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.

For block quantum codes, the Hamiltonian can be local, i.e., 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).

Ground states of infinite families geometrically local block-code Hamiltonians can sometimes be said to be a particular phase of (quantum) matter. A phase is a "region in some parameter space in which the ... states possess properties that can be distinguished from those in other phases" [1]. For a large collection of similar subsystems, a phase is a region in some parameter space in which the thermal equilibrium states possess some properties in common that can be distinguished from those in other phases.

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 [25].

Parent

Children

Cousins

  • Fractal liquid code — Fractal liquid codewords form the ground-state space of a class of spin-glass models with three-body interactions.
  • Newman-Moore code — Newman-Moore codewords form the ground-state space of a class of exactly solvable spin-glass models with three-body interactions.
  • Two-component cat code — The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian [6].
  • 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 \(H\).
  • Topological code — Codespace if a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase. Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes).
  • Tetron Majorana code — The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.
  • 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 [7].
  • Bacon-Shor code — The 2D Bacon-Shor code Hamiltonian is the compass model [8,9].
  • Abelian topological code — Subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram. For example, the Kitaev honeycomb Hamiltonian admits the anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) abelian non-chiral non-modular anyon theory [10][11; Footnote 25].
  • GNU permutation-invariant code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [12].

References

[1]
N. Read, “Topological phases and quasiparticle braiding”, Physics Today 65, 38 (2012) DOI
[2]
F. Ticozzi and L. Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”, (2008) arXiv:0809.0613
[3]
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
[4]
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
[5]
V. V. Albert, “Lindbladians with multiple steady states: theory and applications”, (2018) arXiv:1802.00010
[6]
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
[7]
Aleksander Kubica, private communication, 2019
[8]
K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
[9]
J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
[10]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
[11]
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
[12]
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
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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
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}
}
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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.