arXiv:2601.10683

Optimal lower bound for quantum channel tomography in away-from-boundary regime

Kean Chen, Zhicheng Zhang, Nengkun Yu

quant-phcs.ITSubmitted Jan 15, 2026• Announced Jan 16, 2026• 0 citations

Abstract

Consider quantum channels with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. Any such channel must satisfy the constraint $rd_2\geq d_1$, and the parameter regime $rd_2=d_1$ is called the boundary regime. In this paper, we show an optimal query lower bound $Ω(rd_1d_2/\varepsilon^2)$ for quantum channel tomography to within diamond norm error $\varepsilon$ in the away-from-boundary regime $rd_2\geq 2d_1$, matching the existing upper bound $O(rd_1d_2/\varepsilon^2)$. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions $d_1=d_2=d$ with $r\geq 2$, in sharp contrast to the unitary case $r=1$ where Heisenberg scaling $Θ(d^2/\varepsilon)$ is achievable.

Comments: 23 pages

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Strict Hierarchy for Quantum Channel Certification to Unitary

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quant-pharXiv:2604.26900Leibniz international proceedings in informaticsApr 30, 20260 citations

arXiv:2601.10683

Optimal lower bound for quantum channel tomography in away-from-boundary regime

Kean Chen, Zhicheng Zhang, Nengkun Yu

quant-phcs.ITSubmitted Jan 15, 2026• Announced Jan 16, 2026• 0 citations

arXiv HTML PDF

Abstract

Consider quantum channels with input dimension

d_1

, output dimension

d_2

and Kraus rank at most

r

. Any such channel must satisfy the constraint

rd_2\geq d_1

, and the parameter regime

rd_2=d_1

is called the boundary regime. In this paper, we show an optimal query lower bound

Ω(rd_1d_2/\varepsilon^2)

for quantum channel tomography to within diamond norm error

\varepsilon

in the away-from-boundary regime

rd_2\geq 2d_1

, matching the existing upper bound

O(rd_1d_2/\varepsilon^2)

. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions

d_1=d_2=d

with

r\geq 2

, in sharp contrast to the unitary case

r=1

where Heisenberg scaling

Θ(d^2/\varepsilon)

is achievable.

Comments: 23 pages

Related papers

Quantum channel tomography: optimal bounds and a Heisenberg-to-classical phase transition

Kean Chen, Filippo Girardi, Aadil Oufkir +2 more

How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood. In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$ , 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.How many black-box queries to a quantum channel are needed to learn its full classical description? This question lies at the heart of quantum channel tomography (also known as quantum process tomography), a fundamental task in the characterization and validation of quantum hardware. Despite extensive prior work, the optimal query complexity for quantum channel tomography is far from fully understood. In this paper, we study tomography of an unknown quantum channel with input dimension $d_1$, 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.

quant-pharXiv:2604.17369Apr 21, 20260 citations

Near-optimal incoherent tomography of low-rank quantum channels

Kean Chen, Aadil Oufkir

We study tomography for quantum channels with input dimension $d_1$ , output dimension $d_2$ , and Kraus rank at most $r$ , to within diamond norm error $\varepsilon$ , using adaptive experiments that retain no quantum memory between channel queries. - For quantum channels whose non-zero Choi eigenvalues are bounded below by $Ω(d_1/r)$ , we establish optimal query upper and lower bounds $Θ(d_1d_2r^2/ε^2)$ . The upper bound is achieved by a nonadaptive algorithm that uses the estimator from [Surawy-Stepney et al., Quantum (2022)], together with a new diamond-norm analysis. The lower bound applies to arbitrary adaptive incoherent protocols and follows from a new local family of channels and a uniform one-query Fisher-information bound. - For general channels, we establish an upper bound $O(d_1d_2r^2\log(2d_1)/ε^2)$ , nearly matching the above lower bound $Ω(d_1d_2r^2/\varepsilon^2)$ . To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over $O(\log(2d_1))$ rounds with the Matrix Multiplicative Weight Update algorithm.We study tomography for quantum channels with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, to within diamond norm error $\varepsilon$, using adaptive experiments that retain no quantum memory between channel queries. - For quantum channels whose non-zero Choi eigenvalues are bounded below by $Ω(d_1/r)$, we establish optimal query upper and lower bounds $Θ(d_1d_2r^2/ε^2)$. The upper bound is achieved by a nonadaptive algorithm that uses the estimator from [Surawy-Stepney et al., Quantum (2022)], together with a new diamond-norm analysis. The lower bound applies to arbitrary adaptive incoherent protocols and follows from a new local family of channels and a uniform one-query Fisher-information bound. - For general channels, we establish an upper bound $O(d_1d_2r^2\log(2d_1)/ε^2)$, nearly matching the above lower bound $Ω(d_1d_2r^2/\varepsilon^2)$. To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over $O(\log(2d_1))$ rounds with the Matrix Multiplicative Weight Update algorithm.

quant-pharXiv:2609.23752Sep 22, 20260 citations

Quantum channel tomography and estimation by local test

Kean Chen, Nengkun Yu, Zhicheng Zhang

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 $\varepsilon$ .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 $\varepsilon$.

