Topological Quantum Computation—From Basic Concepts to First Experiments

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.

Topological quantum computation provides an elegant way around decoherence, as one encodes quantum information in a non-local fashion that the environment finds difficult to corrupt. Here we establish that one of the key operations---braiding of non-Abelian anyons---can be implemented in one-dimensional semiconductor wire networks. Previous work [Lutchyn et al., arXiv:1002.4033 and Oreg et al., arXiv:1003.1145] provided a recipe for driving semiconducting wires into a topological phase supporting long-sought particles known as Majorana fermions that can store topologically protected quantum information. Majorana fermions in this setting can be transported, created, and fused by applying locally tunable gates to the wire. More importantly, we show that networks of such wires allow braiding of Majorana fermions and that they exhibit non-Abelian statistics like vortices in a p+ip superconductor. We propose experimental setups that enable the Majorana fusion rules to be probed, along with networks that allow for efficient exchange of arbitrary numbers of Majorana fermions. This work paves a new path forward in topological quantum computation that benefits from physical transparency and experimental realism.

Developing quantum systems which are robust against noise are of prime importance to the realisation of quantum technologies. Without fault tolerance, we will never be able to preserve delicate quantum states for macroscopic time scales, a necessary requirement in the construction of scalable quantum computer. Topologically ordered models offer very beautiful mechanisms for preserving quantum states. In such models, quantum information is encoded globally over a degenerate topological Hilbert space which offers a natural robustness against local environmental noise. Remarkably, topological order is present in certain realistic condensed-matter systems. This provides a platform for accessing topological order in a laboratory. Moreover, considering condensed-matter systems enables us to combine topological features with other physical effects to enhance their behaviour. In this Thesis we study the physics of topologically ordered lattice models with defects. We seek practical applications of lattice defects for the realisation of a fault-tolerant quantum computer. The first result we present shows that anyonic data of point-like lattice defects, called twists, can be found by measuring the entanglement of the ground state of the host system. The data we learn relates to the capacity of a twist defect to perform quantum computational tasks. The second result in this Thesis concerns the dynamics of the qudit toric code model with line-like defects coupled to a thermal bath. We show we can entropically inspire fragile glassy dynamics in the system. Such dynamics qualitatively improve the coherence times of quantum information encoded in the ground state of the lattice. A novel way of achieving fault-tolerant quantum computation is by producing and manipulating twist defects of topologically ordered systems. In many ways, this paradigm of topological quantum computation is analogous to quantum computation using anyons. The first area of study in this Thesis extends the analogy between anyons and twists. Specifically, we show that the anyonic data of twists in Kitaev’s toric code model can be extracted

Hybrid Architectures for Topological Quantum Computation: Platform Integration, Device Engineering, and Scalability

Quantum state engineering and state characterization is a key task in quantum information processing in both discrete and continuous variable systems in the optical domain. In particular, quantum states with non-Gaussian (i.e., non-positive) Wigner quasiprobability distribution functions are crucial to universal, fault-tolerant quantum computing with continuous variables. In this talk, we present our recent results on single-photon Fock state tomography using Photon-Number-Resolving (PNR) measurements. We generated a highly pure narrow-band single-photon Fock state by heralding cavity-enhanced spontaneous-parametric-downconversion from a PPKTP optical parametric oscillator. The Wigner function was reconstructed with photon statistics obtained using superconducting transition-edge sensors with an overall system efficiency of 58(2)%. We then discuss quantum state engineering for pure displaced single-photon Fock states, optical cat states, and approximate GKP states using coherent states and single-photon states along with linear optics and PNR measurements. We report our experimental progress for the same.

We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.

In the pattern-of-zeros approach to quantum Hall states, a set of data ${n;m;{S}_{a}|a=1,\dots{},n;n,m,{S}_{a}∊\mathbb{N}}$ (called the pattern of zeros) is introduced to characterize a quantum Hall wave function. In this paper we find sufficient conditions on the pattern of zeros so that the data correspond to a valid wave function. Some times, a set of data ${n;m;{S}_{a}}$ corresponds to a unique quantum Hall state, while other times, a set of data corresponds to several different quantum Hall states. So in the latter cases, the pattern of zeros alone does not completely characterize the quantum Hall states. In this paper, we find that the following expanded set of data ${n;m;{S}_{a};c|a=1,\dots{},n;n,m,{S}_{a}∊\mathbb{N};c∊\mathbb{R}}$ provides a more complete characterization of quantum Hall states. Each expanded set of data completely characterizes a unique quantum Hall state, at least for the examples discussed in this paper. The result is obtained by combining the pattern of zeros and ${Z}_{n}$ simple-current vertex algebra which describes a large class of Abelian and non-Abelian quantum Hall states ${\ensuremath{\Phi}}_{{Z}_{n}}^{\mathrm{sc}}$. The more complete characterization in terms of ${n;m;{S}_{a};c}$ allows us to obtain more topological properties of those states, which include the central charge $c$ of edge states, the scaling dimensions and the statistics of quasiparticle excitations.

