![]() In topology, braid theory is an abstract geometric theory studying the concept of braid. In the following figure, the two dashed paths shown above are homotopic relative to their endpoints and the animation represents one possible homotopy (from Wikipedia): Such a deformation is called a homotopy between the two functions. Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other. Important topological properties include connectedness and compactness.Ī traditional joke is that a topologist cannot distinguish a coffee mug from a donut (it is said that they are topologically equivalent), since a sufficiently pliable donut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle (from Wikipedia):Īs such, homeomorphism can be considered the most basic topological equivalence. Topology is formally defined as the study of qualitative properties of certain objects ( topological spaces) that are invariant under a certain kind of transformation ( continuous map), especially those properties that are invariant under a certain kind of invertible transformation ( homeomorphism). Intuitively, topology is concerned with the properties of space that are preserved under continuous deformations (such as stretching and bending, but not tearing or gluing). For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. In his model, Kitaev thought that these braids could be used to form the logic gates that make up his topological quantum computer. Small, cumulative perturbations can cause quantum states based on spin or polarization to decohere and introduce errors in the computation, but such small perturbations do not change the braids’ topological properties. If their paths are thought of as shoelaces meandering through space and time, then when the particles rotate around each other, the shoelaces tie in knots (illustration from Wikipedia): In this approach, the state of a pair of non-abelian anyons is determined by its topology: how the paths of the two anyons have been braided around each other. He imagined a topological quantum computer built on anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (one temporal plus two spatial dimensions). In 1997, having this problem in mind, Russian physicist Alexei Kitaev imagined a different approach to quantum computing: stable qubits could theoretically be formed from pairs of hypothetical particles called “non-abelian anyons”. If you want a hundred logical qubits you’d need tens of thousands of physical qubits in the computer.” ![]() Enormous progress has been achieved in this direction, but, to quote John Preskill: “That gives you a big overhead cost. ![]() For this, a quantum computer requires each unit of information to be shared among an elaborate network of many qubits cleverly arranged to prevent an environmental disturbance of one from leading to the collapse of them all. We have already introduced these problems when speaking about quantum error correction: to fight decoherence, one has to completely isolate the computer from its environment, and careful eliminate of noise, and protocols for quantum correction of unavoidable errors. Decoherence arises when the quantum system that encodes the qubits becomes entangled with its environment, which is a bigger, uncontrolled system. ![]() This effect, called decoherence, abruptly ends quantum computations. The outstanding problem with entangled superpositions of spinning electrons, polarized photons or most other particles that might serve as qubits is that they are very unstable: even a slight interaction with the environment will collapse qubit’s superposition, forcing it into a definite state of or. Yet, we haven’t talk much about practicalities. We have talked about its theoretical principles ( quantum entanglement, no-cloning theorem, …) and its applications, especially in the contexts of cryptography and complexity. We have been talking about quantum computing for a few weeks now.
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