The birth of topological quantum computation took place when Alexei Kitaev (2003) made the ingenious step of turning a quantum error correcting code into a many-body interacting system. In particular, he defined a Hamiltonian whose eigenstates are also states of a quantum error correcting code. Beyond the inherited error correcting characteristics, topological systems protect the encoded information with the presence of the Hamiltonian that energetically penalises transformations between states. This opens the door for employing a large variety of many-body effects to combat errors. Storing or manipulating information with a real physical system is naturally subject to errors. To obtain a reliable outcome from a computation we need to be certain that the processed information remains resilient to errors at all times. To overcome errors we need to detect and correct them. The error detection process is based on an active monitoring of the system and the possibility of identifying errors without destroying the encoded information. Error correction employs the error detection outcome and performs the appropriate steps to correct it, thus reconstructing the original information. Classical error correction uses redundancy to spread information in many copies so that errors can be detected, for example by majority voting, and then corrected. Similarly, quantum error correction aims to detect and correct errors of stored quantum information. Quantum states cannot be cloned (Wootters and Zurek, 1982), so the repetition encoding cannot be employed.
The birth of topological quantum computation took place when Alexei Kitaev (2003) made the ingenious step of turning a quantum error correcting code into a many-body interacting system. In particular, he defined a Hamiltonian whose eigenstates are also states of a quantum error correcting code. Beyond the inherited error correcting characteristics, topological systems protect the encoded information with the presence of the Hamiltonian that energetically penalises transformations between states. This opens the door for employing a large variety of many-body effects to combat errors. Storing or manipulating information with a real physical system is naturally subject to errors. To obtain a reliable outcome from a computation we need to be certain that the processed information remains resilient to errors at all times. To overcome errors we need to detect and correct them. The error detection process is based on an active monitoring of the system and the possibility of identifying errors without destroying the encoded information. Error correction employs the error detection outcome and performs the appropriate steps to correct it, thus reconstructing the original information. Classical error correction uses redundancy to spread information in many copies so that errors can be detected, for example by majority voting, and then corrected. Similarly, quantum error correction aims to detect and correct errors of stored quantum information. Quantum states cannot be cloned (Wootters and Zurek, 1982), so the repetition encoding cannot be employed.
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 mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.
The quantum computing must be the initial mark of the second technological revolution caused by the deepening knowledge of the humankind about the peculiar features of the quantum mechanics. If we want to unlock all the potential of this new technology, a development of a fault-tolerant quantum computing will be needed. In order to reach this stage, it is necessary an implementation of quantum error correcting codes that guarantee an exponential suppression of the errors at each step of the execution of an algorithm. Besides that, we need to learn how to isolate our qubits aiming a suppression of the decoherence effects. Quantum mechanics imposes various constraints to the process of construction of the error correction codes, one of them is the impossibility of using copies as redundancy mechanism, once no-cloning theorem does not allow the creation of an universal copy machine of quantum states. Another issue to be noted within this context, is that the measurement process causes a collapse in the quantum state, thus leading to a corruption of the information encoded in the state. Fortunately, there are some ways to bypass the problems pointed out through the entanglement and the stabilizer formalism, the first one allows us to recover the redundancy through the delocalization of the information in the Hilbert spaces of the parts that share quantum correlations, while the second one provides a process of gathering information about the errors that can occur in the quantum state without measuring it. The surface codes are one of the quantum codes that use the features highlighted to correct the errors, this type of code is one of the main candidates to be experimentally implemented on a large scale and it presents promising results in the preliminary studies.
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
This paper investigates the application of topological protection to enhance quantum error correction, ultimately aiming for increased reliability in quantum information transmission and storage. The core concept revolves around encoding quantum bits (qubits) within topological defects, specifically leveraging the robust protection offered by these defects against local disturbances. We explore theoretical frameworks where topological protection fundamentally alters the error correction landscape, moving beyond traditional approaches reliant on local control and measurement. The study details how the inherent symmetries and conserved quantities associated with topological defects create a resilient environment for qubits, effectively mitigating the detrimental effects of decoherence and noise. Mathematical models are presented to illustrate the principles of topological error correction, including the concept of topological codes and their ability to detect and correct errors without directly measuring the qubits themselves. The paper concludes with a discussion on the challenges and future directions in realizing practical topological quantum computers, emphasizing the need for controlled manipulation and characterization of topological defects.
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.