Classic computing can take credit for technology ranging from mobile phones to supercomputers. But in recent years, a budding counterpart to these conventional devices has emerged: quantum computers. Whereas classic computing sometimes fails to solve complex calculations, such as factoring hundred-digit numbers, quantum computing holds the potential to easily tackle such problems. The field of quantum computing has attracted researchers, such as National Academy of Sciences member Christopher Monroe. An experimental atomic physicist at the University of Maryland and fellow of the Joint Quantum Institute and the Joint Center for Quantum Information and Computer Science in Maryland, Monroe uses lasers to exploit atomic particles to study complex computational problems. Monroe spoke to PNAS about how quantum computing might evolve. Christopher Monroe. > PNAS:Can you describe some of the basic mechanics of quantum computing? > Monroe:It all comes down to the superposition principle, in which a quantum system can exist in multiple states at the same time. Consider the fundamental unit of information: the bit. A bit is binary information that can be a 0 or a 1. A quantum bit, or qubit, can be in a superposition of both 0 and 1 as long as it’s isolated and unobserved. A single qubit is pretty trivial, but when you put many qubits together, there become exponentially many possibilities. > > Let’s consider 300 qubits. That’s an interesting number because the number of possible configurations, 2300, is more than the number of particles in the universe. You could never simulate what happens with 300 qubits on a classical machine; there’s not enough stuff in the universe to do it. The magic with quantum is the power of exponential growth: every time you add a qubit, you double the number of configurations. > > But to control these qubits and exploit this massive storage, we …

Topological quantum computing (TQC) represents a paradigm shift in quantum information processing, leveraging the topological properties of quantum states to create inherently fault-tolerant qubits. Unlike conventional qubits (superconducting or trapped ions), topological qubits are protected by the global geometry of their wavefunctions, making them robust against local noise and decoherence. This approach, rooted in the work of Alexei Kitaev (1997), promises to solve the scalability crisis of quantum computers. As of December 10, 2025, TQC is transitioning from theory to prototypes, with breakthroughs from Microsoft, Cornell–IBM, and others marking a pivotal year

As quantum computing development advances closer toward achieving fault-tolerant error-corrected realisation, debates on applications, impacts, risks, and benefits of quantum computing are timely and due. While there is awareness of the potential power and complexity of quantum computers, there has been relatively little attention on the social impacts and ethical implications of this technology. In grappling with the social impacts of quantum computing, some stakeholders have turned to applied ethics, specifically (classical) computer and Artificial Intelligence (AI) ethics, for guidance. While computer and AI ethics are useful starting points for evaluating the ethical issues posed by quantum computing, uncritically applying existing sets of ethical principles risks overlooking how quantum computing differs from these technologies. We argue that borrowing of ethical principles and guidelines from AI and computing is inappropriate for several reasons: (1) quantum computing, classical computing, and AI are different technologies with significant material differences; (2) unlike AI and classical computers which have become established, quantum computing is an emerging technology, which has implications on the levels of accessibility and immediate impact in society; and (3) there are significant differences in the way AI and quantum computing have been developed and by whom. We also briefly summarise some of the unique risks and societ al impacts posed by quantum computing. We posit that these reasons support the argument for a new sub-field of quantum computing ethics, which would allow a relevant scholarship to develop and provide guidance as quantum computing technology continues to mature.

Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.

Generation and preservation of quantum entanglement are among the primary tasks in quantum information processing. State stabilization via quantum bath engineering offers a resource-efficient approach to achieve this objective. However, current methods for engineering dissipative channels to stabilize target entangled states often require specialized hardware designs, complicating experimental realization and hindering their compatibility with scalable quantum computation architectures. In this work, we propose and experimentally demonstrate a stabilization protocol readily implementable in the mainstream integrated superconducting quantum circuits. The approach utilizes a Raman process involving a resonant (or nearly resonant) superconducting qubit array and the qubits' dedicated readout resonators to effectively develop nonlocal dissipative channels. Leveraging individual controllability of the qubits and resonators, the protocol stabilizes two-qubit Bell states with a fidelity of 90.7%, marking the highest reported value in solid-state platforms to date. Furthermore, by extending this strategy to include three qubits, an entangled W state is achieved with a fidelity of 86.2%. This demonstrates the protocol's extensibility beyond two-qubit systems. Notably, the protocol is of practical interest since it only utilizes existing hardware common to standard operations in the underlying superconducting circuits, thereby facilitating the exploration of many-body quantum entanglement with dissipative resources.

In the world of computation, the invention of Computers has reduced human effort for accomplishing task. Depending on the type, size and nature of the work the computers are manufactured in different magnitudes. As the technology grows, it facilitated to manufacture high performing, high throughput, and high memory Classical computers. Classical computers store any information in the form of bits in the memory device, which can be either 1 or 0 (binary system) and all the processing will be done in the same logic i.e., high voltage/low voltage. Because of this logic the classical computers solve some problems by using more resources. Quantum computer are the next level computers which uses quantum theory principles for solving problems. The quantum computer uses Qubit as basic unit which uses principles of quantum computing such as superposition and entanglement, which gives qubit, the efficiency of exhibiting many logical states at a time. This paper shows a comparative study on classical and quantum computers and illustrates various problems in classical computers and respective solutions with use of quantum computers.

Chapter 9 - Quantum Information Science from the Perspective of a Device and Materials Engineer