Three-fermion (3F) Walker-Wang model code

Three-fermion (3F) Walker-Wang model code[1,2]

Description

A 3D lattice stabilizer code whose bulk realizes a 3D time-reversal SPT order [1] and whose gapped boundary supports the 2D three-fermion (3F) topological order. The code can be used as a resource state for fault-tolerant MBQC [2].

Encoding

3F QCA encoder [3,4], which can be simplified using bosonization [5] and can be extended to SPTs in higher dimensions based on an exact bosonization duality [6].

Gates

Clifford gates can be performed by braiding and fusing symmetry defects in the MBQC model.

Fault Tolerance

Fault-tolerant MBQC protocol by encoding in, braiding, and fusing symmetry defects.

Cousins

3D bosonization code— The 3F Walker-Wang QCA encoder [3,4] can be simplified using bosonization [5].
Bosonization code— The 3F Walker-Wang QCA encoder [3,4] can be extended to SPTs in higher dimensions based on an exact bosonization duality [6].
Abelian topological code— The gapped boundary of the 3F Walker-Wang model supports the 3F topological order [1,2].
Three-fermion (3F) subsystem code— The (three-dimensional) 3F Walker-Wang model cluster-like state encodes the temporal gate operations on the (two-dimensional) 3F subsystem code into a third spatial dimension [2].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Parents

Qubit QLDPC code

3D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Walker-Wang model codeTopological Hamiltonian-based QECC Quantum

The Walker-Wang model code reduces to the 3F model code when the input category \(\mathcal{C}=3F\) [2]. When treated as ground states of the code Hamiltonian, 3F Walker-Wang model code states realize a 3D time-reversal SPT order [1], while the gapped boundary supports the 3F anyon theory.

Symmetry-protected topological (SPT) codeTopological Hamiltonian-based QECC Quantum

When treated as ground states of the code Hamiltonian, 3F Walker-Wang model code states realize a 3D time-reversal SPT order [1]. The 3F Walker-Wang QCA encoder [3,4] can be extended to SPTs in higher dimensions based on an exact bosonization duality [6].

Three-fermion (3F) Walker-Wang model code

References

[1]: F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath, “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order”, Physical Review B 90, (2014) arXiv:1302.7072 DOI
[2]: S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
[3]: J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics 398, 469 (2022) arXiv:1812.01625 DOI
[4]: J. Haah, “Topological Phases of Unitary Dynamics: Classification in Clifford Category”, Communications in Mathematical Physics 406, (2025) arXiv:2205.09141 DOI
[5]: L. Fidkowski and M. B. Hastings, “Pumping Chirality in Three Dimensions”, (2024) arXiv:2309.15903
[6]: L. Fidkowski, J. Haah, and M. B. Hastings, “A quantum cellular automaton for every symmetry protected topological phase”, Physical Review B 112, (2025) arXiv:2407.07951 DOI

Page edit log

Victor V. Albert (2023-03-28) — most recent

Cite as:

“Three-fermion (3F) Walker-Wang model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/three_fermion

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/three_fermion.yml.