quant-pharXiv:2512.13614Dec 16, 20250 citations

Optimal learning of quantum channels in diamond distance

Antonio Anna Mele, Lennart Bittel

Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions $d_{\mathrm{in}},d_{\mathrm{out}}$ 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.Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions $d_{\mathrm{in}},d_{\mathrm{out}}$ 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.

quant-pharXiv:2512.10214Dec 12, 20250 citations

Improved Lower Bounds for Learning Quantum Channels in Diamond Distance

Aadil Oufkir, Filippo Girardi

We prove that learning an unknown quantum channel with input dimension $d_A$ , output dimension $d_B$ , and Choi rank $r$ to diamond distance $\varepsilon$ requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A= rd_B$ , and $Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$ . These lower bounds improve upon the best previous $Ω(d_A d_B r)$ bound by introducing explicit, near-optimal $\varepsilon$ -dependence. Moreover, when $d_A\le rd_B/2$ , the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\varepsilon$ requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A= rd_B$, and $Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$. These lower bounds improve upon the best previous $Ω(d_A d_B r)$ bound by introducing explicit, near-optimal $\varepsilon$-dependence. Moreover, when $d_A\le rd_B/2$, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

quant-pharXiv:2601.04180Jan 8, 20260 citations

Strict Hierarchy for Quantum Channel Certification to Unitary

Kean Chen, Qisheng Wang, Zhicheng Zhang

We consider the problem of quantum channel certification to unitary, where one is given access to an unknown $d$ -dimensional channel $\mathcal{E}$ , and wants to test whether $\mathcal{E}$ is equal to a target unitary channel or is $\varepsilon$ -far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that: (i) $Θ(d/\varepsilon^2)$ queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii) $Θ(d/\varepsilon)$ queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii) $Θ(\sqrt{d}/\varepsilon)$ queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026). This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.We consider the problem of quantum channel certification to unitary, where one is given access to an unknown $d$-dimensional channel $\mathcal{E}$, and wants to test whether $\mathcal{E}$ is equal to a target unitary channel or is $\varepsilon$-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that: (i) $Θ(d/\varepsilon^2)$ queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii) $Θ(d/\varepsilon)$ queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii) $Θ(\sqrt{d}/\varepsilon)$ queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026). This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.

quant-pharXiv:2604.26900Leibniz international proceedings in informaticsApr 30, 20260 citations

Consider quantum channels with input dimension

d_1

, output dimension

d_2

and Kraus rank at most

r

. Any such channel must satisfy the constraint

rd_2\geq d_1

, and the parameter regime

rd_2=d_1

is called the boundary regime. In this paper, we show an optimal query lower bound

Ω(rd_1d_2/\varepsilon^2)

for quantum channel tomography to within diamond norm error

\varepsilon

in the away-from-boundary regime

rd_2\geq 2d_1

, matching the existing upper bound

O(rd_1d_2/\varepsilon^2)

. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions

d_1=d_2=d

with

r\geq 2

, in sharp contrast to the unitary case

r=1

where Heisenberg scaling

Θ(d^2/\varepsilon)

is achievable.

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.

We study tomography for quantum channels with input dimension

d_1

, output dimension

d_2

, and Kraus rank at most

r

, to within diamond norm error

\varepsilon

, using adaptive experiments that retain no quantum memory between channel queries. - For quantum channels whose non-zero Choi eigenvalues are bounded below by

Ω(d_1/r)

, we establish optimal query upper and lower bounds

Θ(d_1d_2r^2/ε^2)

. The upper bound is achieved by a nonadaptive algorithm that uses the estimator from [Surawy-Stepney et al., Quantum (2022)], together with a new diamond-norm analysis. The lower bound applies to arbitrary adaptive incoherent protocols and follows from a new local family of channels and a uniform one-query Fisher-information bound. - For general channels, we establish an upper bound

O(d_1d_2r^2\log(2d_1)/ε^2)

, nearly matching the above lower bound

Ω(d_1d_2r^2/\varepsilon^2)

. To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over

O(\log(2d_1))

rounds with the Matrix Multiplicative Weight Update algorithm.

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_{\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 prove that learning an unknown quantum channel with input dimension

d_A

, output dimension

d_B

, and Choi rank

r

to diamond distance

\varepsilon

requires $ Ω\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when

d_A= rd_B

, and

Ω\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)

channel queries when

d_A\le rd_B/2

. These lower bounds improve upon the best previous

Ω(d_A d_B r)

bound by introducing explicit, near-optimal

\varepsilon

-dependence. Moreover, when

d_A\le rd_B/2

, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm.

We consider the problem of quantum channel certification to unitary, where one is given access to an unknown

d

-dimensional channel

\mathcal{E}

, and wants to test whether

\mathcal{E}

is equal to a target unitary channel or is

\varepsilon

-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that: (i)

Θ(d/\varepsilon^2)

queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023). (ii)

Θ(d/\varepsilon)

queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008). (iii)

Θ(\sqrt{d}/\varepsilon)

queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026). This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.