Description
Lattice stabilizer code in two Euclidean dimensions.
Any translation-invariant 2D prime-qudit lattice stabilizer code can be converted to several copies of the prime-qudit 2D surface code along with some trivial codes [1]. Any 2D topological order requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on 2D lattices of qubits or qutrits [2–4].
Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth Clifford circuit [1]. There are algorithms which determine the fusion and braiding rules [5] as well as boundaries and twist defects [6] of a 2D translationally invariant modular-qudit stabilizer code for any qudit dimension.
Encoding
The geometric entanglement measure of a 2D stabilizer codeword with sufficiently high distance \(d\) scales as order \(\Omega(d^2)\) [7].Decoding
Renormalization group (RG) decoder [8].Tensor-network based decoder for 2D codes subject to correlated noise [9].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [10–12]; see also Refs. [13–15].Real-time geometrically local decoder based on introducing an ancillary buffer and confining spacetime interactions between anyons [16].Code Capacity Threshold
Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [17], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [9,18–20]. Namely, the noise threshold for a noise channel \(\cal{E}\) acting on a 2D stabilizer state \(|\psi\rangle\) can be obtained from the properties of the resulting (mixed) state \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [17,21–24].Cousins
- Kitaev surface code— Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth qudit Clifford circuit [25–27].
- Abelian quantum-double stabilizer code— Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth qudit Clifford circuit [1].
- Holographic code— 2D lattice stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom [28].
- Qubit QLDPC code— Chain complexes describing qubit QLDPC codes can be converted to 2D lattice stabilizer codes [29].
- Self-correcting quantum code— 2D stabilizer codes [30] and encodings of frustration-free code Hamiltonians [31] admit only constant-energy excitations, and so do not have an energy barrier.
Primary Hierarchy
Parents
2D lattice stabilizer code
Children
The Majorana color code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
The Majorana surface code is a 2D qubit stabilizer code with respect to the Majorana operator basis.
The 2D bosonization code encodes fermionic modes into a 2D qubit stabilizer code.
Bivariate bicycle codes are defined on 2D lattices with periodic boundary conditions, and versions with open boundary conditions have been investigated [6,32]. Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators. BB codes have been investigated in terms of their anyons and topological order [33].
All Abelian TQD codes can be realized as modular-qudit lattice stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [34]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
References
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Page edit log
- Victor V. Albert (2024-01-27) — most recent
Cite as:
“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer