Classical computation is based on algorithms that manipulate binary variables (bits), which convey the information being processed. Such algorithms are formed by logic units (logic gates), which are basic functions of binary variables yielding an output that will be used in the next step of the algorithm, and so on. Algorithms of classical computation make extensive use of irreversible gates, which can only be used in one direction and, consequently, loose information. Quantum computation, conversely, uses the quantum states of a given Hilbert space (qubits) to store the information [290, 291, 292]. Then, it takes advantage of the fact that these states have a unitary evolution to build data-processing algorithms, which only possess reversible, unitary logic gates. An important point in this scheme is that coherence of the quantum states storing the information is a crucial requirement; otherwise, the information is lost. Nevertheless, we know that quantum systems rapidly loose their coherence as we increase their size. A clever method of coherence protection, therefore, must be devised if we want to have a quantum computer operating on a human scale. For this reason, quantum computation, since its inception, has been always related to the principles of quantum mechanics. The main method of coherence protection involves excitations having the so-called non-abelian statistics. Such excitations were shown to occur in the ν = 5 / 2 plateaus of systems presenting the quantum Hall effect, which is described by the Moore–Read Pfaffian wave-function. Such a state is associated to a Landau–Ginzburg type field theory consisting in the level 2 non-abelian SU(2) Chern–Simons theory in the same way the odd denominator plateaus described by the Laughlin wave-function are associated to the Zhang–Hansson–Kivelson abelian Chern–Simons–Landau–Ginzburg theory. We have shown that the vortex excitations of this theory correspond to the (abelian) anyon excitations of Laughlis's wave-function. In an analogous way, the vortex excitations of the non-abelian Chern–Simons theory are associated to the Pfaffian wave-function excitations, which present non-abelian statistics. Such gapped bulk excitations produce gapless Dirac excitations on the edges, which may combine in specific ways to reproduce the non-abelian statistics exhibited by the bulk vortex states.
Classical computation is based on algorithms that manipulate binary variables (bits), which convey the information being processed. Such algorithms are formed by logic units (logic gates), which are basic functions of binary variables yielding an output that will be used in the next step of the algorithm, and so on. Algorithms of classical computation make extensive use of irreversible gates, which can only be used in one direction and, consequently, loose information. Quantum computation, conversely, uses the quantum states of a given Hilbert space (qubits) to store the information [290, 291, 292]. Then, it takes advantage of the fact that these states have a unitary evolution to build data-processing algorithms, which only possess reversible, unitary logic gates. An important point in this scheme is that coherence of the quantum states storing the information is a crucial requirement; otherwise, the information is lost. Nevertheless, we know that quantum systems rapidly loose their coherence as we increase their size. A clever method of coherence protection, therefore, must be devised if we want to have a quantum computer operating on a human scale. For this reason, quantum computation, since its inception, has been always related to the principles of quantum mechanics. The main method of coherence protection involves excitations having the so-called non-abelian statistics. Such excitations were shown to occur in the ν = 5 / 2 plateaus of systems presenting the quantum Hall effect, which is described by the Moore–Read Pfaffian wave-function. Such a state is associated to a Landau–Ginzburg type field theory consisting in the level 2 non-abelian SU(2) Chern–Simons theory in the same way the odd denominator plateaus described by the Laughlin wave-function are associated to the Zhang–Hansson–Kivelson abelian Chern–Simons–Landau–Ginzburg theory. We have shown that the vortex excitations of this theory correspond to the (abelian) anyon excitations of Laughlis's wave-function. In an analogous way, the vortex excitations of the non-abelian Chern–Simons theory are associated to the Pfaffian wave-function excitations, which present non-abelian statistics. Such gapped bulk excitations produce gapless Dirac excitations on the edges, which may combine in specific ways to reproduce the non-abelian statistics exhibited by the bulk vortex states.
