Covariant Approximate Quantum Codes for Protected Analog Computation
Abstract
Quantum error correction compatible with continuous symmetries is a fundamental problem in quantum information and a possible route to robust analog quantum simulation. Because the Eastin-Knill theorem forbids exact codes with continuous transversal symmetries, we construct explicit SU(d)-covariant approximate codes that exploit permutation symmetry to spread logical information uniformly across all physical subsystems. For one-, two-, and three-qudit erasures at known locations, we prove worst-case purified-distance scaling O(1/N), which matches known approximate Eastin-Knill lower bounds up to constants, and we extend the reduced-state analysis to general flagged local noise. Exact permutation symmetry of this kind is available only up to three erased sites; for an arbitrary fixed number k of erased sites, we instead show that Haar-random and efficiently sampled encodings achieve worst-case distance O(√k/N) with probability exponentially close to one. For single-qudit erasure, we construct an explicit near-optimal decoder from the Petz recovery map. We then use these codes as building blocks for encoded analog dynamics. Symmetry-preserving Hamiltonians generate block-structured dynamical Lie algebras implementable transversally, while controlled symmetry-breaking terms serve as non-transversal resources for universal dynamics. These results provide explicit non-Abelian covariant codes and a framework for robust analog quantum simulation.
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BibTeX
@misc{elovenkova2026covariant,
title = {{Covariant Approximate Quantum Codes for Protected Analog Computation}},
author = {Mariia Elovenkova and Hong-Ye Hu and Susanne F. Yelin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.07607},
eprint = {2607.07607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07607},
}
RIS
TY - UNPB
TI - Covariant Approximate Quantum Codes for Protected Analog Computation
AU - Mariia Elovenkova
AU - Hong-Ye Hu
AU - Susanne F. Yelin
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2607.07607
UR - https://arxiv.org/abs/2607.07607
AB - Quantum error correction compatible with continuous symmetries is a fundamental problem in quantum information and a possible route to robust analog quantum simulation. Because the Eastin-Knill theorem forbids exact codes with continuous transversal symmetries, we construct explicit SU(d)-covariant approximate codes that exploit permutation symmetry to spread logical information uniformly across all physical subsystems. For one-, two-, and three-qudit erasures at known locations, we prove worst-case purified-distance scaling O(1/N), which matches known approximate Eastin-Knill lower bounds up to constants, and we extend the reduced-state analysis to general flagged local noise. Exact permutation symmetry of this kind is available only up to three erased sites; for an arbitrary fixed number k of erased sites, we instead show that Haar-random and efficiently sampled encodings achieve worst-case distance O(√k/N) with probability exponentially close to one. For single-qudit erasure, we construct an explicit near-optimal decoder from the Petz recovery map. We then use these codes as building blocks for encoded analog dynamics. Symmetry-preserving Hamiltonians generate block-structured dynamical Lie algebras implementable transversally, while controlled symmetry-breaking terms serve as non-transversal resources for universal dynamics. These results provide explicit non-Abelian covariant codes and a framework for robust analog quantum simulation.
ER -