Computing with Quantum Knots

1. 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

2. Disclaimer • The presenter does NOT claim to know anything about physics. There is a high probability (close to a 100%) that he will be winging it today. • The opinions expressed today are not necessarily those shared by the university. CSULA assumes no responsibility for any statements made by the unnamed presenter. • Any true statements found in this presentation are purely coincidental. • Names will be withheld to protect the guilty or ignorant (whichever term you feel is more appropriate for the presenter).

3. What language is being used below? • Knot - Kopoilua • Unknot- Eztu kopoilua • Hint:

4. Moore’s Law and the future of computers • In 1965 Intel co-founder Gordan Moore noted that processing power (number of transistors and speed) of computer chips was doubling each 18 months or so. This trend has continued for nearly 4 decades.

5. Increase in Processing Power=Reduction in Transistor size • Currently thousands of electrons are used to drive each transistor. As the processing power increases, the size of each transistor reduces. If Moore's law continues unabated, then each transistor is predicted to be as small as a hydrogen atom by about 2030. At that size the quantum nature of electrons in the atoms becomes significant.

6. What is a quantum computer? • Any device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. • Classical computer- amount of data is measured in bits • Quantum Computer- qubits (quantum bits)

7. Quantum computers : the next frontier • Virtually all encryption methods used for highly sensitive data are vulnerable to one quantum algorithm or another. • These encryption methods can be could be cracked given a computer capable of breaking a large number into its component factors within a reasonable length of time. • Q.C. will be able to create incredibly detailed simulations of the behavior of the universe at the tiniest scale. • It would simulate the interactions of individual molecules and atoms.

8. Basic Principle of Quantum Computation • Quantum properties of particles (such as superposition) can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform operations with this data.

9. Particle Wave Duality • We normally think of electrons, atoms and molecules as particles. But each of these objects can also behave as waves. This dual particle-wave behavior was first suggested in the 1920's by Louis de Broglie. • This duality is seen in quantum computing in the following way. A wave is spread out in space. In particular, a wave can spread out over two different places at once. This means that a particle can also exist at two places at once. This concept is called the superposition principle - the particle can be in a superposition of two places.

10. A | 0 | | 1 X/---------------B | | H A light source (H) emits a photon along a path towards a half-silvered mirror(X). This mirror splits the light reflecting half vertically toward detector A and transmitting half toward detector B. A photon, however, is a single quantized packet of light and cannot be split, so it is detected with equal probability at either A or B. Intuition would say that the photon randomly leaves the mirror in either the vertical or horizontal direction. However, quantum mechanics predicts that the photon actually travels both paths simultaneously! A photon example of superposition

11. Superposition of electrons in relation to qubits • The top figure shows how two atoms would represent the binary number 10 in classic bit form. • The bottom figure demonstrates bits in the quantum level. The electron can be in a superposition of both orbits. • Each atom represents 0 and 1 simultaneously. • Together they represent the 4 binary numbers 00, 01, 10, 11, simultaneously.

12. Quantum vs Classical • In a quantum computer, the fundamental unit of information is not binary but rather more quaternary in nature. The qubit’s property arises as a direct consequence of its adherence to the laws of quantum mechanics which differ radically from the laws of classical physics. • A qubit can exist not only in a state corresponding to the logical state 0 or 1 as in a classical bit, but also in states corresponding to a blend or superposition of these classical states. In other words a qubit can exist as a 0, a 1, or simultaneously as both 0 and 1, with a numerical coefficient representing the probability for each state. This may seem counter-intuitive b/c everyday phenomenon are governed by classical physics, not quantum mechanics-which take over at the atomic level.

13. Kibbles and bits (and qubits)

14. Power ofQuantum computers • A one bit memory can store one of the numbers 0 and 1. Likewise a two bit memory can store one of the binary numbers 00,01,10,11. But these memories can only store a single number. • Just 300 photons can store more numbers than there are atoms in the universe, and calculations can be performed simultaneously on each of these numbers!

