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Topology, DNA, and Quantum Computing

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Topology, DNA, and Quantum Computing

Hsin-hao Su

2007.2.7

- I heard about geometry, numbers, trigonometry, statistics, and probability (Las Vegas, hehehe!). But, what did you say? 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/talbert/clayjugs.htm

- It has been said that a topologist doesn’t know the difference between a donut and a mug!
- A donut and coffee don’t just taste good. They are the same thing!

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- 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

- 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/BoySurface.html

- 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

- 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

- 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.

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

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

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

- 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/conferences/conf11/c11_project4.shtml

- 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/reference/Torus

- 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”.

- Simply starts with a strip. Twist it and glue edges together.
- If you put your pencil on the strip (don’t 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.

- 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

- 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

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

- 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/mini_dna.html

Picture courtesy to www.q-pharm.com/home/contents/backup/quantum_comp

- 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

- 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

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

Picture courtesy to www.cbs.dtu.dk/staff/dave/roanoke

Picture courtesy to biology200.gsu.edu/houghton/4595%20

Picture courtesy to http://www.maths.uq.edu.au/~infinity/Infinity7/supercoiling.html

- 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/staff/sotirios/topo.html

- 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

- 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.

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.

Picture courtesy to http://www.c3.lanl.gov/mega-math/workbk/knot/knbkgd.html

Picture courtesy to http://www.c3.lanl.gov/mega-math/workbk/knot/knbkgd.html

- 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.html/

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

- 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/333.jpg

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

- 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

- 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/whorisk.htm

- 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.

- 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.

- 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.

- “A mathematician is a device for turning coffee into theorems“
-- Alfréd Rényi (Hungarian mathematician, 1921-1970)

(It is often attributed to Paul Erdös)

- 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

- 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/qubit.html

- 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/qubit.html

- 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

- 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.

- 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

- Topological Quantum Computer works its calculations on braided strings.
- Each anyon’s 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.

- 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/braids.php

Picture courtesy to www.math.neu.edu/poster_session/suciu.html

Courtesy to “Computing with Quantum Knots” 2006 Scientific American

- 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

- 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.

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

- 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/DNAknot.html
- J. Preskill, Topological Quantum Computation, Lecture Notes.
- S.D. Sarma, M. Freedman, and C. Nayak, Topological Quantum Computation, Physics Today, July 2006.