Frustration-free Hamiltonian code 

Description

Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term.

Encoding

Lindbladian-based dissipative encoding can be constructed for a codespace that is the ground-state subspace of a frustration-free Hamiltonian [14].

Parent

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.
  • Circuit-to-Hamiltonian approximate code — Circuit-to-Hamiltonian approximate codes form the ground-state space of a frustration-free non-commuting projector Hamiltonian whose projectors are constant weight, but such that each physical qubit is acted on by order \(O(\text{polylog}(n))\) projectors.
  • 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.
  • Magnon code — Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian [5].
  • Valence-bond-solid (VBS) code — VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [6,7].

Cousins

  • Commuting-projector Hamiltonian code — Frustration-free Hamiltonians can contain non-commuting projectors; an example is the AKLT model [8]. On the other hand, commuting-projector Hamiltonians can be frustrated; an example is the 1D classical Ising model on a circle for odd \(n\).
  • 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.
  • Eigenstate thermalization hypothesis (ETH) code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH, and many example codes are eigenstates of frsutration-free Hamiltonians.
  • Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian. Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [9].
  • GNU PI code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [10].

References

[1]
F. Ticozzi and L. Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”, (2008) arXiv: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) arXiv:1112.4860 DOI
[3]
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
[4]
V. V. Albert, “Lindbladians with multiple steady states: theory and applications”, (2018) arXiv:1802.00010
[5]
M. Gschwendtner et al., “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
[6]
D.-S. Wang et al., “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
[7]
D.-S. Wang et al., “Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes”, New Journal of Physics 24, 023019 (2022) arXiv:2105.14777 DOI
[8]
I. Affleck et al., “Rigorous Results on Valence-Bond Ground States in Antiferromagnets”, Condensed Matter Physics and Exactly Soluble Models 249 (2004) DOI
[9]
R. Movassagh and Y. Ouyang, “Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians”, (2020) arXiv:2012.01453
[10]
Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
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Zoo Code ID: frustration_free

Cite as:
“Frustration-free Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/frustration_free
BibTeX:
@incollection{eczoo_frustration_free, title={Frustration-free Hamiltonian code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/frustration_free} }
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Cite as:

“Frustration-free Hamiltonian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/frustration_free

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