15. Quantum braids • Quantum computers promise to greatly exceed the abilities of classical computers, but to function at all, they must have very low error rates. Achieving the required low error rates with conventional designs is far beyond current technological capabilities. • An alternative design is the so-called topological quantum computer, which would use a radically different physical system to implement quantum computation. Topological properties are unchanged by small perturbations, leading to a built-in resistance to errors. • A topological quantum computers works its calculations on braided strings. These strings are referred to as world lines (trajectories) which are representations of particles as they move through time and space. It would make use of theoretically postulated excitations called ANYONS, bizarre particlelike structures that are possible in a two-dimensional world. Experiments have recently indicated that anyons exist.

16. What is a Braid? • Concept first introduced explicitly by E. Artin • Set of n strings, all of which are attached to a horizontal bar at the top and at the bottom • We will look mostly at examples of where n = 3 aka 3-braids • Each string always heads downward as we move along from the top bar to the bottom

17. Not a braid • Each string intersects any horizontal plane between the two bars exactly once. • Each string always heads downwards as we move along any one of the strings from the top bar to the bottom bar

18. Birkhauser Definition of Braid • On the top and base of a cube, B, mark out n points, and respectively • The coordinates of B in R^3

19. Choose as follows • .. • Join these points by means of n curves in such a way that they do not mutually intersect each other. • To obtain the regular diagram of a braid, project the braid onto the yz-plane

20. Further Examples • a,b are 1-braids • c not a 1-braid • d,e are 2-braids

21. Equivalent Braids • If we can rearrange the strings in the 2 braids to look the same without passing any strings through one another or themselves while keeping the bars fixed and keeping the strings attached to the bars • Two braids whose endpoints we keep fixed can be said to be equivalent if we can continuously deform one to the other without causing any of the strings to intersect each other.

22. Same and Different Braids • Top pair are the same braid • Bottom pair are different braids

23. Connections with knots and links We can always pull the bottom bar around and glue it to the top bar, so that the resulting strings form a knot or link, called the closure of the braid. • Every braid corresponds to a particular knot or link. • Every knot or link is a closed braid. • This was first proven by J.W. Alexander in 1923

24. Knot and link invariants using braids • Braid index and Braid permutation • Braid index of a link= the least number of strings in a braid corresponding to a closed-braid representation of the link • Braid index of a link of the unknot is 1, and the braid index of the trefoil knot is 2, Figure 8 is 3 • It is often difficult to compute. • Yamada proved that the braid index of knot is equal to the least number of Seifert circles in any projection of a knot. • Bridge number

25. Braid permutations • If two braids are equivalent, their permutations are equal • The trivial braid corresponds to the identity permutation • … • ….

26. The product of a braid • It is possible to define the product for two n braids as follows • Glue the base of the cube that contains to the top face of the cube that contains (top-down)

27. Example 2 of the product of two braids

28. Are braids commutative? • Does ? • In general, this is not true • Remember that the braid permutation is an invariant

29. Are braids associative? • Although not necessarily commutative, braids are associative i.e

30. The trivial braid • The trivial braid sends • The identity element, e, is simply the trivial braid.

31. Do braids have inverses? • To find an inverse for an arbitrary alpha, consider the mirror image, alpha star of alpha. • Consider the base of the cube to be a mirror, then the mirror image is the image of alpha reflected in this mirror. • Thus

32. The set is a group • Braids are associative • The trivial braid is the identity element • Every braid has an inverse • This group is called the n-braid group

33. WORD UP!!!! • A projection of the braid can be described by listing which of the strings cross over and under each other as we move down the braid. • We can arrange it so that no two crossings in the braid occur at the same height. • This description is called the WORD of the braid

34. Advantages to this notation • A simple Type II Reidemeister move eliminates both crossings but leaves an equivalent braid. Equivalent to trivial braid. • Similarly, collapses down to nothing, meaning that the braid it represents is equivalent to the trivial braid of four vertical strings that do not cross.

