# Eigenstate thermalization hypothesis (ETH) code[1]

## Description

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 or frustration-free systems such as Motzkin chains and 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

Approximately protects against erasure errors at known locations. Translation invariance alone is sufficient for good approximate error-correcting properties in a many-body spectrum, including in integrable models [1]. The ETH code generated from the spectrum of the translation-invariant 1D Heisenberg spin chain [1] has recovery infidelity (against the erasure of a constant number of sites) scale as \(\epsilon_\text{worst}=O(1/n)\) [2].'

The ETH code defined on a Heisenberg spin chain has unbouldable codespace complexity [3].

## Decoding

## Parents

- Qubit code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH.
- Hamiltonian-based code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH, and many example codes are eigenstates of frsutration-free Hamiltonians.
- 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.
- PI qubit code — Several instances of ETH codes contain PI qubit codewords.
- Spin code — Relevant many-body systems housing ETH codes include 1D non-interacting spin chains or frustration-free systems such as Motzkin chains and Heisenberg models.
- Frustration-free Hamiltonian code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH, and many example codes are eigenstates of frsutration-free Hamiltonians.
- Covariant block quantum code — ETH codes consisting of Dicke states are approximately \(U(1)\)-covariant and nearly saturate certain covariance-performance bounds [2,5].
- Magnon code — Magnon 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]
- J. Yi et al., “Complexity and order in approximate quantum error-correcting codes”, Nature Physics (2024) arXiv:2310.04710 DOI
- [4]
- N. Bao and N. Cheng, “Eigenstate thermalization hypothesis and approximate quantum error correction”, Journal of High Energy Physics 2019, (2019) arXiv:1906.03669 DOI
- [5]
- 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