We naturally obtain the NOT and CNOT logic gates, which are key pieces of quantum computing algorithms, in the framework of the non-Abelian Chern–Simons–Higgs theory in two spatial dimensions. For that, we consider the anyonic quantum vortex topological excitations and show that Majorana anyons, namely, self-adjoint combinations of these vortices and anti-vortices, have in general non-Abelian statistics. The associated unitary monodromy braiding matrices become the required logic gates in the special case when the vortex spin is s = 1/4, which corresponds to the case of Ising non-Abelian anyons, found in different quantum computing systems. We explicitly construct the vortex field operators, show that they carry both magnetic flux and charge and obtain their Euclidean correlation functions by using the method of quantization of topological excitations, which is based on the order-disorder duality. These correlators are in general multivalued, the number of sheets being determined by the vortex spin. This, by its turn, is proportional to the vacuum expectation value of the Higgs field and therefore can be tuned by both the free parameters of the Higgs potential and the temperature.
Quantum computers have shown great promise as a resource providing exponential speedup over classical computers for certain problems.Akin to their classical counterparts, quantum computers will be prone to errors.There are two major sources of errors.First, the most formidable source of error is decoherence.To perform quantum computation, quantum information must be stored, processed and read out while protecting it from the debilitating effects of decoherence.Second, even if the system is protected from decoherence, it is almost certain that all the operations performed on the quantum information during its processing will be imperfect.These errors will accumulate over the duration of the computation, eventually causing failure.Thus, for a quantum computing scheme to be feasible, it needs to be fault-tolerant.What this means is that the quantum computer can still perform its task effectively while its components are imperfect.A major breakthrough in this field is the threshold theorem, which asserts that ideal quantum circuits can be simulated efficiently by noisy ones provided the error rates of individual gates are below a certain threshold [1,2,3].Topological quantum computing is an approach to fault-tolerant quantum computation, where the protection from errors occurs not from active intervention, but at a hardware level.The advantage of this approach is that it is robust to localized imperfections.First envisioned by Alexei Kitaev [4], this scheme makes use of a non-abelian exchange statistics of elementary excitations of some two-dimensional quantum many-body system.These excitations form the degenerate ground-space of this topologically ordered system, and all other states are separated from this space by a finite excitation gap.In order to perform computation in this scheme, one needs to create pairs of these non-abelian excitations (called non-abelian anyons) from vacuum, separating them spatially, transporting them around each other to implement the logical gates and finally, fusing them together for measurements.In order for this scheme to be viable, there are two conditions that must be fulfilled.First, there could be errors due to quantum tunneling between the non-abelian anyons.These processes can take place even at zero temperature.But, the amplitude of these processes are exponentially suppressed with the spatial separation of these particles and goes as e -L/l 0 , where L is the spatial separation and l 0 is some characteristic length scale of topological ordering of the system.Thus, to prevent such tunneling during quantum information processing and storage, L has to be much larger than l 0 .Second, at finite temperature, undesired thermally excited quasiparticles can lead to errors.The generation rate of these particles are exponentially suppressed by a Boltzmann factor e -∆/T , where ∆ is the excitation energy and T is the temperature.Therefore, the temperature must be low enough so that there are sufficiently low number of these undesired excitations.There are several physical systems where these excitations could, in principle, be found, manipulated and measured.Non-abelian anyons were proposed as elementary excitations of the ν = 5/2 state in the fractional quantum hall effect [5].The braiding statistics of these particles and their possible use in quantum computing has been investigated thoroughly.For more information on this topic, the reader is invited to consult [6,7].In a pathbreaking work [8], it was proposed that the non-abelian statistics of the excitations of the ν = 5/2 state are shared by the Majorana excitations of a 2D spinless p + ip superconductor.Since then, there have been other theoretical models [9,10] which also support these non-abelian excitations.Moreover, several experimental realizations of these Majorana excitations in solid state systems have also been proposed (for a comprehensive review, see [11]).In this lecture note, we will focus on implementation of topological quantum computation with
