{"id":2226,"date":"2018-05-07T15:13:52","date_gmt":"2018-05-07T14:13:52","guid":{"rendered":"https:\/\/www.quantum-bits.org\/?p=2226"},"modified":"2022-08-12T17:16:01","modified_gmt":"2022-08-12T16:16:01","slug":"on-topological-quantum-computers","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=2226","title":{"rendered":"On topological quantum computers"},"content":{"rendered":"

We have been talking about quantum computing<\/a> for a few weeks now. We have talked about its theoretical principles (quantum entanglement<\/a>, no-cloning theorem, …) and its applications, especially in the contexts of cryptography<\/a> and complexity<\/a>. Yet, we haven’t talk much about practicalities.<\/p>\n

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\u2019s superposition, forcing it into a definite state of |0\\rangle or |1\\rangle. This effect, called decoherence<\/a>, abruptly ends quantum computations. Decoherence arises when the quantum system that encodes the qubits becomes entangled with its environment, which is a bigger, uncontrolled system.<\/p>\n

We have already introduced these problems when speaking about quantum error correction<\/a>: 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. 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. Enormous progress has been achieved in this direction, but, to quote John Preskill<\/a>: “That gives you a big overhead cost. If you want a hundred logical qubits you\u2019d need tens of thousands of physical qubits in the computer.\u201d<\/p>\n

In 1997, having this problem in mind, Russian physicist Alexei Kitaev<\/a> imagined a different approach to quantum computing: stable qubits could theoretically be formed from pairs of hypothetical particles called “non-abelian anyons”. 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 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. 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):<\/p>\n

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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. In his model, Kitaev thought that these braids could be used to form the logic gates that make up his topological quantum computer.<\/p>\n

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What is topology ?<\/strong><\/p>\n

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. 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.<\/p>\n

Intuitively, topology<\/strong> is concerned with the properties of space that are preserved under continuous deformations<\/strong> (such as stretching and bending, but not tearing or gluing).<\/p>\n

Topology is formally defined as the study of qualitative properties of certain objects (topological spaces<\/a>) that are invariant under a certain kind of transformation (continuous map<\/a>), especially those properties that are invariant under a certain kind of invertible transformation (homeomorphism<\/a>). Important topological properties include connectedness<\/a> and compactness<\/a>.<\/p>\n

A 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):<\/p>\n

\"\"As such, homeomorphism<\/strong> can be considered the most basic topological equivalence. Another one is the homotopy equivalence<\/a>. Two continuous functions from one topological space to another are called homotopic<\/strong> if one can be “continuously deformed” into the other. Such a deformation is called a homotopy between the two functions. 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):<\/p>\n

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What are braids ?<\/strong><\/p>\n

In topology, braid theory<\/b> is an abstract geometric theory studying the concept of braid. The idea is that braids can be organized into groups<\/a>, in which the group operation is “do the first braid on a set of strings, and then follow it with a second on the twisted strings”.<\/p>\n

A braid group on n<\/i><\/span> strands (denoted Bn<\/sub><\/i><\/span>) is the group whose elements are equivalence classes of n-braids<\/a>, and whose group operation is composition of braids.<\/b><\/p>\n

In this introduction we will set n<\/em> = 4 (the generalization to other values of n<\/em> is straightforward). Let’s consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. In the following illustration, these will be figured as black dots.<\/p>\n

Using our four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a braid<\/strong>.<\/p>\n

Often some strands will have to pass over or under others, and this is crucial. The following two connections are different braids:<\/p>\n

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On the other hand, two such connections which can be made to look the same by “pulling the strands” are considered the same braid:<\/p>\n

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All strands are required to move from left to right. Knots like the following are not considered braids:<\/p>\n

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Any two braids can be composed by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:<\/p>\n

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The set of all braids on four strands is denoted by B<\/i>4<\/sub><\/span>. The above composition of braids is a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which “undoes” whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its center:<\/p>\n

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What are anyons ?<\/strong><\/p>\n

In space of three or more dimensions, elementary particles<\/a> are either fermions<\/a> or bosons<\/a>: fermions obey Fermi\u2013Dirac statistics<\/a>, while bosons obey Bose\u2013Einstein statistics<\/a>.<\/p>\n

In the language of quantum mechanics<\/a>, for a 2-particle state with indistinguishable particles P1 and P2 of states |\\psi_1\\rangle and |\\psi_2\\rangle:<\/p>\n