Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains

Abstract

Quantum error correction was invented to allow for fault-tolerant quantum computation. Systems with topological order turned out to give a natural physical realization of quantum error correcting codes (QECC) in their groundspaces. More recently, in the context of the AdS/CFT correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs.These two examples raise the question of how generally eigenstates of many-body models form quantum codes. In this work we establish new connections between quantum chaos and translationinvariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems exhibiting the Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming approximate quantum errorcorrecting codes. Then we show that AQECC can be obtained probabilistically from translationinvariant energy eigenstates of every translation-invariant spin chain, including integrable models. Applying this result to 1D classical systems, we describe a method for using local symmetries to construct parent Hamiltonians that embed these codes into the low-energy subspace of gapless 1D quantum spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding non-stabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.

Eigenstate thermalization hypothesis and approximate quantum error correction

“…Comparing (2.23) to (2.1), we arrive at the observation made in [8]: regions of finite energy density in the spectrum of theories that satisfy ETH form -correctable [[n, k, d]] approximate quantum error correcting codes with…”

“…In this work, we continue in this direction by studying the error correcting properties of CFTs in the AdS/CFT correspondence. More generally, we analyze the observation made in [8] that systems satisfying the eigenstate thermalization hypothesis (ETH) contain approximate error correcting codes in their spectrums. In doing so, we find a connection between generic chaotic theories, of which holographic CFTs are an example, and quantum error correction.…”

“…where P is the projector onto C, C ij are components of a density matrix, and d(N , N +B) ≤ , where N (ρ) = i,j Tr(ρ)C ij |i j| and (N + B)(ρ) = N (ρ) + i,j Tr(ρB ij ) |i j|. The condition (2.22) leads to a corollary [8]: let {|ψ 1 , . .…”

“…Therefore, the error correction must be approximate, rather than exact. Exploration in this direction has yielded some success [33], and it is this direction that formed the original motivation for this work, and it was suggested in [8] that the presence of AQECCs in ETH systems is support for the QEC picture of AdS/CFT. However, as we will now show, the two codes are very different, but their combination yields the possibility of total bulk reconstruction against erasure errors for black holes.…”

Quantum error-detection at low energies

“…A first step towards showing the ubiquity of such features is the work of Brandao, et. al. [52]. There, it is shown that quantum chaotic systems satisfying the Eigenstate Thermalization Hypothesis (ETH) have energy eigenstates that form approximate quantum error-correcting codes.…”