Topological Quantum Matter & Protected Information

Topology offers a way to encode information in global properties that cannot be changed by every local disturbance. Turning that principle into a useful quantum device requires identifying the actual protected quantity, the perturbations that preserve it, and the operations that access it. Agarwal's work spans quantum Hall domain walls, Majorana control, isolated superconductors, and topological electromagnetic response. These are related by their use of constraints, but they do not all demonstrate the same form of protection.

Figure to read

A double exchange changes the signs of both selected Majorana operators. Figure 1 in the source paper. Martin, Ivar; Agarwal, Kartiek. Double Braiding Majoranas for Quantum Computing and Hamiltonian Engineering, PRX Quantum 1, 020324 (2020). Paper. CC BY 4.0. Original manuscript graphic; no alterations.

Track each labeled mode through both exchanges. A coupling with one flipped operator changes sign; a coupling with two flipped operators does not. This distinction determines which terms can be canceled by the sequence.

From an invariant to an operating device

The theory of non-Abelian anyons describes how exchanging excitations can implement transformations within a degenerate state space. A real implementation must create the relevant excitations, maintain their separation and gap, perform operations without unwanted transitions, and read out the result. An appealing braid algebra alone does not establish all those steps. This distinction is central to the wider topological-computation literature and provides the standard against which proposals in this branch should be assessed. 1

Quantum Hall systems supply an intrinsically interacting setting. Different valley-polarized states can meet along domain walls, forming one-dimensional channels even inside a two-dimensional sample. In the bismuth experiments involving Randeria, Agarwal, and collaborators, scanning tunneling microscopy resolves boundary modes and changes in their spectra. The comparison between conducting and spectroscopically gapped walls illustrates why interactions and allowed scattering processes matter alongside topology. 2

Defects, memory, and manipulation

Agarwal and collaborators' theoretical analysis of quantum Hall nematic walls asks which combinations of channels can be gapped without violating relevant symmetries. This moves beyond a picture in which every domain wall automatically conducts. The valley structure constrains possible scattering, while interactions determine the resulting one-dimensional phases. The map's quantum Hall page connects that analysis to the broader theory of edge modes and defect engineering. 3

A later proposal considers non-Abelian defects in fractional quantum Hall valley ferromagnets and strain-controlled operations. The appeal is a route to a protected memory using a material's internal degrees of freedom. The required fractional states, engineered gapping interfaces, and control scheme remain physical requirements, rather than consequences of observing an integer quantum Hall domain wall. The proposal should therefore be understood as a possible architecture, not an experimental realization of a functioning memory. 4

Superconducting Majoranas pose a complementary challenge. The familiar theoretical description uses mean-field pairing and localized zero-energy operators. Martin and Agarwal's double-braiding work uses controlled transformations of those operators to average unwanted couplings. It connects topological operations to dynamical decoupling: protection can be actively improved, but its improvement depends on control assumptions and the unwanted Hamiltonian terms present. 5

Why fixed charge changes the question

An isolated superconductor has a definite total particle number, whereas a conventional mean-field wavefunction need not. This is not a minor notational difference for a proposed qubit. One must identify which states remain available at fixed total charge, how they split, and whether operations connect them coherently. Martin and Agarwal investigate that issue using multiple wires and number-projected states; the result exposes both possibilities for braiding-equivalent gates and limits on their protection. 6

Recent work with Thomas-Markarian develops correlation-based diagnostics in interacting, number-conserving wires. The significance is methodological as well as physical: a many-body wavefunction need not resemble a single-particle edge orbital, yet its correlations can encode edge structure. Spectral evidence, spatial correlations, and finite-size scaling should be compared together. 7

A broader topological landscape

The branch's materials-response page also considers dynamical axions and driven topological-insulator surfaces. These studies concern collective response and phase engineering rather than directly demonstrating protected logical operations. Their inclusion makes the broader context visible without conflating distinct achievements. Across the branch, the open challenge is to connect elegant effective descriptions to realistic gaps, disorder, charge constraints, and readout. The most informative experiments or simulations are those that distinguish topological protection from a merely long-lived or nearly zero-energy feature.

References

1. Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Sarma, Sankar Das. Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80, 1083-1159 (2008). Open manuscript.
2. Randeria, Mallika T.; Agarwal, Kartiek; Feldman, Benjamin E.; Ding, Hao; Ji, Huiwen; Cava, R. J.; Sondhi, S. L.; Parameswaran, Siddharth A.; Yazdani, Ali. Interacting multi-channel topological boundary modes in a quantum Hall valley system. Nature 566, 363–367 (2019). Open manuscript.
3. Agarwal, Kartiek; Randeria, Mallika T.; Yazdani, A.; Sondhi, S. L.; Parameswaran, S. A.. Topology- and symmetry-protected domain wall conduction in quantum Hall nematics. Physical Review B 100, 165103 (2019). Open manuscript.
4. Agarwal, Kartiek. Quantum Hall valley Ferromagnets as a platform for topologically protected quantum memory. Physical Review B 107, 125163 (2023). Open manuscript.
5. Martin, Ivar; Agarwal, Kartiek. Double Braiding Majoranas for Quantum Computing and Hamiltonian Engineering. PRX Quantum 1, 020324 (2020). Open manuscript.
6. Martin, Ivar; Agarwal, Kartiek. Understanding Majorana braiding in superconductors with a fixed total number of particles. Physical Review B 113, 155149 (2026). Open manuscript.
7. Thomas-Markarian, Jaden; Agarwal, Kartiek; Martin, Ivar. Majorana Edge Modes in Isolated Wires. Physical Review Letters 137, 086504 (2026). Open manuscript.

Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.