# Quantum Error-Detection at Low Energies

## TL;DR

This paper explores the construction of approximate quantum error-detecting codes within matrix product states, revealing that both gapped and gapless many-body systems can inherently contain such codes with significant error-detection capabilities.

## Contribution

It introduces a method to construct error-detecting codes from excited states in MPS, overcoming previous limitations of boundary-to-bulk encoding maps, and demonstrates their existence in gapped and gapless models.

## Key findings

- Error-detecting codes with distance $oldsymbol{ ext{Omega}(n^{1-
u})}$ in gapped systems.
- Codes encoding $oldsymbol{ ext{Omega}( ext{log } n)}$ qubits.
- Existence of error-detecting codes in low-energy eigenspaces of gapless models.

## Abstract

Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance $\Omega(n^{1-\nu})$ for any $\nu\in (0,1)$ and $\Omega(\log n)$ encoded qubits. This shows that gapped systems contain $-$ within isolated energy bands $-$ error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains $-$ within its low-energy eigenspace $-$ an error-detecting code with the same parameter scaling. All these codes detect arbitrary $d$-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features.