Hamiltonian-based code

Description

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.

Protection

Often determined from the underlying physical properties of the Hamiltonian.

Encoding

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].

Parent

Children

Cousins

  • 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)

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
BibTeX:
@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:
https://errorcorrectionzoo.org/c/hamiltonian

References

[1]
Francesco Ticozzi and Lorenza Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”. 0809.0613
[2]
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
[3]
Frank Verstraete, Michael M. Wolf, and J. Ignacio Cirac, “Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation”. 0803.1447
[4]
Victor V. Albert, “Lindbladians with multiple steady states: theory and applications”. 1802.00010
[5]
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
[6]
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021). DOI; 1904.01458
[7]
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.