We present a scheme for universal topological quantum computation based on Clifford-complete braiding and fusion of symmetry defects in the 3-fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realization of this computational scheme on the lattice using ground states of the Walker-Wang model for the 3-fermion anyon theory with symmetry defects. The Walker-Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-fermion anyon theory can be realized in a two-dimensional subsystem code due to Bombín—making contact with an alternative implementation of the 3-fermion defect-computation scheme via code deformations. Published by the American Physical Society 2024
We present a scheme for universal topological quantum computation based on Clifford-complete braiding and fusion of symmetry defects in the 3-fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realization of this computational scheme on the lattice using ground states of the Walker-Wang model for the 3-fermion anyon theory with symmetry defects. The Walker-Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-fermion anyon theory can be realized in a two-dimensional subsystem code due to Bombín—making contact with an alternative implementation of the 3-fermion defect-computation scheme via code deformations. Published by the American Physical Society 2024
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.
One potential route toward fault-tolerant universal quantum computation is to use non-Abelian topological codes. In this work, we investigate how to achieve this goal with the quantum double model D(S3) . By embedding each on-site Hilbert space into a qubit-qutrit pair, we explicitly construct circuits for creating, moving, and locally measuring all non-trivial anyons. We also design a specialized anyon interferometer to remotely measure the total charge of well-separated anyons; this avoids fusion, which compromises fault tolerance. These protocols enable the implementation of a universal gate set proposed by Cui et al. and active correction of the circuit-level noise during computation. To further suppress errors, we encode each physical degree of freedom of D(S3) into a novel error-correcting code, enabling fault-tolerant realization, at the logical level, of all gates in the anyon manipulation circuits. Our proposal offers a promising path to realize robust universal topological quantum computation in the NISQ era.
There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.
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.
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.