If one could be built, a quantum computer would be capable of storing and manipulating quantum states with sufficient accuracy to carry out computations that no classical computer can do (most notably factoring integers in polynomial time). The greatest obstacle to building such a device is the problem of error and decoherence. Classical computers can exploit the physical robustness of ordered states to protect classical information (as in, for example, the magnetically ordered state of a hard drive). Remarkably, a type of quantum order known as topological order can, in principle, play the same role for quantum information. The best studied topologically ordered states are quantum Hall states. These states arise when a two-dimensional electron gas is placed in a strong magnetic field and cooled to low temperatures. Under the right conditions, the electrons condense into an incompressible quantum liquid whose excitations are particle-like objects with fractional charge (quasiparticles). Certain quantum Hall states are thought to be non Abelian. This means that when a finite number of quasiparticles are present and fixed in space there is a low energy Hilbert space with finite dimension, rather than a unique state. Unitary operations can then be carried out on this Hilbert space by adiabatically dragging quasiparticles around one another so that their world-lines sweep out braids in 2+1 dimensional space time. A quantum computer which stores quantum information in this Hilbert space and computes by braiding is known as a topological quantum computer. In this thesis I review our work on determining precisely how one would carry out a computation on a topological quantum computer. I focus on the so-called Fibonacci anyons--quasiparticles which may exist in the experimentally observed quantum Hall state at Landau level filling fraction ν = 12/5. I give explicit prescriptions for encoding qubits (quantum bits) using Fibonacci anyons, and show how one would carry out a universal set of quantum gates (the quantum analogs of Boolean logic gates) by braiding them. I then focus in particular on my work developing algorithms for performing brute force searches over the space of braids to find braids which produce unitary operations close to any desired operation. These brute force searches are a crucial part of our quantum gate construction, and I show that by using a so-called "load balanced" bidirectional search I can find braids which approximate any desired operation to an accuracy of 1 part in 10 5 . I then turn to my work calculating the Berry's phase obtained when quasiparticles are moved around one another in the Moore-Read state, a non Abelian state generally believed to describe the ν = 5/2 quantum Hall effect. This work is done using variational Monte Carlo, a method which allows one to numerically evaluate the Berry's phase for finite size systems. By exploiting certain properties of the Moore-Read state I have been able to study systems consisting of as many as 150 electrons. In so doing I have verified the conjectured connection between the Berry's phase produced by physically moving quasiparticles around one another and the mathematical phase one obtains by simply analytically continuing the quasiparticle coordinates. An added benefit of these calculations is that we can deduce the length scale which determines the size of the quasiparticles. This length scale dictates how far apart the quasiparticles must be in order to prevent errors when they are used for topological quantum computation.
Certain exotic states of matter, so-called non-Abelian states, have the potential to provide a natural medium for the storage and manipulation of quantum information. In these states, localized particle-like excitations (quasiparticles) possess quantum numbers which are in many ways analogous to ordinary spin quantum numbers. However, unlike ordinary spins, the quantum information associated with these quantum numbers is stored globally, throughout the entire system, and so is intrinsically protected against decoherence. Quantum computation can then be carried out by dragging these quasiparticles around one another so that their trajectories sweep out world-lines in 2+1-dimensional space-time. The resulting computation depends only on the topology of the braids formed by these world-lines, and thus is robust against error. In these lectures I will review the theory of non-Abelian states, including the necessary mathematical background for describing the braiding of their quasiparticles. I will then introduce the basic ideas behind topological quantum computation and demonstrate explicitly that certain non-Abelian quasiparticles can indeed by used for universal quantum computation by showing how any quantum algorithm can be "compiled" into a braiding pattern for them. I will also discuss the most promising experimental systems for realizing non-Abelian quasiparticles, focusing primarily on fractional quantum Hall states.
A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.
We show that one of the fundamental operations of topological quantum computation, namely the non-Abelian braiding of identical particles, can be physically realized in a general system of Dirac quasiparticles in 1 + 1D. Our method is based on the study of the analytic structure of the different Euclidean correlation functions of Dirac fields, which are conveniently expressed as functions of a complex variable. When the Dirac field is an (Abelian) anyon with statistics parameter s (2 s not an integer), we show that the associated Majorana states of such a field present non-Abelian statistics. The explicit form of the unitary, non-commuting (monodromy) matrices generated upon braiding is derived as a function of s and is shown to satisfy the Yang–Baxter algebra. For the special case of s = 1/4, we show that the braiding matrices become the logic gates NOT, CNOT,… required in the algorithms of universal quantum computation. We suggest that maybe polyacetylene, alternately doped with alkali and halogen atoms, is a potential candidate for a physical material realization of the system studied here.