Hamiltonian-based code


Encoding corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy.

When the physical space is a tensor product of subsystems, the Hamiltonian is typically local, consisting of operators acting on a number of subsystems that is independent of the total number of subsystems (e.g., QLDPC codes). When the physical space is endowed with a geometry, the Hamiltonian is typically geometrically local, consisting of operators acting on subsystems that occupy a region whose size is independent of the number of subsystems (e.g., topological codes). When the terms in a geometrically local Hamiltonian commute and can be expressed as projectors (i.e., having eigenvalues 0 or 1), the Hamiltonian is called commuting-projector.


Often determined from the underlying physical properties of the Hamiltonian.


Lindbladian-based dissipative encoding can be constructed for a codespace that is the ground-state subspace of a frustration-free Hamiltonian [1][2][3][4].




  • Two-component cat code — The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian [5].
  • Pair-cat code — Two-legged pair-cat codewords form ground-state subspace of a multimode Kerr Hamiltonian.
  • Error-corrected sensing code — Metrologically optimal codes admit a \(U(1)\) set of gates generated by a signal Hamiltonian \(H\), meaning that there exists a basis of codewords that are eigenstates of the \(H\).
  • Tetron Majorana code — The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.
  • Qubit 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 commuting projector Hamiltonian.
  • Quantum repetition code — Bit-flip codespace is the ground-state space of a one-dimensional classical Ising model with nearest-neighbor interactions.
  • Five-qubit perfect code — The five-qubit code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [6].
  • Bacon-Shor code — The 2D Bacon-Shor code Hamiltonian is the compass model [7][8].
  • GNU permutation-invariant code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [9].


F. Ticozzi and L. Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”, (2008) arXiv:0809.0613
F. Ticozzi and L. Viola, “Stabilizing entangled states with quasi-local quantum dynamical semigroups”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 5259 (2012) arXiv:1112.4860 DOI
F. Verstraete, M. M. Wolf, and J. I. Cirac, “Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation”, (2008) arXiv:0803.1447
V. V. Albert, “Lindbladians with multiple steady states: theory and applications”, (2018) arXiv:1802.00010
S. Puri, S. Boutin, and A. Blais, “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving”, npj Quantum Information 3, (2017) arXiv:1605.09408 DOI
Aleksander Kubica, private communication, 2019
K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: hamiltonian

Cite as:
“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian
@incollection{eczoo_hamiltonian, title={Hamiltonian-based code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hamiltonian} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/hamiltonian/hamiltonian.yml.