Nearly Optimal T-Count for Symmetry-Based Quantum State Purification
Abstract
Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For n copies of a d-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most ε using Õ(n) T gates, where the tilde-O notation suppresses polylogarithmic factors in n, d, and 1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<1/2√(1−1/n) requires at least 2n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
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BibTeX
@misc{tang2026nearly,
title = {{Nearly Optimal T-Count for Symmetry-Based Quantum State Purification}},
author = {Jialiang Tang and Junhong Nie and Keming He and Xin Wang and Xiaoming Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12098},
eprint = {2610.12098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12098},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nearly Optimal T-Count for Symmetry-Based Quantum State Purification
AU - Jialiang Tang
AU - Junhong Nie
AU - Keming He
AU - Xin Wang
AU - Xiaoming Sun
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12098
UR - https://arxiv.org/abs/2610.12098
AB - Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For n copies of a d-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most ε using Õ(n) T gates, where the tilde-O notation suppresses polylogarithmic factors in n, d, and 1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<1/2√(1−1/n) requires at least 2n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
ER -