Topology, DNA, and Quantum Computing

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Topology, DNA, and Quantum Computing

• Hsin-hao Su
• 2007.2.7

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What? Topology?

• I heard about geometry, numbers, trigonometry,
  statistics, and probability (Las Vegas, hehehe!).
  But, what did you say? Topology?

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Topology

• Topology is also called rubber band geometry or
  clay geometry. It is the study of geometric
  figures that are shrunk, stretched, twisted or
  somehow distorted. One of the main parts of
  topology is that of classifying surfaces.

Picture courtesy to www.lz95.org/sl/finearts/talbe
rt/clayjugs.htm
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Topologist

• It has been said that a topologist doesnt know
  the difference between a donut and a mug!
• A donut and coffee dont just taste good. They
  are the same thing!

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Smooth Deformation

• If an object can be smoothly deformed (think of
  morphing) to another one, topologically, they are
  the same.

Picture courtesy to http//www-xray.ast.cam.ac.uk/
jgraham/hypo/h12/topology.html
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Rules of Morphing
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Homotopy Equivalent

• Two spaces are homotopy equivalent if they can be
  transformed into one another by bending,
  shrinking and expanding operations.

Picture courtesy to mathworld.wolfram.com/BoySurfa
ce.html
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Which One Is Allowed?
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Holes!!!

• Without tearing, cutting and gluing, we cannot
  create a hole or eliminate a hole.
• Topologists use the number of holes to
  distinguish geometric objects.

Pictures courtesy to http//www.math.toronto.edu/
drorbn/People/Eldar/thesis/SurAndHome.htm
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Genus

• The number of holes (or handles) is called
  genus in topology.
• Or, how many cuts do you need to eliminate all
  the holes?

Pictures courtesy to http//www.math.toronto.edu/
drorbn/People/Eldar/thesis/SurAndHome.htm
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Euler Characteristic Number

• Another useful invariant in topology is Euler
  Characteristic Number.

• The classical Euler Characteristic Number is
  defined by , where V is
  the number of vertices, E is the number of edges,
  and F is the number of faces of a polyhedron.

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Euler Characteristic of a Cube

• For a cube, we have 8 vertices, 12 edges, and 6
  faces. Thus, we have that the Euler
  Characteristic number equals

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Euler Formula

• In general, for a polyhedron, the Euler
  Characteristic Number is always 2. This is called
  the Euler Formula.
• Tetrahedron
• Octahedron

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Euler Characteristic of a Sphere

• According to our previous discussion, a sphere is
  the same thing as a cube. We should expect the
  same Euler Characteristic, 2.

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Euler Characteristic of a Donut

• To find the Euler Characteristic of a torus
  (donut), We start from how to make one.
• According to my secret undercover in the Dunkin
  Donuts, we roll out a dough and then form a ring
  to make a donut.

Picture courtesy to http//motivate.maths.org/conf
erences/conf11/c11_project4.shtml
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Euler Characteristic of a Torus

• By using a rectangular paper to form a torus, we
  can count to get 1 vertex, 2 edges, and 1 face.
  Therefore, the Euler Characteristic is

Picture courtesy to http//www.search.com/referenc
e/Torus
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Möbius Band

• Möbius Band is a surface which has only one side
  and one edge.
• If you are walking on a Möbius Band, you cannot
  tell that you are in the inside or outside.

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How to Make a Möbius Band

• Simply starts with a strip. Twist it and glue
  edges together.
• If you put your pencil on the strip (dont lift
  if off) and then draw a line, you will see that
  one line covers the whole strip!
• If cut it through this line, you get a larger
  Möbius Band.

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Euler Characteristic of a Möbius Band

• From previous discussion, a Möbius Band has only
  1 edge and 1 face.
• Look at the paper.
• We can see that there are two vertices. But, we
  create two more edges from these two vertices.
  Therefore, the Euler Characteristic of a Möbius
  Band is

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Relation with Genus

• Can we use Euler Characteristic to distinguish
  geometric objects?
• Yes! Euler Characteristic is an invariant of
  topological objects. Also, we have a relation
  between Euler Characteristic and the genus.

Picture courtesy to www.indiana.edu/minimal/maze/
handles.html
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Euler Characteristic and Genus
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Its Fun But, Why Topology?

• Topology plays a central role in mathematics
  since it is the tool to study continuity.
• It is looked like a mathematicians personal
  habit. Any real-world application?

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Applications

• Biologists use it to understand how enzymes cut
  and recombine DNA.
• Quantum Computer Scientists use it to construct a
  fault-tolerant bit.

Picture courtesy to www.ericharshbarger.org/lego/m
ini_dna.html
Picture courtesy to www.q-pharm.com/home/contents/
backup/quantum_comp
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DNA

• The basic structure of duplex DNA consists of two
  molecular strands that are twisted together in a
  right-handed helix, while the two strands are
  joined together by bonds.

