Professor protagonist
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.'" Parallel transport for qubits
*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
z_{0}**<0|**
+ z_{1}**<1|** of its two base states
corresponds to the point [z_{0}:
z_{1}] in complex projective 1-space, which can be identified with
the Riemann sphere by [z_{0}:
z_{1}] --> z_{0}/z_{1} 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. Physical Chemistry in 4D
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." Physical insight into a hard combinatorial problem
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* E**76** 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 |