We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a ${2}^{N/4}$ -fold degeneracy, where $N$ is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink) ${Z}_{2}$ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the ${Z}_{2}$ flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a ${2}^{N/4}$-fold degeneracy, where $N$ is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink) ${Z}_{2}$ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the ${Z}_{2}$ flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.

arXiv:1007.4091Physical Review BNov 8, 201022 citations

Flux-based superconducting quantum computing

M. J. Feldman

Optical Fiber Communication ConferenceJun 13, 20010 citations

Reviving Majorana zero modes in the spin- 1 2 Kitaev ladder model

Haoting Xu, Hae‐Young Kee

The one-dimensional $p$ -wave superconductor, characterized by boundary Majorana modes, has attracted significant interest owing to its potential application in topological quantum computation. Similarly, spin-1/2 Kitaev ladder systems with bond-dependent Ising interactions, featuring Majorana fermions coupled with ${Z}_{2}$ flux, exhibit boundary Majorana modes when in a topological phase. However, the ground-state degeneracy, inherent in these systems, may result in the annihilation of Majorana modes due to the superposition of the degenerate states. To avoid this issue, here we introduce a projective measurement that selects one of the degenerate ${Z}_{2}$ sectors, enabling the revival of Majorana modes. Once the state is selected, we show that the application of the local spin operators on a bond flips the sign of the adjacent ${Z}_{2}$ flux. Repeating such operators enables the system to reach a desired ${Z}_{2}$ flux configuration. The Majorana zero modes can be manipulated and fused by tuning the flux sectors achievable through applying local spin operators. We also discuss the engineering of the Kitaev ladder and open questions for future studies.The one-dimensional $p$-wave superconductor, characterized by boundary Majorana modes, has attracted significant interest owing to its potential application in topological quantum computation. Similarly, spin-1/2 Kitaev ladder systems with bond-dependent Ising interactions, featuring Majorana fermions coupled with ${Z}_{2}$ flux, exhibit boundary Majorana modes when in a topological phase. However, the ground-state degeneracy, inherent in these systems, may result in the annihilation of Majorana modes due to the superposition of the degenerate states. To avoid this issue, here we introduce a projective measurement that selects one of the degenerate ${Z}_{2}$ sectors, enabling the revival of Majorana modes. Once the state is selected, we show that the application of the local spin operators on a bond flips the sign of the adjacent ${Z}_{2}$ flux. Repeating such operators enables the system to reach a desired ${Z}_{2}$ flux configuration. The Majorana zero modes can be manipulated and fused by tuning the flux sectors achievable through applying local spin operators. We also discuss the engineering of the Kitaev ladder and open questions for future studies.

arXiv:2404.16099Physical review. B./Physical review. BNov 14, 20241 citation

Kitaev chain in synthetic dimension with cavity-controlled Majorana modes

Adel Ali, Alexey Belyanin

We introduce a tunable synthetic-dimension platform for realizing Kitaev-chain physics with high degree of control over Majorana zero modes. It is based on a generic Landau-quantized two dimensional electron system coupled to the magnetic flux of a superconducting LC circuit. The structured vector potential of a superconducting LC inductor induces attractive interactions between electron angular-momentum states at the lowest Landau level. These states serve as a synthetic dimension for the coveted fermionic Kitaev chain, with Majorana zero modes existing at the boundaries of the angular-momentum lattice. The crucial advantage of this proposal is the possibility of a robust, nonlocal readout and control of the Majorana states by a LC resonator. The platform relies on mature circuit QED and semiconductor technologies and provides a promising pathway to topological quantum computing.We introduce a tunable synthetic-dimension platform for realizing Kitaev-chain physics with high degree of control over Majorana zero modes. It is based on a generic Landau-quantized two dimensional electron system coupled to the magnetic flux of a superconducting LC circuit. The structured vector potential of a superconducting LC inductor induces attractive interactions between electron angular-momentum states at the lowest Landau level. These states serve as a synthetic dimension for the coveted fermionic Kitaev chain, with Majorana zero modes existing at the boundaries of the angular-momentum lattice. The crucial advantage of this proposal is the possibility of a robust, nonlocal readout and control of the Majorana states by a LC resonator. The platform relies on mature circuit QED and semiconductor technologies and provides a promising pathway to topological quantum computing.

cond-mat.mes-hallCross-listedarXiv:2605.04384May 7, 2026

Fundamentals of flux-based quantum computing

Timothy L. Robertson

Jan 1, 20051 citation

Scalable Designs for Topological Quantum Computation with Majorana Zero Modes: Part 1, Overview

Torsten Karzig, Christina Knapp, Roman M. Lutchyn +9 more

Bulletin of the American Physical SocietyMar 15, 20170 citations

We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a

{2}^{N/4}

-fold degeneracy, where

N

is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink)

{Z}_{2}

flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the

{Z}_{2}

flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.

The one-dimensional

p

-wave superconductor, characterized by boundary Majorana modes, has attracted significant interest owing to its potential application in topological quantum computation. Similarly, spin-1/2 Kitaev ladder systems with bond-dependent Ising interactions, featuring Majorana fermions coupled with

{Z}_{2}

flux, exhibit boundary Majorana modes when in a topological phase. However, the ground-state degeneracy, inherent in these systems, may result in the annihilation of Majorana modes due to the superposition of the degenerate states. To avoid this issue, here we introduce a projective measurement that selects one of the degenerate

{Z}_{2}

sectors, enabling the revival of Majorana modes. Once the state is selected, we show that the application of the local spin operators on a bond flips the sign of the adjacent

{Z}_{2}

flux. Repeating such operators enables the system to reach a desired

{Z}_{2}

flux configuration. The Majorana zero modes can be manipulated and fused by tuning the flux sectors achievable through applying local spin operators. We also discuss the engineering of the Kitaev ladder and open questions for future studies.