Optimal learning of covariant quantum states and channels
Abstract
We give collective tomography protocols for quantum states and channels with known symmetries, using random purification and dilation to reduce learning to pure-state estimation. For states commuting with a compact-group representation with multiplicities mλ, the optimal copy complexity is Θ((∑λmλ2+logη−1)/ε2) for sufficiently small trace-distance error ε and failure probability η for the nontrivial case ∑λmλ2>1. For G-covariant channels with finite-dimensional unitary representations of a compact group G, parallel Õ(DG/ε2) queries achieve a diamond-distance error $\varepsilon $ at fixed success probability 2/3, where DG counts the real parameters of covariant Choi operators before imposing trace preservation. For permutation-covariant channels on k qudits of fixed local dimension d, at sufficiently small fixed error and fixed success probability 2/3, we obtain the optimal query scalings Θd(kd4−1) for diamond distance and Θd(kd4−d2) for Choi trace distance. Furthermore, we resolve an open problem in quantum tomography of constructing efficient quantum circuits that approximate the Hayashi measurement for optimal pure-state estimation. We encode states in the symmetric subspace into bosonic occupation modes, realize the measurement by heterodyne detection and normalization, and approximate this procedure on qubits using the quantum Hermite transform. Combined with our symmetry-compatible purification and dilation circuits, this yields query-optimal and gate-efficient learners of permutation-covariant states and channels with gate complexity Od(poly(k,ε−1,logη−1)).
Related
- From Symmetry to Secrecy: Covariant Classical--Quantum Wiretap Channels
- Near-Optimal Learning of Quantum Many-Body Systems from Gibbs and Thermal Metastable States via the Stationarity Test
- Near-optimal incoherent tomography of low-rank quantum channels
- Private Capacity of Quantum Channels Induced by Non-stabilizer Environmental States
- Joint GKP Encoding for GHZ States Transmission over Bosonic Quantum MIMO Channels
BibTeX
@misc{yoshida2026optimal,
title = {{Optimal learning of covariant quantum states and channels}},
author = {Satoshi Yoshida and Kazuki Okigami and Pietro M. Posta and Dmitry Grinko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39280},
eprint = {2609.39280},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39280},
}
RIS
TY - UNPB
TI - Optimal learning of covariant quantum states and channels
AU - Satoshi Yoshida
AU - Kazuki Okigami
AU - Pietro M. Posta
AU - Dmitry Grinko
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39280
UR - https://arxiv.org/abs/2609.39280
AB - We give collective tomography protocols for quantum states and channels with known symmetries, using random purification and dilation to reduce learning to pure-state estimation. For states commuting with a compact-group representation with multiplicities m_λ, the optimal copy complexity is Θ((∑_λm_λ^2+logη^−1)/ε^2) for sufficiently small trace-distance error ε and failure probability η for the nontrivial case ∑_λm_λ^2>1. For G-covariant channels with finite-dimensional unitary representations of a compact group G, parallel Õ(D_G/ε^2) queries achieve a diamond-distance error $\varepsilon $ at fixed success probability 2/3, where D_G counts the real parameters of covariant Choi operators before imposing trace preservation. For permutation-covariant channels on k qudits of fixed local dimension d, at sufficiently small fixed error and fixed success probability 2/3, we obtain the optimal query scalings Θ_d(k^d^4−1) for diamond distance and Θ_d(k^d^4−d^2) for Choi trace distance. Furthermore, we resolve an open problem in quantum tomography of constructing efficient quantum circuits that approximate the Hayashi measurement for optimal pure-state estimation. We encode states in the symmetric subspace into bosonic occupation modes, realize the measurement by heterodyne detection and normalization, and approximate this procedure on qubits using the quantum Hermite transform. Combined with our symmetry-compatible purification and dilation circuits, this yields query-optimal and gate-efficient learners of permutation-covariant states and channels with gate complexity O_d(poly(k,ε^−1,logη^−1)).
ER -