## Description

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

## Protection

Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order (LTQO) condition (cf. the TQO conditions), meaning that a notion of a phase can be defined [1,2].

## Encoding

Lindbladian-based dissipative encoding can be constructed for a codespace that is the ground-state subspace of a frustration-free Hamiltonian [3–6].

## 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 [7].
- Valence-bond-solid (VBS) code — VBS codewords are eigenstates of the frustration-free VBS Hamiltonian [8,9].

## Cousins

- Commuting-projector Hamiltonian code — Frustration-free Hamiltonians can contain non-commuting projectors; an example is the AKLT model [10]. On the other hand, commuting-projector Hamiltonians can be frustrated; an example is the 1D classical Ising model on a circle for odd \(n\) with one two-body interaction having the opposite sign.
- Topological code — Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the TQO conditions), meaning that a notion of a phase can be defined [1,2].
- 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 [11].
- GNU PI code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [12].

## References

- [1]
- S. Michalakis and J. P. Zwolak, “Stability of Frustration-Free Hamiltonians”, Communications in Mathematical Physics 322, 277 (2013) arXiv:1109.1588 DOI
- [2]
- S. Bachmann et al., “Stability of invertible, frustration-free ground states against large perturbations”, Quantum 6, 793 (2022) arXiv:2110.11194 DOI
- [3]
- F. Ticozzi and L. Viola, “Analysis and synthesis of attractive quantum Markovian dynamics”, (2008) arXiv:0809.0613
- [4]
- 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
- [5]
- 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
- [6]
- V. V. Albert, “Lindbladians with multiple steady states: theory and applications”, (2018) arXiv:1802.00010
- [7]
- M. Gschwendtner et al., “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
- [8]
- D.-S. Wang et al., “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
- [9]
- 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
- [10]
- I. Affleck et al., “Rigorous Results on Valence-Bond Ground States in Antiferromagnets”, Condensed Matter Physics and Exactly Soluble Models 249 (2004) DOI
- [11]
- 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
- [12]
- Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI

## Page edit log

- Victor V. Albert (2024-06-06) — most recent

## Cite as:

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