Topological quantum computer
Encyclopedia
A topological quantum computer is a theoretical quantum computer
that employs two-dimensional quasiparticle
s called anyon
s, whose world line
s cross over one another to form braids
in a three-dimensional spacetime
(i.e., one temporal plus two spatial dimensions). These braids form the logic gate
s that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. The smallest perturbations can cause a quantum particle to decohere
and introduce errors in the computation, but such small perturbations do not change the topological properties of the braids
. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) simply bumping into a wall. While the elements of a topological quantum computer originate in a purely mathematical realm, recent experiments indicate these elements can be created in the real world using semiconductor
s made of gallium arsenide near absolute zero
and subjected to strong magnetic fields.
s are quasiparticles in a two-dimensional space. Anyons are not strictly fermion
s or boson
s, but do share the characteristic of fermions in that they cannot occupy the same state. Thus, the world lines of two anyons cannot cross or merge. This allows braids to be made that make up a particular circuit. In the real world, anyons form from the excitations in an electron gas in a very strong magnetic field, and carry fractional units of magnetic flux in a particle-like manner. This phenomenon is called the fractional quantum Hall effect
. The electron "gas" is sandwiched between two flat plates of gallium arsenide, which create the two-dimensional space required for anyons, and is cooled and subjected to intense transverse magnetic fields.
When anyons are braided, the transformation of the quantum state of the system
depends only on the topological class of the anyons' trajectories (which are classified
according to the braid group
). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories.
In 2005, Sankar Das Sarma
, Michael Freedman
, and Chetan Nayak proposed a quantum Hall device which would realize a topological qubit.
In a key development for topological quantum computers, in 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou were said to have created the first experimental evidence for using fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. It should also be noted that nonabelian
anyons, a species required for topological quantum computers, have yet to be experimentally confirmed.
The original proposal for topological quantum computation is due to Alexei Kitaev
in 1997.
model and to the quantum Turing machine
model. That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.
They found that conventional quantum computer device, given a flawless (error-free) operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by edge states of fractional quantum Hall effect. This make models one dimensional anyons important. In one space dimension anyons are defined algebraically.
Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...
that employs two-dimensional quasiparticle
Quasiparticle
In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
s called anyon
Anyon
In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
s, whose world line
World line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
s cross over one another to form braids
Braid theory
In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...
in a three-dimensional spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
(i.e., one temporal plus two spatial dimensions). These braids form the logic gate
Logic gate
A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and...
s that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. The smallest perturbations can cause a quantum particle to decohere
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
and introduce errors in the computation, but such small perturbations do not change the topological properties of the braids
Braid theory
In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...
. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) simply bumping into a wall. While the elements of a topological quantum computer originate in a purely mathematical realm, recent experiments indicate these elements can be created in the real world using semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s made of gallium arsenide near absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
and subjected to strong magnetic fields.
Introduction
AnyonAnyon
In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
s are quasiparticles in a two-dimensional space. Anyons are not strictly fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s or boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s, but do share the characteristic of fermions in that they cannot occupy the same state. Thus, the world lines of two anyons cannot cross or merge. This allows braids to be made that make up a particular circuit. In the real world, anyons form from the excitations in an electron gas in a very strong magnetic field, and carry fractional units of magnetic flux in a particle-like manner. This phenomenon is called the fractional quantum Hall effect
Fractional quantum Hall effect
The fractional quantum Hall effect is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations...
. The electron "gas" is sandwiched between two flat plates of gallium arsenide, which create the two-dimensional space required for anyons, and is cooled and subjected to intense transverse magnetic fields.
When anyons are braided, the transformation of the quantum state of the system
depends only on the topological class of the anyons' trajectories (which are classified
according to the braid group
Braid group
In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
). Therefore, the quantum information which is stored in the state of the system is impervious to small errors in the trajectories.
In 2005, Sankar Das Sarma
Sankar Das Sarma
Sankar Das Sarma is an Indian-born American theoretical condensed matter physicist, who has worked in the areas of strongly correlated materials, graphene, semiconductor physics, low-dimensional systems, topological matter, quantum Hall effect, nanoscience, spintronics, collective properties of...
, Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
, and Chetan Nayak proposed a quantum Hall device which would realize a topological qubit.
In a key development for topological quantum computers, in 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou were said to have created the first experimental evidence for using fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. It should also be noted that nonabelian
Nonabelian group
In mathematics, a non-abelian group, also sometimes called a non-commutative group, is a group in which there are at least two elements a and b of G such that a * b ≠ b * a...
anyons, a species required for topological quantum computers, have yet to be experimentally confirmed.
The original proposal for topological quantum computation is due to Alexei Kitaev
Alexei Kitaev
Alexei Kitaev is a professor of physics and computer science at the California Institute of Technology. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at the Landau Institute for Theoretical Physics. For...
in 1997.
Topological vs. standard quantum computer
Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuitQuantum circuit
In quantum information theory, a quantum circuit is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an n-bit register...
model and to the quantum Turing machine
Quantum Turing machine
A quantum Turing machine , also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer. It provides a very simple model which captures all of the power of quantum computation. Any quantum algorithm can be expressed formally as a particular quantum...
model. That is, any of these models can efficiently simulate any of the others. Nonetheless, certain algorithms may be a more natural fit to the topological quantum computer model. For example, algorithms for evaluating the Jones polynomial were first developed in the topological model, and only later converted and extended in the standard quantum circuit model.
Computations
To live up to its name, a topological quantum computer must provide the unique computation properties promised by a conventional quantum computer design, which uses trapped quantum particles. Fortunately in 2002, Michael H. Freedman along with Zhenghan Wang, both with Microsoft, and Michael Larsen of Indiana University proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do.They found that conventional quantum computer device, given a flawless (error-free) operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by edge states of fractional quantum Hall effect. This make models one dimensional anyons important. In one space dimension anyons are defined algebraically.