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
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
The superposition principle of quantum mechanics implies that the amount of classical (i.e., non-quantum) information needed to describe a quantum system is in general exponentially large in the size of the system. This makes simulating quantum physics on a classical computer quickly infeasible, even for moderately-sized systems and using the best present-day supercomputers in the world. This is unfortunate, since systems for which quantum effects are relevant are of interest in many areas of science and engineering, ranging from drug design to materials research. In 1982, Richard Feynman gave birth to the idea of a quantum computer, when he realized that simulating quantum physics could be achieved much more efficiently with a computer that was itself a quantum system. Since then, the number of problems for which a quantum computer is known to enjoy an advantage over a classical computer has steadily increased, ranging now far beyond the task of simulating quantum physics. A quantum computation of any interest will need to create complicated superposition states, for otherwise it would not achieve anything that could not be simulated on a classical computer. These superposition states, however, are highly fragile, and this fragility is the main obstacle that the scientific community faces on the road towards a useful quantum computer. Quantum computing will only be possible if it can be performed in a fault-tolerant way. It is due to seminal work by Alexei Kitaev from 1997 that the modern theory of quantum fault-tolerance is closely related to the term topology. Topology, as a sub-field of mathematics, is concerned with the properties of space that are preserved under continuous deformations. Topological order, a purely quantum phenomenon, refers to states which cannot be distinguished or evolved into each other locally, yet are globally distinct. The information that distinguishes between these states is stored in non-local degrees of freedom. If it were possible to store quantum information in these non-local degrees of freedom, this would be hugely attractive from a practical perspective, as it would mean that the information stored this way is immune to many forms of local errors. Two-dimensional topologically ordered states support excitations known as anyons. These are exotic quasi-particles that defy the dichotomy between fermions and bosons that applies to quantum particles in three spatial dimensions. In a topological quantum computer, the non-locally stored quantum information is processed by braiding these anyons around each other. Such a computation would be insensitive to small perturbations of the path along which anyons are braided, but would only depend on its topological properties. An open problem is whether topological order can persist at finite temperature. Equivalently, one can ask whether it is possible to build a system in which quantum information can be stored in a stable manner for arbitrarily long times, without performing active error correction and despite constant influence of a thermal environment. Such a system would constitute a self-correcting quantum memory and would extend the concept of a hard disk drive to the quantum realm. Whether nature allows for such a system to be built is of tremendous interest from both a fundamental and a practical perspective. Two-dimensional topologically ordered systems – those hosting anyons – do not qualify as self-correcting quantum memories; any finite temperature corrupts them in a time that is independent of the size of the system. In the present thesis, we propose and study systems in which the thermal stability of a simple toy model of an anyonic system, Kitaev’s toric code, is enhanced in various ways. We consider coupling it to optical cavity modes, to bosonic particles, or to a ferromagnet. These auxiliary systems then induce long-ranged interactions between the anyons, which allow to increase the finite-temperature lifetime of the stored quantum information arbitrarily by increasing the size of the system. While a topological quantum computer is naturally immune to many forms of imperfections and perturbations, accidental creation of anyonic quasi-particles due to coupling to an external environment is a form of error that requires active correction. This problem has seen surprisingly little attention until recently and is still poorly understood. In this thesis, we develop algorithms that are able to perform this task and provide the first proofs of its in-principle feasibility. Furthermore, we develop a system with the rare property that it both supports anyons that can be used for topological quantum computing, and allows for error correction with well-established techniques. Historically, the first proposals for building a fault-tolerant quantum \ncomputer involved an array of qubits (quantum bits) and a set of elementary operations that can be performed on individual and pairs of qubits. Fault-tolerant qubit-based quantum computing has been inspired tremendously from the insights gained in the study of topological quantum information processing. The surface code, which combines a qubit-based architecture with topological methods, is now at the forefront of the quest towards a fault-tolerant, scalable quantum computer. In the surface code, a large number of measurements are performed on a continuous basis to get some information about what errors have occurred. This information then needs to be converted by a classical algorithm into a prescription for performing error correction. In this thesis, we develop such an algorithm that, at the time of its publication, was the best efficient algorithm known for the surface code. Error correction for qubit-based quantum computers is typically studied with simplistic error models in which the errors on each qubit are independent from each other. We study what kinds of spatial and temporal correlations between the errors arise when a surface code is coupled to a typical model of an environment, and how they affect its correctability.
In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.
Topological quantum computation relies on the concepts of non-Abelian anyons and topological superconductors, holding great promise for noise-protected computation scalable since it stores quantum information coherently, irrespective of position. Included in this review are fundamentals like Majorana fermion zero modes, current innovations at Microsoft regarding nanowires, along with challenges and plans for scaling quantum computation at the level of million-qubits by the year 2030. It employs the use of anyons, a kind of quasi particle that exists in a two-dimensional setting. The world lines of the anyons cross each other in a braid fashion in the three-dimensional spacetime (one time-dimensional and two space-dimensional). The braids correspond to the logic gates in a computer. The one big advantage that the use of the quantum braid has over the trapped quantum particles has to do with stability. A small disturbance in a system may cause the decoherence of the quantum states and errors in calculation in a regular quantum computer, but in the braids, the disturbance does not affect the topological properties because it is like the difference between cutting and reconnecting a string to create a new braid compared to a ball (standing in place for a regular trapped quantum particle in four-dimensional spacetime) hitting a wall. 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. 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.
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.
The construction of a large-scale fault-tolerant quantum computer is an outstanding scientific and technological goal. It holds the promise to allow us to solve a variety of complex problems such as factoring large numbers, quick database search, and the quantum simulation of many-body quantum systems in fields as diverse as condensed matter, quantum chemistry, and even high-energy physics. Sophisticated theoretical protocols for reliable quantum information processing under imperfect conditions have been de-veloped, when errors affect and corrupt the fragile quantum states during storage and computations. Arguably, the most realistic and promising ap-proach towards practical fault-tolerant quantum computation are topologi-cal quantum error-correcting codes, where quantum information is stored in interacting, topologically ordered 2D or 3D many-body quantum systems. This approach offers the highest known error thresholds, which are already today within reach of the experimental accuracy in state-of-the-art setups. A combination of theoretical and experimental research is needed to store, protect and process fragile quantum information in logical qubits effectively so that they can outperform their constituting physical qubits. Whereas small-scale quantum error correction codes have been implemented, one of the main theoretical challenges remains to develop new and improve existing efficient strategies (so-called decoders) to derive (near-)optimal error cor-rection operations in the presence of experimentally accessible measurement information and realistic noise sources. One main focus of this project is the development and numerical implementation of scalable, efficient decoders to operate topological color codes. Additionally, we study the feasibility of im-plementing quantum error-correcting codes fault-tolerantly in near-term ion traps. To this end, we use realistic modeling of the different noise sources, computer simulations, and most modern quantum information approaches to quantum circuitry and noise suppression techniques.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.