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Robust Universal Braiding with Non-semisimple Ising Anyons

Filippo Iulianelli, Sung Kim, Joshua Sussan, Aaron D. Lauda

Abstract

Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter $α$, the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter $α$ where the computational subspace remains positive-definite. We identify special values of $α$ where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of $α$ is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.

Robust Universal Braiding with Non-semisimple Ising Anyons

Abstract

Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter , the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter where the computational subspace remains positive-definite. We identify special values of where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.

Paper Structure

This paper contains 27 sections, 6 theorems, 111 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Unrolled quantum groups and modified traces
  3. Renormalizing the quantum trace
  4. Data for non-semisimple Ising anyons
  5. Non-semisimple fusion rules
  6. Normalization, $F$ and $R$ symbols
  7. Hermitian conjugates and bubble pops
  8. Non-semisimple Topological Quantum Computation
  9. Traditional Ising encoding
  10. Non-semisimple encoding
  11. Normalization
  12. Braiding and Jucys-Murphy operators
  13. Multiqubit operations
  14. Decoupling from the indefinite space
  15. Robustness of the generators
  16. ...and 12 more sections

Key Result

Theorem 5

Single-qubit universality is achieved by affine braiding of anyons in $\mathcal{H}_1$ via eq:J1b2_normalized for all values of $\alpha$ for which $\mathsf{b}_1^2$ and $\mathsf{b}_2$ are unitary, except for when $\alpha = 2\pm \frac{2}{3}\quad (\text{mod } 4)$ or $\alpha = 2\pm \frac{3}{5} \quad (\te

Figures (8)

  • Figure 1: Encoding computational qubits into the collective state of a group of anyons. Unitaries are implemented on the computational basis by braiding the anyons around one another.
  • Figure 2: Norms of the basis vectors $\ket{0}$ and $\ket{1}$. The graph is periodic with period 8, but only $\alpha \in (0,4)$ is shown for clarity. The graph for $\alpha \in (4,8)$ is the same as in the figure, but flipped upside-down.
  • Figure 3: The Jucys-Murphy braid $J_3$ acts as the identity on the first qubit, and as $\mathsf{b}_1^2$ on the second qubit.
  • Figure 4: Distance $D(\mathsf{b}_2, \widetilde{\mathsf{b}}_2)$ for select values of $\varepsilon$. The rapid growth on the right-hand side of the graph occurs when $\alpha+\varepsilon$ approaches $3$. The growth is finite and goes to zero as $\varepsilon$ goes to zero.
  • Figure 5: The Hilbert-Schmidt norm distance between target matrix and effective matrix at $\alpha = \alpha_0 + \varepsilon$ (blue) and norm of the non-unitary mixing term (orange) as a function of $\varepsilon$ are bounded by $|\varepsilon|< \frac{\delta}{14.3}$ (dashed)
  • ...and 3 more figures

Theorems & Definitions (25)

  • Example 1
  • Example 2
  • Example 3
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6: Comparison with semisimple Ising
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 15 more