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arXiv:2503.08736

A Note on Clifford Stabilizer Codes for Ising Anyons

Dutta, Sanchayan

Published Mar 11, 2025• 0 citations
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Abstract

We provide a streamlined elaboration on existing ideas that link Ising anyon (or equivalently, Majorana) stabilizer codes to certain classes of binary classical codes. The groundwork for such Majorana-based quantum codes can be found in earlier works (including, for example, Bravyi (arXiv:1004.3791) and Vijay et al. (arXiv:1703.00459)), where it was observed that commuting families of fermionic (Clifford) operators can often be systematically lifted from weakly self-dual or self-orthogonal binary codes. Here, we recast and unify these ideas into a classification theorem that explicitly shows how q-isotropic subspaces in $\mathbb{F}_2^{2n}$ yield commuting Clifford operators relevant to Ising anyons, and how these subspaces naturally correspond to punctured self-orthogonal codes in $\mathbb{F}_2^{2n+1}$.

DOI
10.48550/arxiv.2503.08736

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We provide a streamlined elaboration on existing ideas that link Ising anyon (or equivalently, Majorana) stabilizer codes to certain classes of binary classical codes. The groundwork for such Majorana-based quantum codes can be found in earlier works (including, for example, Bravyi (arXiv:1004.3791) and Vijay et al. (arXiv:1703.00459)), where it was observed that commuting families of fermionic (Clifford) operators can often be systematically lifted from weakly self-dual or self-orthogonal binary codes. Here, we recast and unify these ideas into a classification theorem that explicitly shows how q-isotropic subspaces in F22n\mathbb{F}_2^{2n}F22n​ yield commuting Clifford operators relevant to Ising anyons, and how these subspaces naturally correspond to punctured self-orthogonal codes in F22n+1\mathbb{F}_2^{2n+1}F22n+1​.
In quantum information science, Clifford operators and stabilizer codes play a central role for systems of qubits (or qudits). In this paper, we study their analogues for systems composed of Majorana fermions. In this case, a crucial role is played by fermion parity symmetry, which is an unbreakable symmetry present in any system with fundamentally fermionic degrees of freedom. We prove that the subgroup of parity-preserving Majorana Cliffords can be represented by the orthogonal group over the binary field $\F$, and we show how it can be generated by braiding operators and used to construct any (even-parity) Majorana stabilizer code. We also analyze the frame potential for this so-called p-Clifford group when acting on a fixed-parity sector of the Hilbert space, proving that it is equivalent to the frame potential of the ordinary Clifford group acting on the same sector.
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
To implement a quantum error-correction protocol, we first need a scheme to prepare our state in the correct subspace of the code, and this can be done using a unitary encoding circuit. Majorana codes are special since any gates that transform such codes must preserve fermionic parity. In this paper, we present an algorithm that uses the stabilizer matrix to compute unitary encoding circuits for Majorana codes. We present two approaches, both of which use a version of Gaussian elimination with row operations replaced with elementary fermionic Clifford operations. One approach uses an additional ancilla mode and works for all Majorana stabilizer codes, while the second approach does not use an ancilla but does not work if the total parity is inside the stabilizer group.
Stabilizer states hold a special place in quantum information science due to their connection with quantum error correction and quantum circuit simulation. In the context of classical simulations of many-body physics, they are an example of states that can be both highly entangled and efficiently represented and transformed under Clifford operators. Recently, Clifford operators have been discussed in the context of fermionic quantum computation through their extension, the Majorana Clifford group. Here, we document the phase-sensitive form of the corresponding Majorana stabilizer states, as well as the algorithms for computing their amplitudes, their inner products, and update rules for transforming Majorana stabilizer states under Majorana Clifford gates.
We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. Our stabilizer codes generalize the Pauli codes: they are extracted from local commuting projector Hamiltonians whose terms are made of local alphabets more general than the Pauli operators. In particular, not all such codes can be entangled/disentangled with Clifford Quantum Cellular Automata. We compute their Witt-equivalence classes in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. In four spatial dimensions we obtain a group of order two, conjecturally corresponding to the non-trivial invertible phase. Additionally, we propose a representing spectrum for the generalized cohomology theory corresponding to the invertible stabilizer codes.
A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for nnn-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of nnn-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the (n+1)(n+1)(n+1)D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted Z23\mathbb{Z}_2^3Z23​ gauge theory (equivalent to D4\mathbb{D}_4D4​ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in O(d)O(d)O(d) rounds via code switching. In 3D, we construct a transversal logical T\sqrt{\text{T}}T​ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted Z24\mathbb{Z}_2^4Z24​ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the (n+1)(n+1)(n+1)-th level of Clifford hierarchy in nnn spatial dimension.