieving. er, they are what physicists refer to as is encapsulated in the particular braid so A few researchers are pursuing a very world lines, representations of particles formed. The final states of the anyons, different way to build a quantum com- as they move through time and space. which embody the result of the computa- puter. In their approach the delicate (Imagine that the length of one of these tion, are determined by the braid and quantum states depend on what are strings represents a particle’s movement not by any stray electric or magnetic in- known as topological properties of a through time and that its thickness rep- teraction. And because the braid is topo- physical system. Topology is the math- resents the particle’s physical dimen- logical—nudging the threads a little bit ematical study of properties that are un- sions.) Moreover, even the particles in- this way and that does not change the changed when an object is smoothly de- volved are unlike the electrons and pro- braiding—it is inherently protected from formed, by actions such as stretching, tons that one might first imagine. They outside disturbances. The idea of using anyons to carry out computations in this Overview/Quantum Braids fashion was proposed in 1997 by Alexei Y. Kitaev, now at Microsoft. ■ Quantum computers promise to greatly exceed the abilities of classical Michael H. Freedman, now at Mi- computers, but to function at all, they must have very low error rates. crosoft, lectured at Harvard University Achieving the required low error rates with conventional designs is far beyond in the fall of 1988 on the possibility of current technological capabilities. using quantum topology for computa- ■ An alternative design is the so-called topological quantum computer, which tion. These ideas, published in a research would use a radically different physical system to implement quantum paper in 1998, built on the discovery computation. Topological properties are unchanged by small perturbations, that certain mathematical quantities leading to a built-in resistance to errors such as those caused by stray known as knot invariants were associ- interactions with the surrounding environment. ated with the quantum physics of a two- ■ Topological quantum computing would make use of theoretically postulated dimensional surface evolving in time. If excitations called anyons, bizarre particlelike structures that are possible in an instance of the physical system could a two-dimensional world. Experiments have recently indicated that anyons be created and an appropriate measure- exist in special planar semiconductor structures cooled to near absolute zero ment carried out, the knot invariant and immersed in strong magnetic fields. would be approximately computed auto- matically instead of via an inconvenient- 58 SCIENTIFIC AMERICAN A PRIL 2006 COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC. TOPOLOGY AND KNOTS Topology of a closed loop (a) is unaltered if the string is pushed by moving around the string. Instead one must cut the string, tie the around to form another shape (b) but is different from that of knot and rejoin the ends. Consequently, the topology of the loop a closed loop with a knot tied in it (c). The knot cannot be formed just is insensitive to perturbations that only push the string around. a b c = = ly long calculation on a conventional it at various velocities, and so on. For when two particles are swapped. You computer. Equally diffi cult problems of example, a particle is most likely to be might say that their wave functions are more real-world importance would have found in a region where the wave func- multiplied by a factor of plus one. similar shortcuts. tion has a large amplitude. Deep mathematical reasons require Although it all sounds like wild theo- A pair of electrons is described by a that quantum particles in three dimen- rizing quite removed from reality, recent joint wave function, and when the two sions must be either fermions or bosons. experiments in a fi eld known as frac- electrons are exchanged, the resulting In two dimensions, another possibility tional quantum Hall physics have put joint wave function is minus one times arises: the factor might be a complex the anyon scheme on firmer footing. the original. That changes peaks of the phase. A complex phase can be thought Further experiments have been proposed wave into troughs, and vice versa, but it of as an angle. Zero degrees corresponds to carry out the rudiments of a topolog- has no effect on the amplitude of the to the number one; 180 degrees is minus ical quantum computation. oscillations. In fact, it does not change one. Angles in-between are complex any measurable quantity of the two numbers. For example, 90 degrees cor- Anyons electrons considered by themselves. responds to i, the square root of minus as previously mentioned, a to- What it does change is how the elec- one. As with a factor of minus one, mul- pological quantum computer braids trons might interfere with other elec- tiplying a wave function by a phase has world lines by swapping the positions of trons.

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