Majorana Fermion Codes

Abstract

We initiate the study of Majorana fermion codes. These codes can be viewed as extensions of Kitaev's 1D model of unpaired Majorana fermions in quantum wires to higher spatial dimensions and interacting fermions. The purpose of Majorana fermion codes (MFCs) is to protect quantum information against low-weight fermionic errors, that is, operators acting on sufficiently small subsets of fermionic modes. We examine to what extent MFCs can surpass qubit stabilizer codes in terms of their stability properties. A general construction of 2D MFCs is proposed which combines topological protection based on a macroscopic code distance with protection based on fermionic parity conservation. Finally, we use MFCs to show how to transform any qubit stabilizer code to a weakly self-dual CSS code.

22 pages, 1 figure. References added + extended discussion

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Cited by in corpus (76)

• Quantum Low-Density Parity-Check Codes
• Entanglement Transition in the Projective Transverse Field Ising Model
• Strong Resilience of Topological Codes to Depolarization
• Balanced Product Quantum Codes
• Roadmap to Majorana surface codes
• Majorana Fermion Surface Code for Universal Quantum Computation
• Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates
• Majorana braiding with thermal noise
• Quantum Computing with Majorana Fermion Codes
• Combining Topological Hardware and Topological Software: Color Code Quantum Computing with Topological Superconductor Networks
• Clifford Gates by Code Deformation
• Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
• Majorana loop stabilizer codes for error correction of fermionic quantum simulations
• Parity-controlled $2\pi$ Josephson effect mediated by Majorana Kramers pairs
• Coherence requirements for quantum communication from hybrid circuit dynamics
• Topological phases in two-dimensional arrays of parafermionic zero modes
• Braiding by Majorana Tracking and Long-Range CNOT Gates with Color Codes
• Interacting Majorana Fermions
• Jordan-Wigner Dualities for Translation-Invariant Hamiltonians in Any Dimension: Emergent Fermions in Fracton Topological Order
• Effective and exact holographies from symmetries and dualities
• Physical Implementation of a Majorana Fermion Surface Code for Fault-Tolerant Quantum Computation
• Low-field Topological Threshold in Majorana Double Nanowires
• Equivalence between fermion-to-qubit mappings in two spatial dimensions
• Foliated fracton order in the Majorana checkerboard model
• Monte Carlo studies of the properties of the Majorana quantum error correction code: is self-correction possible during braiding?
• Majorana dimers and holographic quantum error-correcting codes
• Optimal error correction in topological subsystem codes
• Modeling noise and error correction for Majorana-based quantum computing
• Majorana Spin Liquids, Topology and Superconductivity in Ladders
• Majorana fermions at the edge of superconducting islands
• Unwinding Short-Range Entanglement
• Local Integrals of Motion for Topologically Ordered Many-Body Localized Systems
• Generalized Toric Codes Coupled to Thermal Baths
• Fault-tolerant fermionic quantum computation based on color code
• Fault-tolerant gates via homological product codes
• Noise threshold and resource cost of fault-tolerant quantum computing with Majorana fermions in hybrid systems
• Quantum memories with zero-energy Majorana modes and experimental constraints
• Optimizing Clifford gate generation for measurement-only topological quantum computation with Majorana zero modes
• Engineering Yang-Lee anyons via Majorana bound states
• Fermion-Parity-Based Computation and its Majorana-Zero-Mode Implementation
• Scalable Fermionic Error Correction in Majorana Surface Codes
• Localization and oscillations of Majorana fermions in a two-dimensional electron gas coupled with $d$-wave superconductors
• Quantum XYZ Product Codes
• Complexity of Fermionic Dissipative Interactions and Applications to Quantum Computing
• Strategy for implementing stabilizer-based codes on solid-state qubits
• CSS code surgery as a universal construction
• Universal quantum computing with parafermions assisted by a half fluxon
• Non-Abelian defects in Fracton phase of matter
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Quantum information spreading in random spin chains with topological order
• Combating quasiparticle poisoning with multiple Majorana fermions in a periodically-driven quantum wire
• On the Hardness of the Minimum Distance Problem of Quantum Codes
• NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians
• Parafermion stabilizer codes
• Stability of topologically-protected quantum computing proposals as seen through spin glasses
• Implementation of standard quantum error correction codes for solid-state qubits
• Encoding Majorana codes
• Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata
• Free-Fermion Subsystem Codes
• A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
• A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians
• A graph-based formalism for surface codes and twists
• Approximate Quantum Codes From Long Wormholes
• Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma
• A fermionic code related to the exceptional group $E_8$
• Erasure conversion in Majorana qubits via local quasiparticle detection
• Triangular lattice Majorana-Hubbard model: Mean field theory and DMRG on a width-4 torus
• Designing ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological orders in networks of Majorana bound states
• 1D Error Correcting Code for Majorana Qubits
• Majorana braiding simulations with projective measurements
• Quantum error correction in a time-dependent transverse field Ising model
• Edge Modes and Teleportation in a Topologically Insulating Quantum Wire
• Clifford Transformations for Fermionic Quantum Systems: From Paulis to Majoranas to Fermions
• Quantum operation of fermionic systems and process tomography using Majorana fermion gates
• Partons from stabilizer codes
• Unified Framework for Quantum Code Embedding