Tony Phillips' Take on Math in the Media A monthly survey of math news

This month's topics:

• Professor protagonist
• Parallel transport for qubits
• Physical Chemistry in 4D
• Physical insight into a hard combinatorial problem

NPR's Fresh Air broadcast on February 5, 2008 the review of a book about a mathematics professor, Susan Choi's "A Person of Interest." Choi, whose last, successful, novel took its inspiration from the Patty Hearst saga, this time starts her story with the explosion of a mail bomb in the office of a popular computer science professor. In real life, this would have been David Gelernter at Yale; in Choi's story the victim's name is Hendley, he teaches at a midwestern state university, and he dies. The "person of interest" is Professor Lee, the mathematician in the adjoining office; the story relates how Lee becomes a suspect, and how the investigation changes his life.

Fresh Air's reviewer, Maureen Corrigan, explains how Lee is presented to us: "It doesn't add up. Whatever the sum total of factors is that's needed to render the ordinary American normal, Professor Lee ... doesn't have enough of them." "[He] likes to go home after teaching his trigonometry classes to students he lightly disdains, pop open a beer, scrape together dinner, and fiddle around with mathematics problems far into the night. He's a loner and an intellectual or, as we say in America, a weirdo." And it turns out that Lee couldn't stand Hendley; Corrigan reads us the book's first sentence: "It was only after Hendley was bombed that Lee was forced to admit to himself just how much he'd disliked him." It sounds like Choi has stacked the odds against herself by chooosing an essentially unattractive main character, but apparently she pulls it off: "In her hands even a misanthropic, second-rate mathematician like Professor Lee becomes 'a person of interest.'"

Science for December 21, 2007 ran "Observation of Berry's Phase in a Solid-State Qubit." The authors are a team of ten from ETH, Sherbrooke and Yale, headed by Peter Leek and Andreas Wallraff. Their work falls under the rubric of what Seth Loyd called "holonomic quantum computation" (Science 292 5222): they use Berry phase changes produced by motion along paths (here the paths are in parameter space) to systematically manipulate the state of a qubit.

Fig. 1. Parallel transport of a vector around a geodesic triangle on the unit sphere, beginning (initial state v_i) and ending (final state v_f) at the North Pole. Image courtesy of Peter Leek and Andreas Wallraff.

Fig. 1 could come from a differential geometry text: it shows the parallel translation of a tangent vector around a (φ, π/2, π/2) geodesic triangle on the unit sphere. The final state v_f is rotated with respect to the initial state v_i by an angle which, when measured in radians, is exactly equal to the area enclosed by the path of the transport: in this case, φ. "The analogy of the quantum geometric phase with the above classical picture is particularly clear in the case of a two-level system (a qubit) in the presence of a bias field that changes in time," the authors write. In fact the set of states of a qubit may be represented as a sphere: an arbitrary superposition z₀<0| + z₁<1| of its two base states corresponds to the point [z₀: z₁] in complex projective 1-space, which can be identified with the Riemann sphere by [z₀: z₁] --> z₀/z₁ and stereographic projection.

Fig. 2. The state space of a qubit can be represented as a sphere, with pure state <1| at the North Pole and <0| at the South Pole. Here the state s precesses at fixed speed about a vector R, and R itself is moving, at much slower speed, along a path of its own. Image courtesy of Peter Leek and Andreas Wallraff.

Suppose that as in Fig. 2, "the qubit state s continually precesses about the vector R, acquiring dynamic phase δ(t) at a rate R =|R|." "When the direction of R is now changed adiabatically in time (i.e., at a rate slower than R), the qubit additionally acquires Berry's phase while remaining in the same superposition of eigenstates with respect to the quantization axis R." When the axis R has been brought back to its original position after traversing a path C in its parameter space (via a two-stage maneuver that results in zero dynamic phase accumulation), "the geometric phase acquired by an eigenstate is ±Θ_C/2, where Θ_C is the solid angle of the cone subtended by C at the origin." In the example illustrated, that cone is geometrically the same cone traced out by R in Fig. 2. As the authors remind us, its solid angle (i.e. the enclosed area intercepted on the unit sphere) "is given by Θ_C = 2π(1-cosθ), depending only on the cone angle θ." [Note that the phase change is only 1/2 of the solid angle, in contrast with the purely geometric example. So a 360^o planar rotation--enclosed area 2π--reverses the sign of the qubit.]

