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A **topological quantum computer** is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates 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. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. 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) bumping into a wall.

- Introduction
- Topological vs. standard quantum computer
- Computations
- Error correction and control
- Example: Computing with Fibonacci Anyons
- State preparation
- Gates
- See also
- References
- Further reading

While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.

Anyons are quasiparticles in a two-dimensional space. Anyons are neither fermions nor bosons, but like fermions, they cannot occupy the same state. Thus, the world lines of two anyons cannot intersect or merge, which allows their paths to form stable braids in space-time. Anyons can form from excitations in a cold, two-dimensional electron gas in a very strong magnetic field, and carry fractional units of magnetic flux. This phenomenon is called the fractional quantum Hall effect. In typical laboratory systems, the electron gas occupies a thin semiconducting layer sandwiched between layers of aluminium gallium arsenide.

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.^{ [1] } In 2005, Sankar Das Sarma, Michael Freedman, and Chetan Nayak proposed a quantum Hall device that would realize a topological qubit. In a key development for topological quantum computers, in 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou claimed to have created and observed the first experimental evidence for using a fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. Non-abelian anyons, a species required for topological quantum computers, have yet to be experimentally confirmed. Possible experimental evidence has been found,^{ [2] } but the conclusions remain contested.^{ [3] }

Topological quantum computers are equivalent in computational power to other standard models of quantum computation, in particular to the quantum circuit model and to the quantum Turing machine model.^{ [4] } 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.

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 2000, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do, and vice versa.^{ [4] }^{ [5] }^{ [6] }

They found that a conventional quantum computer device, given an 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 the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.

Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. Simulating the dynamics of a topological quantum computer may be a promising method of implementing fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf, Harrington, and Goyal have studied one model, with promising simulation results.^{ [7] }

One of the prominent examples in topological quantum computing is with a system of Fibonacci anyons. In the context of conformal field theory, fibonacci anyons are described by the Yang–Lee model, the SU(2) special case of the Chern–Simons theory and Wess–Zumino–Witten models.^{ [8] } These anyons can be used to create generic gates for topological quantum computing. There are three main steps for creating a model:

- Choose our basis and restrict our Hilbert space
- Braid the anyons together
- Fuse the anyons at the end, and detect how they fuse in order to read the output of the system.

Fibonacci anyons are defined by three qualities:

- They have a topological charge of . In this discussion, we consider another charge called which is the ‘vacuum’ charge if anyons are annihilated with each-other.
- Each of these anyons are their own antiparticle. and .
- If brought close to each-other, they will ‘fuse’ together in a nontrivial fashion. Specifically, the ‘fusion’ rules are:
- Many of the properties of this system can be explained similarly to that of two spin 1/2 particles. Particularly, we use the same tensor product and direct sum operators.

The last ‘fusion’ rule can be extended this to a system of three anyons:

Thus, fusing three anyons will yield a final state of total charge in 2 ways, or a charge of in exactly one way. We use three states to define our basis.^{ [9] } However, because we wish to encode these three anyon states as superpositions of 0 and 1, we need to limit the basis to a two-dimensional Hilbert space. Thus, we consider only two states with a total charge of . This choice is purely phenomenological. In these states, we group the two leftmost anyons into a 'control group', and leave the rightmost as a 'non-computational anyon'. We classify a state as one where the control group has total 'fused' charge of , and a state of has a control group with a total 'fused' charge of . For a more complete description, see Nayak.^{ [9] }

Following the ideas above, adiabatically braiding these anyons around each-other will result in a unitary transformation. These braid operators are a result of two subclasses of operators:

- The F matrix
- The R matrix

The R matrix can be conceptually thought of as the topological phase that is imparted onto the anyons during the braid. As the anyons wind around each-other, they pick up some phase due to the Aharonov-Bohm effect.

The F matrix is a result of the physical rotations of the anyons. As they braid between each-other, it is important to realize that the bottom two anyons—the control group—will still distinguish the state of the qubit. Thus, braiding the anyons will change which anyons are in the control group, and therefore change the basis. We evaluate the anyons by always fusing the control group (the bottom anyons) together first, so exchanging which anyons these are will rotate the system. Because these anyons are non-abelian, the order of the anyons (which ones are within the control group) will matter, and as such they will transform the system.

The complete braid operator can be derived as:

In order to mathematically construct the F and R operators, we can consider permutations of these F and R operators. We know that if we sequentially change the basis that we are operating on, this will eventually lead us back to the same basis. Similarly, we know that if we braid anyons around each-other a certain number of times, this will lead back to the same state. These axioms are called the pentagonal and hexagonal axioms respectively as performing the operation can be visualized with a pentagon/hexagon of state transformations. Although mathematically difficult,^{ [10] } these can be approached much more successfully visually.

