# 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

## Encoding

## Parent

## Children

- Constant-excitation (CE) code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
- Eigenstate thermalization hypothesis (ETH) code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH.
- Fracton code — Codespace is the ground-state subspace of a geometrically local commuting-projector Hamiltonian admitting a fracton phase.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian.
- Self-correcting quantum code
- Topological code — Codespace is either the ground-state or low-energy subspace of a geometrically local commuting-projector Hamiltonian admitting a topological phase.

## 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

## 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.