@misc{10481/110641,
year = {2025},
month = {12},
url = {https://hdl.handle.net/10481/110641},
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 worstcase
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.},
organization = {Ministry for Digital Transformation and of Civil Service of the Spanish Government through projects, PID2024-162155OB-I00, PID2021-128970OA-I00 10.13039/501100011033 and QUANTUM ENIA project call - Quantum Spain project},
organization = {European Union through the Recovery, Transformation and Resilience Plan - NextGenerationEU within the framework of the Digital Spain 2026 Agenda},
organization = {EU HORIZON RIA FoQaCiA GA 101070558},
publisher = {Springer Nature},
title = {A new approximate Eastin-Knill theorem},
doi = {10.1038/s41534-025-01156-0},
author = {Alexander-Turner, Rhea},
}