Universal Quantum Computation with Nonsemisimple Ising Anyons

Filippo Iulianelli, Sung Kim, Joshua Sussan, Aaron Lauda

The big picture

Non-semisimple

Advances here

Can we do something here?

Definitions

Topological Quantum Computing

• In (3+1)-D, only bosonic and fermionic statistics are allowed:
• When two particles are swapped (braided) the wavefunction picks up a phase of ±1.
• In (2+1)-D the story is different:
• In topological systems, the braiding two particle-like excitations multiplies the wavefunction by a unitary.
• Unitary braid group representation – Only depends on the topology of the braid
• Idea: use these unitaries as gates - Built-in resistance to noise!
• Braiding two particles is a macroscopic process: it’s unlikely to happen by mistake!

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Topological quantum computers and knots

B. Field, T. Simula (2018)

Mathematical description – Modular Tensor Categories

Topological systems are described by TQFTs. To describe TQFTs we need the following algebraic data

Ising Anyons

Using the full category

Renormalize quantum dimensions

New qubit space

Single qubit gates - success

Multiple qubits – more difficult

• Spaces don’t have the right dimension
• e.g. 2-qubit space is 6-dimensional�
• Entangling gates “leak” into the non-computational subspace�
• Standard issue in TQC, can be fixed by known techniques to produce low-leakage entangling gates

Our “best” model for two qubits:

Multiple qubits – more difficult

Our “best” model for two qubits:

Robustness results

Robustness of decoupling from negative-definite space

Physical models – non hermiticity

Non Hermitian Kitaev Chain

Non Hermitian Kitaev Chain

References

• Iulianelli, F., Kim, S., Sussan, J. et al. Universal quantum computation using Ising anyons from a non-semisimple topological quantum field theory. Nat Commun 16, 6408 (2025). Filippo Iulianelli, Sung Kim, Joshua Sussan, and Aaron D. Lauda, “Robust Universal Braiding with Non-semisimple Ising Anyons,” Physical Review A (accepted March 27, 2026).
• Negron, C. Log-modular quantum groups at even roots of unity and the quantum Frobenius I. Comm. Math. Phys. 382, 773–814 (2021).
• Geer, N., Lauda, A.D., Patureau-Mirand, B. and Sussan, J. (2024), Non-semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups. J. London Math. Soc., 109: e12853.

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References