With these braid operators, we can finally formalize the notion of braids in terms of how they act on our Hilbert space and construct arbitrary universal quantum gates.^{ [11] }

**Quantum entanglement** is a physical phenomenon that occurs when a pair or group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

In physics, an **anyon** is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as *abelian* or *non-abelian*. Abelian anyons play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected, although this is an active area of research.

In quantum computing, a **quantum algorithm** is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement.

The **fractional quantum Hall effect** (**FQHE**) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of . It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer, and Daniel Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations" However, Laughlin's explanation was a phenomenological guess and only applies to fillings where is an odd integer. The microscopic origin of the FQHE is a major research topic in condensed matter physics.

The **Peres–Horodecki criterion** is a necessary condition, for the joint density matrix of two quantum mechanical systems and , to be separable. It is also called the **PPT** criterion, for *positive partial transpose*. In the 2x2 and 2x3 dimensional cases the condition is also sufficient. It is used to decide the separability of mixed states, where the Schmidt decomposition does not apply.

In quantum mechanics, **separable quantum states** are states without quantum entanglement.

In physics, **topological order** is a kind of order in the zero-temperature phase of matter. Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders cannot change into each other without a phase transition.

A **Majorana fermion**, also referred to as a **Majorana particle**, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

**Sankar Das Sarma** is an India-born American theoretical condensed matter physicist, who has worked in the broad research topics of theoretical physics, condensed matter physics, statistical mechanics, and quantum information. He has been a member of the Department of Physics at University of Maryland, College Park since 1980.

The **one-way** or **measurement-based quantum computer** (**MBQC**) is a method of quantum computing that first prepares an entangled *resource state*, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.

The **toric code** is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice It is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—*Z*_{2} topological order (first studied in the context of *Z*_{2} spin liquid in 1991). The toric code can also be considered to be a *Z*_{2} lattice gauge theory in a particular limit. It was introduced by Alexei Kitaev.

In condensed matter physics, a **quantum spin liquid** is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

In quantum many-body physics, **topological degeneracy** is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations.

The **Kibble–Zurek mechanism** (**KZM**) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.

The term **Dirac matter** refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter are Graphene, topological insulators, Dirac semimetals, Weyl semimetals, various high-temperature superconductors with -wave pairing and liquid Helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the Dirac matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

A **fracton** is an emergent topological quasiparticle excitation which is immobile when in isolation. Many theoretical systems have been proposed in which fractons exist as elementary excitations. Such systems as known as fracton models. Fractons have been identified in various CSS codes as well as in symmetric tensor gauge theories.

**Anyon fusion** is the process by which multiple anyons behave as one larger composite anyon. Anyon fusion is essential to understanding the physics of non-abelian anyons and how they can be used in quantum information.

In quantum computing, a *qubit* is a unit of information analogous to a bit in classical computing, but it is affected by quantum mechanical properties such as superposition and entanglement which allow qubits to be in some ways more powerful than classical bits for some tasks. Qubits are used in quantum circuits and quantum algorithms composed of quantum logic gates to solve computational problems, where they are used for input/output and intermediate computations.

**Magnetic topological insulators** are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

In many-body physics, the problem of **analytic continuation** is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from quantum Monte Carlo simulations, which often compute Green function values only at imaginary-times or Matsubara frequencies.

- ↑ Castelvecchi, Davide (July 3, 2020). "Welcome anyons! Physicists find best evidence yet for long-sought 2D structures".
*Nature*. Retrieved September 23, 2020.Simon and others have developed elaborate theories that use anyons as the platform for quantum computers. Pairs of the quasiparticle could encode information in their memory of how they have circled around one another. And because the fractional statistics is 'topological' — it depends on the number of times one anyon went around another, and not on slight changes to its path — it is unaffected by tiny perturbations. This robustness could make topological quantum computers easier to scale up than are current quantum-computing technologies, which are error-prone.

- ↑ Willet, R. L. (January 15, 2013). "Magnetic field-tuned Aharonov–Bohm oscillations and evidence for non-Abelian anyons at ν = 5/2".
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- ↑ Explicit braids that perform particular quantum computations with Fibonacci anyons have been given by Bonesteel, N. E.; Hormozi, L.; Zikos, G.; Simon, S. H.; West, K. W. (2005). "Braid Topologies for Quantum Computation".
*Physical Review Letters*.**95**(14): 140503. arXiv: quant-ph/0505065 . Bibcode:2005PhRvL..95n0503B. doi:10.1103/PhysRevLett.95.140503. PMID 16241636.

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