# Eigenstate thermalization hypothesis (ETH) code[1]

## Description

Also called a thermodynamic code [2]. An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains, Motzkin chains, or Heisenberg models.

ETH requires that for ordered energy eigenstates \(|E_l\rangle\) and any local observable \(O\), \begin{align} |\langle E_l|O|E_l\rangle-\langle E_{l+1}|O|E_{l+1}\rangle|\leq\exp(-cn) \tag*{(1)}\end{align} for a constant \(c\). This implies that energy eigenstates around some energy \(\bar E\) are approximately locally indistinguishable from one another, as their reduced density matrices on any subsystem are both approximately thermal at energy \(\bar E\). In this way, global information is protected from local measurements by the environment as \(n\to\infty\).

## Protection

## Decoding

## Parents

- Qubit code
- Hamiltonian-based code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH.
- Approximate quantum error-correcting code (AQECC) — ETH codes approximately protect against erasures in the thermodynamic limit.

## Cousins

- Topological code — ETH codewords, like topological codewords, are locally indistinguishable.
- Permutation-invariant qubit code — Several instances of ETH codes contain permutation-invariant qubit codewords.
- Covariant code — ETH codes consisting of Dicke states are approximately \(U(1)\)-covariant and nearly saturate certain covariance-performance bounds [2,4].
- Matrix-product state (MPS) code — MPS codes have been shown to protect against non-geometrically local noise, while ETH codes protect only against erasures on geometrically local patches.

## References

- [1]
- F. G. S. L. Brandão et al., “Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains”, Physical Review Letters 123, (2019) arXiv:1710.04631 DOI
- [2]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [3]
- N. Bao and N. Cheng, “Eigenstate thermalization hypothesis and approximate quantum error correction”, Journal of High Energy Physics 2019, (2019) arXiv:1906.03669 DOI
- [4]
- Z.-W. Liu and S. Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”, (2023) arXiv:2111.06360

## Page edit log

- Victor V. Albert (2022-01-01) — most recent
- Chris Fechisin (2021-12-13)

## Cite as:

“Eigenstate thermalization hypothesis (ETH) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/eth