Picture courtesy to whyfiles.org/075genome
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DNA

• DNA can exist in nature in linear form (i.e. as a
  long line segment) or in closed circular form
  (i.e. as a simple closed curve).

Pictures courtesy to www.ucalgary.ca/zleonenk
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Supercoiling

• DNA that is circular can be either supercoiled or
  relaxed. The supercoiled form is much more
  compact.

Picture courtesy to biology200.gsu.edu/houghton/45
9520
Picture courtesy to www.cbs.dtu.dk/staff/dave/roan
oke
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Picture courtesy to http//www.maths.uq.edu.au/in
finity/Infinity7/supercoiling.html
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DNA Replication

• Supercoiling allows for easy manipulation and so
  easy access to the information coded in the DNA.
  When a cell is copying a DNA strand, it will
  uncoil a strand, copy it and then recoil it.

Picture courtesy to http//www.maich.gr/natural/st
aff/sotirios/topo.html
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DNA Replication

• DNA replication begins with a partial unwinding
  of the double helix at a part known as the
  replication fork.
• An enzyme known as DNA helicase does this.

Picture courtesy to http//www.virtualsciencefair.
org/2004/mcgo4s0/public_html/t2/dna.html
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Knots

• Let us play a rubber band again.
• A knot is a closed curve in three-dimensional
  space or paths that you can trace round and round
  with your finger. It is as though the two free
  ends of tangled rope have been spliced together.
• Two knots are considered the same if one can be
  moved smoothly through space, without any
  cutting, so that it is identical to the second.

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Picture courtesy to http//www.tiem.utk.edu/gross
/bioed/webmodules/DNAknot.html
The first knot is merely a loop of string that
has been twisted, an "unknot". It could easily be
unknotted by pulling on the string to form a
single loop. The 2nd knot, however, is clearly a
knot. The only way to get rid of the knot would
be to cut through it and retie the string. The
3rd knot is even more complicated.
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Picture courtesy to http//www.c3.lanl.gov/mega-ma
th/workbk/knot/knbkgd.html
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Picture courtesy to http//www.c3.lanl.gov/mega-ma
th/workbk/knot/knbkgd.html
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Crossing Points

• Each crossing point is assigned a or - sign,
  depending on the orientation of the crossing
  point.
• If the strand passing over a crossing point can
  be turned counter-clockwise less than 180 to
  match the direction of the strand underneath,
  then the sign is positive () if the strand on
  top must be rotated clockwise, it is negative
  (-).

Picture courtesy to mathworld.wolfram.com/Writhe.h
tml/
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Writhe

• The writhe of a knot is the sum of all signs of
  its crossing points.

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Unknotting Number

• The only way to untie a mathematical knot is to
  cut through the knot so that the strand that was
  lying on top is now underneath.

• This is equivalent to changing the sign of a
  crossing point.
• The number of times on must allow one strand of a
  knot to pass through another (in order to unknot
  it), is called the unknotting number.

Picture courtesy to www.tricksecrets.com/images/33
3.jpg
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Picture courtesy to http//www.tiem.utk.edu/gross
/bioed/webmodules/DNAknot.html
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Electron Microscopes

• Scientists use electron microscopes to take
  pictures of DNA. Underlying and overlying
  segments are distinguished by using a protein
  coating. The flattened DNA is then visualized as
  a knot.
• The unknotting number and ideal crossing number
  can then be estimated.

Picture courtesy to http//www.tiem.utk.edu/gross
/bioed/webmodules/DNAknot.html
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Difficulties

• It is possible to experimentally determine the
  outcome of an enzyme action.
• But, there is no known method to actually observe
  the action of an enzyme.

Picture courtesy to www.readingwithtlc.com/Pages/w
horisk.htm
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The Action of Enzymes on DNA

• When an enzyme acts on a DNA, one possible result
  is that one (or a pair of) strand(s) of DNA will
  pass through the other strand.
• The DNA become more or less supercoiled.

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Topoisomerase

• An enzyme called topoisomerase is used to unpack
  and pack in DNA by changing the number of twists
  in DNA.
• Brown and Cozzarelli (1979) use topology to
  determine how the topoisomerase works to
  supercoil DNA.

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Oh! Topology!

• Topology gives cell biologists a quantitative and
  invariant way to measure properties of DNA.
• Knot theory has helped to understand the
  mechanisms by which enzymes unpack DNA.
• Measuring changes in crossing number helps to
  understand the termination of DNA replication and
  the role of enzymes in recombination.