The article goes on to describe the experimental setup for implementing this phenomenon in real life. The qubit is a Cooper-pair box, the R-motions are driven by pulse-modulated microwave frequency signals, and the result is measured using quantum-state tomography.

The structure of an alkane-urea channel-inclusion compound. C_host = 1.102 nm at room temperature. The "host" urea subsystem (spirals) and the "guest" alkane have irrationally related periodicity, which leads to the material presenting phase transitions that can only be explained in a 4-dimensional "superspace." Image courtesy of Bertrand Toudic.

"Hidden Degrees of Freedom in Aperiodic Materials" is a report in the January 4 2008 Science. The first author is Bertrand Toudic (Rennes); of the other nine, seven are based in France, one at Kansas State and one in Bilbao. The geometric structure of a crystal is described by listing its planes of reflection symmetry, labeled by triples (h,k,l) of integers (Miller indices) that describe their slopes in coordinates (a, b, c) adapted to the crystal. These planes give rise to characteristic patterns of peaks of intensity in photographic (or other) records of how the crystal scatters radiation. Toudic and his team investigate "aperiodic" materials like the alkane-urea compound illustrated above. This compound consists of a framework of nanotubes ("urea molecules are connected by hydrogen bonds to form helical ribbons, which are woven together to form a honeycomb array of linear, nonintersecting, hexagonal tunnels") inside which "Guests such as nonadecane pack end to end within van der Waals contact of each other." In case the repeat length C_host of the helical structure of the tunnels and the packing distance C_guest of the alkane guests are not rationally related [presumably, on an appropriate scale], we need an additional Miller index to explain the diffraction patterns. The sets of Miller indices are now vectors (h,k,l,m) in a "superspace" of which the first three dimensions are the familiar ones.

The authors bring this extra dimension into salience by exhibiting a phase transition that cannot exist without it. As Philip Coppens explains it, in a "Perspectives" piece in the same issue of Science, when the c axis is pointing along the tube, " the average structure of the urea ...is described by the hkl0 reflections, and the average structure of the alkane ... by the hk0m reflections, whereas the remaining hklm reflections are due exclusively to the mutual interaction between the two lattices. This implies that [the urea lattice] imposes a distortion on [the alkane lattice], and vice versa." "At temperatures above 149 K, all nonadecane columns in the crystal distort in an identical way. However, below this temperature, the extra hklm reflections that appear in the diffraction pattern show that the relative modulation of the host and guest lattices alternates from channel to channel in the a-axis direction ... even though the periodicity of the average structures of the host and the guest in this direction does not change, as indicated by the absence of additional hkl0 and hk0m reflections." He concludes "Such a transition, which only affects the mutual interaction, can only be described properly in super-space, even though the physical reality is obviously three-dimensional."

The hitting-set problem. Here the job is to find a minimal-size set of students (discs) representing all five sports. The red discs are a "hitting set." (Image after Selman).

Since work of Mitchell, Selman and Levesque in 1992 it has been understood that some hard computational problems can undergo phase transitions at critical values of their parameters (see news about the 3-SAT problem in this column for November 2002). Recently this statistical mechanical behavior has been harnessed to yield information about the solutions of some hitherto intractable problems. The work, by Marc Mézard and Marco Tarzia, appeared (Phys Rev E76 041124) last year, and was picked up by Bart Selman in a "News and Views" piece for Nature, February 7, 2008. Selman tells us that the authors "demonstrate an innovative approach to solving one well-known NP-complete problem, known as the hitting-set problem." A hitting set picks out from the union of a collection of sets a subset that contains at least one element of each; the problem is to find a hitting set with the smallest number of elements. Mézard and Tarzia, realizing "that tools from statistical physics developed to study physical phase transitions might help in developing more efficient algorithms for solving combinatorial problems," adapted the calculation of ground-state properties of certain condensed-matter systems to give the survey-propagation method. "Mézard and Tarzia use the survey-propagation method to compute statistical properties of the solutions of instances of the hitting-set problem." This is considered even harder than finding a single solution, but survey propagation, working near a phase boundary, can get the information "... by iteratively solving a large set of coupled equations, modelling the local interactions between variables probabilistically. This solution process can be performed in a parallel, distributed fashion using many different processors, and generally converges to an answer extremely quickly --in seconds for equations with thousands of variables."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu