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A new approximate Eastin-Knill theorem

Year: 2025
Type: article
Abstract: Transversal encoded gatesets are highly desirable for fault tolerant quantum computing. However, a quantum error correcting code which exactly corrects for local erasure noise and supports a universal set of transversal gates is ruled out by the Eastin-Knill theorem. Here, we provide a new approximate Eastin-Knill theorem for the single-shot regime when we allow for some probability of error in the decoding. In particular, we show that a quantum error correcting code can support a universal set of transversal gates and approximately correct for local erasure if and only if the conditional min-entropy of the Choi state of the encoding and noise channel is upper bounded by a simple function of the worst-case error probability. Our no-go theorem can be computed by solving a semidefinite program, and, in the spirit of the original Eastin-Knill theorem, is formulated in terms of a condition that is both necessary and sufficient, ensuring achievability whenever it is passed. As an example, we find that with n = 100 physical qutrits we can encode k = 1 logical qubit in the W -state code, which admits a universal transversal set of gates and corrects for single subsystem erasure with error probability of ε = 0.005. To establish our no-go result, we leverage tools from the resource theory of asymmetry, where, in the single-shot regime, a single (output state-dependent) resource monotone governs all state purifications.
DOI: 10.1038/s41534-025-01156-0
Author: Rhea Alexander
Language: English
Open Access status: gold
Estimated APC paid: $4,290
OpenAlex ID: W4417518502
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Cites: 45
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