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].




  • Quantum low-density parity-check (QLDPC) code — Hamiltonian-based codes are not generally defined using Pauli strings. However, codes forming the ground-state subspace of a local Hamiltonain consisting of commuting terms are QLDPC codes in the sense that they satisfy the QLDPC locality requirements.
  • Cat code — Two-legged cat codewords form ground-state subspace of a Kerr Hamiltonian [5].
  • GNU permutation-invariant code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [6].
  • Quantum repetition code — Bit-flip codespace is the ground-state space of a one-dimensional classical Ising model with nearest-neighbor interactions.
  • 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.
  • \([[5,1,3]]\) perfect code — \([[5,1,3]]\) code Hamiltonian is local when expressed in terms of mutually commuting Majorana operators [7].

Zoo code information

Internal code ID: hamiltonian

Your contribution is welcome!

on github.com (edit & pull request)

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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} }
Permanent link:


Francesco Ticozzi and Lorenza Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”. 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). DOI; 1112.4860
Frank Verstraete, Michael M. Wolf, and J. Ignacio Cirac, “Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation”. 0803.1447
Victor V. Albert, “Lindbladians with multiple steady states: theory and applications”. 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). DOI; 1605.09408
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021). DOI; 1904.01458
Aleksander Kubica, private communication, 2019

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.yml.