## Description

Code whose codespace 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].

## Protection

## Encoding

## Parent

## Children

- Matrix-model code — Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.
- Commuting-projector code
- Constant-excitation (CE) code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
- Symmetry-protected self-correcting quantum code
- SYK code — The SYK code Hamiltonian is constructed out of non-commuting few-site terms, and every fermion participates in many interactions.
- Eigenstate thermalization hypothesis (ETH) code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian.
- Matrix-product state (MPS) code — MPS codewords are low-energy excited states of a local Hamiltonian.

## Cousins

- Linear binary code — Hamiltonians whose eigenstates are the canonical basis elements are called classical. One example is the classical Ising model, whose terms are produts of Pauli \(Z\) matrices. Parity-check constraints defining a binary linear code can be encoded in such a model.
- Newman-Moore code — Newman-Moore codewords form the ground-state space of a class of exactly solvable spin-glass models with three-body interactions.
- Classical fractal liquid code — Classical fractal liquid codewords form the ground-state space of a class of 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 of 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).
- 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 [7][8; Footnote 25].
- Lattice stabilizer code — Lattice stabilizer code Hamiltonians are stable with respect to small perturbations [9,10], meaning that the notion of a phase can be defined.
- Quantum low-density parity-check (QLDPC) code — QLDPC code Hamiltonians can be simulated by two-dimensional Hamiltonians with non-commuting terms whose interactions scale with \(n\) [11].
- Majorana box qubit — The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.
- GNU permutation-invariant code — GNU codes lie within the ground state of ferromagnetic Heisenberg models without an external magnetic field [12].
- 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 [13].
- Kitaev surface code — While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [14].
- Bacon-Shor code — The 2D Bacon-Shor gauge-group Hamiltonian is the compass model [15–17].

## 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]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [8]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [9]
- S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
- [10]
- S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
- [11]
- H. Apel and N. Baspin, “Simulating LDPC code Hamiltonians on 2D lattices”, (2023) arXiv:2308.13277
- [12]
- Y. Ouyang, “Quantum storage in quantum ferromagnets”, Physical Review B 103, (2021) arXiv:1904.01458 DOI
- [13]
- Aleksander Kubica, private communication, 2019
- [14]
- C. Fernández-González et al., “Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism”, Physical Review Letters 109, (2012) arXiv:1111.5817 DOI
- [15]
- K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
- [16]
- J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
- [17]
- Z. Nussinov and J. van den Brink, “Compass and Kitaev models -- Theory and Physical Motivations”, (2013) arXiv:1303.5922

## Page edit log

- Victor V. Albert (2022-02-15) — most recent

## Cite as:

“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian