The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any "classical" computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems -- non-Abelian topologically ordered phases of matter -- that would provide intrinsic fault-tolerance at the hardware level. The so-called "Ising-type" non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and -- most importantly -- those applications which no one in our classical era has the prescience to foresee.
The advancement of information processing into the realm of quantum mechanics promises a transcendence in computational power that will enable problems to be solved which are completely beyond the known abilities of any "classical" computer, including any potential non-quantum technologies the future may bring. However, the fragility of quantum states poses a challenging obstacle for realization of a fault-tolerant quantum computer. The topological approach to quantum computation proposes to surmount this obstacle by using special physical systems -- non-Abelian topologically ordered phases of matter -- that would provide intrinsic fault-tolerance at the hardware level. The so-called "Ising-type" non-Abelian topological order is likely to be physically realized in a number of systems, but it can only provide a universal gate set (a requisite for quantum computation) if one has the ability to perform certain dynamical topology-changing operations on the system. Until now, practical methods of implementing these operations were unknown. Here we show how the necessary operations can be physically implemented for Ising-type systems realized in the recently proposed superconductor-semiconductor and superconductor-topological insulator heterostructures. Furthermore, we specify routines employing these methods to generate a computationally universal gate set. We are consequently able to provide a schematic blueprint for a fully topologically-protected Ising based quantum computer using currently available materials and techniques. This may serve as a starting point for attempts to construct a fault-tolerant quantum computer, which will have applications to cryptanalysis, drug design, efficient simulation of quantum many-body systems, solution of large systems of linear equations, searching large databases, engineering future quantum computers, and -- most importantly -- those applications which no one in our classical era has the prescience to foresee.
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 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.
A central challenge at the intersection of quantum condensed-matter physics and quantum information science is to understand how exotic topological phases and quantum resources govern the power of quantum computation and simulation---and to translate that understanding into practical architectures. This thesis contributes to this program along three interconnected directions: enabling fault-tolerant universal quantum computation with non-Abelian topological order, establishing the fundamental computational reach of near-term analog quantum simulators, and delineating the boundary between classical and quantum computational power through the lenses of topology and magic. In the first direction, drawing on the structure of non-Abelian topological order, we address the dominant resource overhead in current quantum computing architectures---magic-state distillation---by moving beyond the standard stabilizer formalism entirely. We construct a comprehensive framework for fault-tolerant universal quantum computation based on the non-Abelian quantum double model D(S3), with all operations realized on qubit--qutrit degrees of freedom in a planar, geometrically local architecture feasible on all current experimental platforms. This work addresses the long-standing open question of realizable, fault-tolerant universal topological quantum computation. In the second direction, we establish the computational universality of analog quantum simulators operating under global control---platforms that circumvent the demanding requirements of digitization and individual addressability. We prove a necessary and sufficient condition for universality in globally controlled qubit chains, showing that widely used platforms such as Rydberg atom arrays and trapped ions are generically universal once lattice reflection symmetry is broken. We further derive sufficient conditions for universality in fermionic and bosonic optical superlattices. As an experimental demonstration of this framework, we realize the dynamics of the cluster-state ZXZ model---an exemplar of symmetry-protected topological order---in Rydberg atom arrays, validating the theoretical foundation. In the third direction, we sharpen the classical--quantum computational boundary from both sides. We show that non-Abelian Majorana braiding realized in classical mechanical metamaterials admits a faithful classical simulation, establishing that braiding-only architectures are classically simulable without supplementary non-Clifford resources. Conversely, we prove that random matrix product states carry magic growing exponentially with system size, demonstrating that generic one-dimensional many-body states harbor ample non-stabilizer resources. Together, these results delineate precisely where genuine quantum advantage emerges in topological and many-body systems. Taken as a whole, this thesis leverages the convergence of topological matter and quantum information to remove key bottlenecks toward fault-tolerant quantum computation, to place near-term quantum simulation on rigorous footing, and to clarify the fundamental resources that separate classical from quantum computational power.
The idea of topological quantum computation (TQC) is to store and manipulate quantum information in an intrinsically fault-tolerant manner by utilizing the physics of topologically ordered phases of matter. Currently, one of the most promising platforms for a topological qubit is in terms of Majorana fermion zero modes (MZMs) in spin-orbit coupled superconducting nanowires. However, the topologically robust operations that are possible with MZMs can be efficiently simulated on a classical computer and are therefore not sufficient for realizing a universal gate set for TQC. Here, we show that an array of coupled semiconductor-superconductor nanowires with MZM edge states can be used to realize a more sophisticated type of non-Abelian defect: a genon in an Ising × Ising topological state. This leads to a possible implementation of the missing topologically protected π/8 phase gate and thus universal TQC based on semiconductor-superconductor nanowire technology. We provide detailed numerical estimates of the relevant energy scales, which we show to lie within accessible ranges.
Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.
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.