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Coffee or Tea Time

• A mathematician is a device for turning coffee
  into theorems
• -- Alfréd Rényi (Hungarian mathematician,
  1921-1970)
• (It is often attributed to Paul Erdös)

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Quantum Computing

• Quantum computing is a new field in computer
  science which relies on quantum physics by taking
  advantage of certain quantum physics properties
  of atoms that allow them to work together as
  quantum bits, or qubits.

• By interacting with each other, qubits can
  perform certain calculations exponentially faster
  than conventional computers.

Picture courtesy to http//www.theteacherplace.com
/gallery.php
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Qubits

• Quantum computer operates on information
  represented as qubits, or quantum bits.

• A qubit, so-called superposition state, can be
  any proportion of 0 and 1.
• We can think of the possible qubit states as
  points on a sphere.

Picture courtesy to www.uni-ulm.de/qiv/forschung/q
ubit.html
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Superposition

• 3 bits have 8 combinations of values. Only one of
  those values can be stored in a digital 3-bit set

• But in a 3-qubit set we can store all of them
  thanks to the superposition property of the
  qubits!

Picture courtesy to www.uni-ulm.de/qiv/forschung/q
ubit.html
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Anyon

• Ordinarily, every particle in quantum theory is
  neatly classified as either a boson--a particle
  happy to fraternize with any number of identical
  particles in a single quantum state--or a
  fermion, which insists on sole occupancy of its
  state.
• Almost 30 years ago researchers proposed a third
  category, "anyons," where a limited number of
  particles could inhabit a single state.
• Frank Wilczek used the term anyons in 1982 to
  describe such particles, since they can have
  "any" phase when particles are interchanged.

Hall of Quantum Effects Four voltage "gates" on
this semiconductor surface created a central disk
with "quasiparticles" having one-fifth of an
electron's charge (red) surrounded by a ring of
one-third charge quasiparticles (blue).
Measurements revealed that the quasiparticles are
neither fermions nor bosons.
Courtesy to http//focus.aps.org/story/v16/st14
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Decoherence

• When a qubit interacts with the environment, it
  will decohere and fall into one of the states.
  This makes the extraction of the results
  calculated by an operation to a set of qubits
  really difficult.
• Furthermore, the problem increases in large qubit
  systems and causes the potential computing
  ability of quantum computers to drastically fall.

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Solution?

• Since the topological properties is not changed
  by actions such as stretching, squashing and
  bending but not by cutting or joining, it
  prevents small perturbations from the environment.

Picture courtesy to http//www-xray.ast.cam.ac.uk/
jgraham/hypo/h12/topology.html
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Topological Quantum Computer

• Topological Quantum Computer works its
  calculations on braided strings.
• Each anyons time line forms a thread.
• When the anyons swap, it produce a braiding of
  all the threads.
• The final states of the anyons encapsulated the
  result which is determined by the braid.

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Braids

• The pictures you see here are braids, which you
  can think of as strings of wire weaving around
  each other, without backing up.

Picture courtesy to www.math.vu.nl/mathanalysis/br
aids.php
Picture courtesy to www.math.neu.edu/poster_sessio
n/suciu.html
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Courtesy to Computing with Quantum Knots 2006
Scientific American
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Topological Quantum Computing

• Uncontrolled exchange of quantum numbers will be
  rare if particles are widely separated, and
  thermal anyons are suppressed.

Courtesy to Topological Quantum Computing for
Beginners by John Preskill
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Anyons Restrictions

• The temperature must be small compared to the
  energy gap, so that stray anyons are unlikely to
  be excited thermally.

• The anyons must be kept far apart from one
  another compared to the correlation length, to
  suppress charge-exchanging virtual processes,
  except during the initial pair creation and the
  final pair annihilation.

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A student was doing miserably on his final
exam in Topology. To make up, the professor asked
the student "So, what do you know about
topology?" The student replied, "I know the
definition of a topologist." The professor asked
him to state the definition, expecting to get the
old saw about someone who can't tell the
difference between a coffee cup and a doughnut.
Instead, the student replied "A topologist is
someone who can't tell the difference between his
ass and a hole in the ground, but who can tell
the difference between his ass and two holes in
the ground."
So,
PASS
FAIL
or
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References

• C.C. Adams, The Knot Book An Elementary
  Introduction to the Mathematical Theory of Knots,
  W.H. Freeman Company, March 1994.
• G.P. Collins, Computing with Quantum Knots,
  Scientific American, 2006.
• Erica Flapan, When Topology Meets Chemistry A
  Topological Look at Molecular Chirality,
  Cambridge University Press, July 2000.
• L.J. Gross, DNA and Knot Theory,
  http//www.tiem.utk.edu/gross/bioed/webmodules/DN
  Aknot.html
• J. Preskill, Topological Quantum Computation,
  Lecture Notes.
• S.D. Sarma, M. Freedman, and C. Nayak,
  Topological Quantum Computation, Physics Today,
  July 2006.