arXiv:2609.39280

Optimal learning of covariant quantum states and channels

Satoshi Yoshida, Kazuki Okigami, Pietro M. Posta, Dmitry Grinko

quant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations

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 $m_λ$, the optimal copy complexity is $Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)$ for sufficiently small trace-distance error $\varepsilon$ and failure probability $η$ for the nontrivial case $\sum_λm_λ^2>1$. For $G$-covariant channels with finite-dimensional unitary representations of a compact group $G$, parallel $\widetilde{O}(D_G/\varepsilon^2)$ 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 $O_d(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))$.

Comments: 95 pages, 6 figures

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arXiv:2609.39280

Optimal learning of covariant quantum states and channels

Satoshi Yoshida, Kazuki Okigami, Pietro M. Posta, Dmitry Grinko

quant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations

arXiv HTML PDF

Abstract

m_λ

, the optimal copy complexity is

Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)

for sufficiently small trace-distance error

\varepsilon

and failure probability

η

for the nontrivial case

\sum_λm_λ^2>1

. For

G

-covariant channels with finite-dimensional unitary representations of a compact group

G

, parallel

\widetilde{O}(D_G/\varepsilon^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(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))

Comments: 95 pages, 6 figures

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m_λ

, the optimal copy complexity is

Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)

for sufficiently small trace-distance error

\varepsilon

and failure probability

η

for the nontrivial case

\sum_λm_λ^2>1

. For

G

-covariant channels with finite-dimensional unitary representations of a compact group

G

, parallel

\widetilde{O}(D_G/\varepsilon^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})

O_d(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))

d_{\mathrm{in}},d_{\mathrm{out}}

and Kraus rank at most

k

can be learned to accuracy

\varepsilon

using

O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})

channel uses. Conversely, we prove that

Ω(d_{\mathrm{in}}d_{\mathrm{out}}k)

uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate

Ω(1/\varepsilon^{2})

contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.

We study the estimation of an unknown quantum channel

\mathcal{E}

with input dimension

d_1

, output dimension

d_2

and Kraus rank at most

r

. We establish a connection between the query complexities in two models: (i) access to

\mathcal{E}

, and (ii) access to a random dilation of

\mathcal{E}

. Specifically, we show that for parallel (possibly coherent) testers, access to dilations does not help. This is proved by constructing a local tester that uses

n

queries to

\mathcal{E}

yet faithfully simulates the tester with

n

queries to a random dilation. As application, we show that: -

O(rd_1d_2/\varepsilon^2)

queries to

\mathcal{E}

suffice for channel tomography to within diamond norm error

\varepsilon

. Moreover, when

rd_2=d_1

, we show that the Heisenberg scaling

O(1/\varepsilon)

can be achieved, even if

\mathcal{E}

is not a unitary channel: -

O(\min\{d_1^{2.5}/\varepsilon,d_1^2/\varepsilon^2\})

queries to

\mathcal{E}

suffice for channel tomography to within diamond norm error

\varepsilon

, and

O(d_1^2/\varepsilon)

queries suffice for the case of Choi state trace norm error

\varepsilon

. -

O(\min\{d_1^{1.5}/\varepsilon,d_1/\varepsilon^2\})

queries to

\mathcal{E}

suffice for tomography of the mixed state

\mathcal{E}(|0\rangle\langle 0|)

to within trace norm error

\varepsilon

d_1

, output dimension

d_2

, and Kraus rank at most

r

, to within error

\varepsilon

. We identify the dilation rate

τ= r d_2 / d_1

(which always satisfies

τ\geq 1

due to the trace preservation of quantum channels) as a key parameter, and establish that the optimal query complexity of channel tomography exhibits distinct scaling laws across three regimes of

τ

. - In the boundary regime (

τ= 1

): we show that the query complexity is

Θ(r d_1 d_2/\varepsilon)

for Choi trace norm error

\varepsilon

, and is upper bounded by

O(\min\{r d_1^{1.5} d_2/\varepsilon, r d_1 d_2/\varepsilon^2\})

and lower bounded by

Ω(r d_1 d_2/\varepsilon)

for diamond norm error

\varepsilon

. - In the away-from-boundary regime (

τ\geq 1+Ω(1)

): we show that the query complexity is

Θ(r d_1 d_2/\varepsilon^2)

for both Choi trace norm and diamond norm errors

\varepsilon

. Our results uncover a sharp Heisenberg-to-classical phase transition in the query complexity of quantum channel tomography: at

τ=1

, the optimal query complexity exhibits Heisenberg scaling

1/\varepsilon

, whereas for

τ\geq 1+Ω(1)

, it exhibits classical scaling

1/\varepsilon^2

. In addition, we show that in the near-boundary regime (

1< τ< 1+o(1)

), the query complexity exhibits a mixture of Heisenberg and classical scaling behaviors.

The recently introduced random purification channel, which converts

n

copies of an arbitrary mixed quantum state into

n

copies of the same uniformly random purification, has emerged as a powerful tool in quantum information theory. Motivated by this development, we introduce a channel-level analogue, which we call the random Stinespring superchannel. This consists in a procedure to transform

n

parallel queries of an arbitrary quantum channel into

n

parallel queries of the same uniformly random Stinespring isometry, via universal encoding and decoding operations that are efficiently implementable. When the channel is promised to have Choi rank at most

r

, the procedure can be tailored to yield a Stinespring environment of dimension

r

. We present two applications of the random Stinespring superchannel, one in quantum Shannon theory and one in quantum learning theory. In quantum Shannon theory, we prove a channel-level analogue of Uhlmann's theorem for quantum divergences. In quantum learning theory, our construction shows that tomography of quantum channels reduces to tomography of isometries. This yields a simple channel learning algorithm, based on existing isometry learning protocols, that matches the performance of the two recently proposed channel tomography algorithms. Complementarily, whereas the optimality of these algorithms had previously been established only up to a logarithmic factor in the dimension, we close this gap by removing this logarithmic factor from the lower bound. Taken together, our results fully establish the optimality of these recently introduced channel learning algorithms, showing that the optimal query complexity of learning a quantum channel with input dimension

d_A

, output dimension

d_B

, and Choi rank

r

Θ(d_A d_B r)

d

-dimensional unitary channel to within

\varepsilon

in diamond distance using

\mathcal{O}(d^2/\varepsilon)

parallel queries, matching the lower bound presented in [Haah, Kothari, O'Donnell, and Tang, FOCS '23]. Our protocol thus closes the gap between parallel and sequential strategies for unitary channel tomography. As applications, it achieves optimal query complexity for boundary-regime quantum channel tomography and improves the query efficiency of the best-known protocol for tomography of fermionic linear optics.