Description
Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other.
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 codes exhibiting check soundness [7] (see also [8]).
2D topological order on qubit manifolds requires weight-four (four-body) Hamiltonian terms, i.e., it cannot be stabilized via weight-two (two-body) or weight-three (three-body) terms on nearly Euclidean geometries of qubits or qutrits [9–11].
Ground-state spaces of commuting-projector Hamiltonians with weight-two (two-body) terms cannot be used to suppress errors in adiabatic quantum computation [12], but this can be circumvented with excited-state subspaces [13] or ground-state subspaces of subsystem code Hamiltonians, e.g., using BBS codes [14,15]. No eigenspace of a weight-two commuting-projector Hamiltonian can simultaneously have \(d > 2\) and dimension greater than 1 [13].
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 codes exhibiting check soundness [7] (see also [8]).
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 codes exhibiting check soundness [7] (see also [8]).
- Subsystem qubit stabilizer code — Ground-state spaces of commuting-projector Hamiltonians with weight-two (two-body) terms cannot be used to suppress errors in adiabatic quantum computation [12], but this can be circumvented with excited-state subspaces [13] or ground-state subspaces of subsystem code Hamiltonians, e.g., using BBS codes [14,15].
- Bravyi-Bacon-Shor (BBS) code — Ground-state spaces of commuting-projector Hamiltonians with weight-two (two-body) terms cannot be used to suppress errors in adiabatic quantum computation [12], but this can be circumvented with excited-state subspaces [13] or ground-state subspaces of subsystem code Hamiltonians, e.g., using BBS codes [14,15].
- 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 [16]. 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 — Qubit 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, M. J. Gullans, V. V. Albert, and M. Barkeshli, “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]
- W. De Roeck, V. Khemani, Y. Li, N. O’Dea, and T. Rakovszky, “LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations”, (2024) arXiv:2411.02384
- [9]
- S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”, (2004) arXiv:quant-ph/0308021
- [10]
- 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
- [11]
- D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
- [12]
- 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
- [13]
- Y. Cao, S. Liu, H. Deng, Z. Xia, X. Wu, and Y.-X. Wang, “Robust analog quantum simulators by quantum error-detecting codes”, (2024) arXiv:2412.07764
- [14]
- Z. Jiang and E. G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression”, Quantum Information Processing 16, (2017) arXiv:1511.01997 DOI
- [15]
- M. Marvian and D. A. Lidar, “Error Suppression for Hamiltonian-Based Quantum Computation Using Subsystem Codes”, Physical Review Letters 118, (2017) arXiv:1606.03795 DOI
- [16]
- I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “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