# Topological Quantum Computation And Quantum Com... Extra Quality

A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, 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.

## Topological Quantum Computation and Quantum Com...

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] In 2018, scientists again claimed to have isolated the required Majorana particles, but the finding was retracted in 2021. Quanta Magazine stated in 2021 that "no one has convincingly shown the existence of even a single (Majorana zero-mode) quasiparticle".[4]

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.[5] 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. 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.[5][6][7]

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.[8]

We provide a current perspective on the rapidly developing field of Majorana zero modes (MZMs) in solid-state systems. We emphasise the theoretical prediction, experimental realisation and potential use of MZMs in future information processing devices through braiding-based topological quantum computation (TQC). Well-separated MZMs should manifest non-Abelian braiding statistics suitable for unitary gate operations for TQC. Recent experimental work, following earlier theoretical predictions, has shown specific signatures consistent with the existence of Majorana modes localised at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental findings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic MZMs in solid-state systems. We also discuss fractional quantum Hall systems (the 5/2 state), which have been extensively studied in the context of non-Abelian anyons and TQC. We describe proposed schemes for carrying out braiding with MZMs as well as the necessary steps for implementing TQC.

It is useful, at this point, to make a distinction between the two computational uses of braiding, for unitary gates and for projective measurement. Braiding-based gates can operate in essentially the same way for quasiparticles in a topological phase and for defects in an ordered (quasi-topological) state. However, braiding-based measurement procedures rely on interferometry, which is only possible if the motional degrees of freedom of the objects being braided are sufficiently quantum mechanical. This will be satisfied by quasiparticles at sufficiently low temperatures, but the motion of defects is classical at any relevant temperature except, possibly, in some special circumstances.

(a) With a two-point contact interferometer in a quantum Hall state, it is possible to detect topological charge and, thereby, read out a qubit by measuring electrical conductance (taken from ref. 3). (b) In a long Josephson junction with two arms, different paths for Josephson vortices can interfere, thereby enabling the detection of topological charge through electrical measurement (taken from ref. 18). Conversely, if two MZMs, 71 and 72, are brought close together, then the right-hand-side of Equation (3) may no longer be small.

Domain walls in nanowires are always classical objects whose position is determined by gate voltages. Abrikosov vortices in 2D topological superconductors are similarly classical in their motion. However, Josephson vortices, whose cores lie in the insulating barriers between superconducting regions, may move quantum mechanically, thereby making possible an interferometer such as that depicted in Figure 4. Moreover, the fermionic excitations at the edge of a superconductor are light and can be used to detect the presence or absence of a MZM (but not to detect the quantum information encoded in a collection of MZMs).

Our results suggest a promising new avenue towards scalable quantum computation and also reveal deep links between topics in statistical mechanics, field theory, quantum Hall physics, and superconductivity.

Following a suggestion of A. Kitaev, we explore the connection between fault-tolerant quantum computation and nonabelian quantum statistics in two spatial dimensions. A suitably designed spin system can support localized excitations (quasiparticles) that exhibit long-range nonabelian Aharonov-Bohm interactions. Quantum information encoded in the charges of the quasiparticles is highly resistant to decoherence, and can be reliably processed by carrying one quasiparticle around another. If information is encoded in pairs of quasiparticles, then the Aharonov-Bohm interactions can be adequate for universal fault-tolerant quantum computation.

In topological quantum computation one aims to make use of quantum systems described by topological quantum field theory for quantum computation, the idea being that the defining invariance of TQFTs under small deformations implements an intrinsic fault tolerance of the quantum computer against noise and decoherence (see also at quantum error correction).

The standard paradigm for potentially realizing topological quantum computation in practice (Kitaev 03, Freedman, Kitaev, Larsen & Wang03) considers adiabatic braiding of defect anyons in effectively 2-dimensional quantum materials, such as in the quantum Hall effect and effectively described by some kind of Chern-Simons theory/Reshetikhin-Turaev theory: 041b061a72