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arXiv:2310.04710 · Nature Physics

Complexity and order in approximate quantum error-correcting codes

Jinmin Yi, Weicheng Ye, Daniel Gottesman, Zi-Wen Liu

Published Sep 3, 2024• 17 citations
arXivHTMLPDF
DOI
10.1038/s41567-024-02621-x

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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.
Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behavior of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We also derive general methods for reducing the computational complexity of calculating the exact effective logical noise by many orders of magnitude. This reduction is achieved by determining when different recovery operations produce equivalent logical noise. These methods could also be used to better approximate soft decoding schemes for concatenated codes or to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error-correcting codes. We give examples of such reductions for the three-qubit, five-qubit, Steane, concatenated, and toric codes.