35. Generators • Given any braid, we can express it as the finite product of the and • Thus, the braids are said to generate the braid group,

36. Generating . • The elements of B2 are equivalent to the two types of braids shown • Any 2-braid may be written as or

37. More on . • B2 is infinite and isomorphic to the infinite cyclic group Z • Two 2-braids are equivalent if and only if they have been twisted in the same direction the same number of times

38. The presentation of the braid group • The braid group has two types of relations called the fundamental relations: • (1) • (2) • We may write in terms of its generators and the fundamental relations

39. Visual Example -Compare B3 with S3 - There is a homomorphism B3>S3

40. Markov’s Theorem • Two braids are Markov equivalent if and only if they are related through a sequence of operations. • These operations are conjugation(M1), stabilization (M2), plus previous operations we have already seen. • As a result, each knot corresponds to only one M-equivalent class.

41. Markov and Jones • Therefore, when a braid a corresponds to a certain value, say o(a) , then if this value is o(a) is the same for any other M-equivalent braid B, it follows that this o(a) is an invariant of the knot, Ka formed from the braid a. • Enter V. Jones • Jones defined a function that was invariant under both Markov moves, M1 and M2.

42. Now to the point of all this:But first more background info • 1980s - Witten described a physical system that should calculate information about the Jones polynomial during the course of its regularly scheduled activities. • Freedman becomes convinced that certain extremely cold electron seas called quantum Hall fluids might obtain that physical system • 1997-Kitaev offers a concrete model and the two join forces • The computer would physically weave braids in space-time and then (according to Witten) nature would take over carrying out the complex Jones polynomial. • The topological computer has a built-in defense against decoherence • f.q.H.f- an exotic form of matter in which the electrons at the flat interface of 2 semiconductors form a disorganized liquid sea of electrons, and if some extra electrons are added then strange quasi-particles called anyons emerge. • Anyons can have a charge that is a fraction of a whole number-something never before seen in physics.

43. Anyon understand this ? (If you do please help me, because I ‘m dying up here) • When anyons are moved around each other, they remember (in a specific physical sense) the knottedness of the paths they followed. • In space-time, in which there is one snapshot of the surface for each moment in time - the paths don’t pass directly through each other, but simply braid around each other. • Different braids leads to different interactions b/w anyons allowing one to calculate what occurs when they collide. • fqHf- has effectively calculated numerical properties of the braid and measuring the anyons gives info about the result of this calculation.

44. Last slide of info before the pretty pictures. • Quantum Hall fluids depend on the fractional charge of the anyons. • Different fluids carry out different calculations. • If the interactions aren’t complicated, then the calculations are not very interesting. • The current goal is to find an anyonic system with complex enough (nonabelian transformations) to carry out Jones polynomial equations. • Kitaev is currently investigating of building memory for a quantum computer from a system in which electrons spinning on the corners of a hexagonal grid give rise to anyons.

45. Back to Braiding • Just two basic moves in a plane (a clockwise swap and a counterclockwise swap) generate all the possible braidings of the world lines (trajectories through spacetime) of a set of anyons

46. Computing • First, pairs of anyons are created and lined up in a row to represent the qubits of the computation. • The anyons are moved around by swapping the positions of adjacent anyons in a particular sequence. • These moves correspond to operations performed on the qubits. • Finally, pairs of adjacent anyons are brought together and measured to produce the output of the computation.

47. Building a logic gate • A logic gate known as a CNOT gate is produced by this complicated braiding of six anyons. A CNOT agate takes two input qubits and produces two output qubits. Those qubits are represented by triplets (green and blue) of so-called Fibonacci anyons. The particular style of braiding-leaving one triplet in place and moving two anyons of the other triplet around its anyons- simplified the calculations involved in designing the gate. This braiding produces a CNOT gate that is accurate to about 10to the -3.

48. References • Scientific American April 2006 • Knot Book by Adams • Knot Theory by Birkhauser • Wikipedia • www.cti.cu.edu.au

49. The Grand Finale • What have we learned today? • Absolutely nothing!!!!! • This is all theory!!!!! • I told you not to believe me, but you did not listen. So tough luck!!!! • Bang, Bang, Shoot, Shoot • I’m outta here!!!!!

50. Facts about the Braid Group • Bi • B2 • B3 • B0 and B1 are trivial;B2 is already infinite and isomorphic to the infinite cyclic group Z • B3 is a non-abelian infinite group