## Description

## Protection

Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [1–5]. This notion can be extended to semi-hyperbolic manifolds [6] and non-geometrically local QLDPC exhibiting check soundness [7].

2D topological order on qubit manifolds requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on nearly Euclidean geometries of qubits or qutrits [8–10]. Hamiltonians with weight-two (two-body) terms cannot be used for suppressing errors [11].

## Parent

- Hamiltonian-based code — Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [1–5]. This notion can be extended to semi-hyperbolic manifolds [6] and non-geometrically local QLDPC exhibiting check soundness [7].

## Children

- Two-gauge theory code — Two-gauge theory codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
- Multi-fusion string-net code — Multi-fusion string-net codes form eigenspaces of frustration-free commuting projector Hamiltonians.
- \(G\)-enriched Walker-Wang model code — \(G\)-enriched Walker-Wang model codewords form ground-state subspaces of frustration-free commuting projector Hamiltonians.
- Quantum locally testable code (QLTC) — Quantum LTC codespaces are ground-state spaces of \(u\)-local frustration-free commuting-projector Hamiltonians.
- Stabilizer code — Codespace is the ground-state space of the code Hamiltonian, which consists of an equal linear combination of stabilizer generators and which can be made into a frustration-free commuting-projector Hamiltonian.
- Quantum repetition code — The codespace of the quantum repetition code is the ground-state space of a frustration-free classical Ising model with nearest-neighbor interactions.

## Cousins

- Topological code — Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [1–5]. This notion can be extended to semi-hyperbolic manifolds [6] and non-geometrically local QLDPC exhibiting check soundness [7].
- Linear binary code — Parity-check constraints defining a binary linear code can be encoded into a classical Ising model Hamiltonian, a commuting-projector model whose terms contain produts of Pauli \(Z\) matrices participating in each parity check. Such Ising models are also frustration-free since the codewords satisfy all parity checks.
- Classical fractal liquid code — Classical fractal liquid codewords form the ground-state space of a class of exactly solvable spin-glass Ising models with three-body interactions.
- Frustration-free Hamiltonian code — Frustration-free Hamiltonians can contain non-commuting projectors; an example is the AKLT model [12]. On the other hand, commuting-projector Hamiltonians can be frustrated; an example is the 1D classical Ising model on a circle for odd \(n\) with one two-body interaction having the opposite sign.
- Quantum LDPC (QLDPC) code — QLDPC codes with check soundness, meaning that every weight-\(m\) stabilizer can be written as a product of order \(O(m)\) stabilizer generators, are robust against few-body perturbations. This means that phases of matter can be defined from certain non-geometrically local QLDPC code Hamiltonians [7].

## References

- [1]
- S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
- [2]
- S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
- [3]
- S. Michalakis and J. P. Zwolak, “Stability of Frustration-Free Hamiltonians”, Communications in Mathematical Physics 322, 277 (2013) arXiv:1109.1588 DOI
- [4]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms”, Journal of Mathematical Physics 60, (2019) arXiv:1810.02428 DOI
- [5]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States”, Annales Henri Poincaré 23, 393 (2021) arXiv:2010.15337 DOI
- [6]
- A. Lavasani et al., “On stability of k-local quantum phases of matter”, (2024) arXiv:2405.19412
- [7]
- C. Yin and A. Lucas, “Low-density parity-check codes as stable phases of quantum matter”, (2024) arXiv:2411.01002
- [8]
- S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”, (2004) arXiv:quant-ph/0308021
- [9]
- D. Aharonov and L. Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”, (2011) arXiv:1102.0770
- [10]
- D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
- [11]
- I. Marvian and D. A. Lidar, “Quantum Error Suppression with Commuting Hamiltonians: Two Local is Too Local”, Physical Review Letters 113, (2014) arXiv:1410.5487 DOI
- [12]
- I. Affleck et al., “Rigorous Results on Valence-Bond Ground States in Antiferromagnets”, Condensed Matter Physics and Exactly Soluble Models 249 (2004) DOI

## Page edit log

- Victor V. Albert (2023-05-15) — most recent

## Cite as:

“Commuting-projector Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/commuting_projector