Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" – anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" – anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Quantum error correction protocols protect against local errors by storing logical information non-locally. This poses a challenge: how to design efficient logical gates on the non-local ``hidden'' logical information, and how to implement these gates using the local physical operations. We develop a general description of planar Pauli stabilizer codes and protocols for logical operations in terms of point-like particles called Majorana fermions. Information is stored in the pairwise fermion parities of spatially separated Majorana fermions. The description in terms of Majorana fermions captures not only large distance asymptotics, but also all scales down to the lattice constants. We exploit this locality to densely pack logical information in spacetime. The simplest application is to a static case: dense memory. More importantly, we implement fault-tolerant Majorana motion and leverage this primitive to design braiding-based logical gates. This approach reduces space overhead of logical operations resulting in an improved logical error rate given fixed number of physical qubits. We illustrate a practical use of our approach by designing and benchmarking of 2-qubit Clifford gates. We find numerically that our protocol outperforms lattice surgery in this setting for near-term error rates and realistic device constraints. More generally, introduction of compact motion of Majorana fermions as an efficient computational primitive opens a promising new route for the design of low overhead error correction protocols.
We show how to absorb fermionic quantum simulation's expensive fermion-to-qubit mapping overhead into the overhead already incurred by surface-code-based fault-tolerant quantum computing. The key idea is to process information in surface-code twist defects, which behave like logical Majorana fermions. Our approach encodes Dirac fermions, a key data type for simulation applications, directly into logical Majorana fermions rather than atop a logical qubit layer in the architecture. Using quantum simulation of the N-fermion 2D Fermi-Hubbard model as an exemplar, we demonstrate two immediate algorithmic improvements. First, by preserving the model's locality at the logical level, we reduce the asymptotic Trotter-Suzuki quantum circuit depth from O(N) in a typical Jordan-Wigner encoding to O(1) in our encoding. Second, by exploiting optimizations manifest for logical fermions but less obvious for logical qubits, we reduce the T-count of the block-encoding \textsc{select} oracle by 20\% over standard implementations, even when realized by logical qubits and not logical fermions.
A 2d×d×d Spacetime Volume Implementation of a Logical S Gate in the Surface Code
The logical S gate implemented via twist defect braiding in the surface code is one of the major sources of overhead in fault-tolerant quantum computing, since an S-gate correction is required in every logical T-gate teleportation. Existing logical S-gate implementations require spacetime volumes of 2d×2d×d or 2d×1.5d×d, where d is the code distance of the surface code. To the best of our knowledge, their circuit-level implementations have not yet been shown, hindering quantitative comparisons of fault distances and logical error rates. In this work, we provide these missing circuit-level implementations. Additionally, we propose a novel twist defect braiding protocol that reduces the spacetime volume to 2d×d×d. First, we construct an implementation of the proposed method using constant-length non-local gates, and then refine it to utilize only nearest-neighbor two-qubit gates on a square grid, without requiring additional two-qubit gate depth beyond that of standard syndrome extraction circuits. Through numerical simulations, we evaluate the fault distances and logical error rates for both existing and proposed methods. Our results show that, although the proposed method reduces the fault distance by one or three, its logical error rates remain comparable to those of existing methods at large code distances (d≥5) and at physical error rates near p=10−3. This demonstrates that the proposed method is promising for near-term fault-tolerant quantum computing.
We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to π-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as ϵ∼exp(−L/ξ), where L is the inter-vortex separation and ξ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits 4π-periodic oscillations with phase jumps of π/2 per braid. Any deviation from this quantized phase beyond the exp(−L/ξ) envelope at large L/ξ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a px+ipy paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to π-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as exp(−L/ξ), and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with π/2 rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi
Surface codes represent a promising route towards universal fault-tolerant quantum computation, allowing for the protection of stored quantum states against many types of errors. This theoretical work shows that, in addition, a significantly enhanced versatility in quantum information processing is expected when qubits are encoded into the Majorana bound states in topological superconductor-semiconductor heterostructures. In view of recent experimental evidence for Majorana states in such heterostructures, a two-dimensional network of interacting Majorana states thus may feature key advantages towards the long-term goal of a functional quantum computer. The authors discuss how topologically protected logical qubits in this Majorana surface code architecture can be defined, initialized, manipulated, and read out. The physical ingredients needed to implement such operations are familiar from topologically trivial quantum devices. In particular, by employing quantum interference terms in conductance measurements, composite single-electron pumping protocols, and gate-tunable tunnel barriers, the full set of gates required for universal quantum computation can be implemented.