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

Brandão, Fernando G. S. L. and Crosson, Elizabeth and Şahinoğlu, M. Burak and Bowen, John (2019) Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains. Physical Review Letters, 123 (11). Art. No. 110502. ISSN 0031-9007. doi:10.1103/PhysRevLett.123.110502.

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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 ground spaces. More recently, in the context of the anti-de Sitter/conformal field theory 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 Letter we establish new connections between quantum chaos and translation invariance 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 obeying the eigenstate thermalization hypothesis have eigenstates forming approximate quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from translation-invariant 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 nonstabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.

Item Type:Article
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URLURL TypeDescription Paper
Brandão, Fernando G. S. L.0000-0003-3866-9378
Şahinoğlu, M. Burak0000-0003-4488-2300
Additional Information:© 2019 American Physical Society. Received 14 February 2018; published 11 September 2019. We thank Xi Dong, Tarun Grover, Nick Hunter-Jones, Robert Koenig, John Preskill for discussions. E. C. is grateful for support provided by the Institute for Quantum Information and Matter, with support of the Gordon and Betty Moore Foundation (GBMF-12500028). M. B. S. acknowledges the support from the Simons Foundation through It from Qubit collaboration; this work was supported by a grant from the Simons Foundation/SFARI (385612, JPP). F. B, E. C. and M. B. S. were supported by an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-12500028
Simons Foundation385612
Issue or Number:11
Record Number:CaltechAUTHORS:20190206-155714600
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:92748
Deposited By: Bonnie Leung
Deposited On:15 Feb 2019 21:00
Last Modified:16 Nov 2021 03:53

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