We study and generalize the class of qubit topological stabilizer codes that arise in the Abelian phase of the honeycomb lattice model. The resulting family of codes, which we call 'matching codes' realize the same anyon model as the surface codes, and so may be similarly used in proposals for quantum computation. We show that these codes are particularly well suited to engineering twist defects that behave as Majorana modes. A proof of principle system that demonstrates the braiding properties of the Majoranas is discussed that requires only three qubits.
A family of stabilizer codes for D(Z2) anyons and Majorana modes
We study and generalize the class of qubit topological stabilizer codes that arise in the Abelian phase of the honeycomb lattice model. The resulting family of codes, which we call 'matching codes' realize the same anyon model as the surface codes, and so may be similarly used in proposals for quantum computation. We show that these codes are particularly well suited to engineering twist defects that behave as Majorana modes. A proof of principle system that demonstrates the braiding properties of the Majoranas is discussed that requires only three qubits.
Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic Z2 topological order in (2+1) dimensions, using Z8 Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for ZDF symmetries for D even. Using this, we realize a stabilizer model for a nontrivial Z8F fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" – anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
Dimer models have long been a fruitful playground for understanding topological physics. Here, we introduce a class, termed Majorana-dimer models, wherein bosonic dimers are decorated with pairs of Majorana modes. We find that the simplest examples of such systems realize an intriguing, intrinsically fermionic phase of matter that can be viewed as the product of a chiral Ising theory, which hosts deconfined non-Abelian quasiparticles, and a topological ${p}_{x}\ensuremath{-}i{p}_{y}$ superconductor. While the bulk anyons are described by a single copy of the Ising theory, the edge remains fully gapped. Consequently, this phase can arise in exactly solvable, frustration-free models. We describe two parent Hamiltonians: one generalizes the well-known dimer model on the triangular lattice, while the other is most naturally understood as a model of decorated fluctuating loops on a honeycomb lattice. Using modular transformations, we show that the ground-state manifold of the latter model unambiguously exhibits all properties of the $\text{Ising}\ifmmode\times\else\texttimes\fi{}({p}_{x}\ensuremath{-}i{p}_{y})$ theory. We also discuss generalizations with more than one Majorana mode per site, which realize phases related to Kitaev's 16-fold way in a similar fashion.
We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the Z4(1) anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.
Stabilizer codes allow for non-local encoding and processing of quantum information. Deformations of stabilizer surface codes introduce new and non-trivial geometry, in particular leading to emergence of long sought after objects known as projective Ising non-Abelian anyons. Braiding of such anyons is a key ingredient of topological quantum computation. We suggest a simple and systematic approach to construct effective unitary protocols for braiding, manipulation and readout of non-Abelian anyons and preparation of their entangled states. We generalize the surface code to a more generic graph with vertices of degree 2, 3 and 4. Our approach is based on the mapping of the stabilizer code defined on such a graph onto a model of Majorana fermions charged with respect to two emergent gauge fields. One gauge field is akin to the physical magnetic field. The other one is responsible for emergence of the non-Abelian anyonic statistics and has a purely geometric origin. This field arises from assigning certain rules of orientation on the graph known as the Kasteleyn orientation in the statistical theory of dimer coverings. Each 3-degree vertex on the graph carries the flux of this “Kasteleyn” field and hosts a non-Abelian anyon. In our approach all the experimentally relevant operators are unambiguously fixed by locality, unitarity and gauge invariance. We illustrate the power of our method by making specific prescriptions for experiments verifying the non-Abelian statistics.