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

Process tomography for unitary quantum channels

Gus Gutoski, Nathaniel Johnston

We study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a $d$ -level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only $O(d^2)$ interactive observables, as opposed to the $O(d^4)$ required for tomography of arbitrary channels. This result generalizes, so that channels with at most $q$ Kraus operators can be identified with only $O(qd^2)$ interactive observables. Slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with $q$ Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.We study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a $d$-level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only $O(d^2)$ interactive observables, as opposed to the $O(d^4)$ required for tomography of arbitrary channels. This result generalizes, so that channels with at most $q$ Kraus operators can be identified with only $O(qd^2)$ interactive observables. Slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with $q$ Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.

Jan 1, 201310 citations

Identification of quantum closed systems via the quantum process tomography

Weiwei Zhou, S. G. Schirmer, Hongwei Xie +1 more

The dynamics of an unknown closed quantum system can be characterized by a Hamiltonian. We discuss the identification of this Hamiltonian from quantum process tomography data at several time points for systems with a time-independent Hamiltonian. Quantum process tomography (QPT) yields a set of unitary operators and allows us to convert the identification problem into an estimation problem for the eigenfrequencies of the system. Spectral decomposition of Hermitian and unitary operators shows that process tomography needs to be performed at least at two different time points and the identifiability of the Hamiltonian depends on the choice of these points. Sufficient conditions on the timing to ensure complete identifiability of the Hamiltonian are presented under the assumption that unitary matrices obtained from QPT are exact without numerical errors.The dynamics of an unknown closed quantum system can be characterized by a Hamiltonian. We discuss the identification of this Hamiltonian from quantum process tomography data at several time points for systems with a time-independent Hamiltonian. Quantum process tomography (QPT) yields a set of unitary operators and allows us to convert the identification problem into an estimation problem for the eigenfrequencies of the system. Spectral decomposition of Hermitian and unitary operators shows that process tomography needs to be performed at least at two different time points and the identifiability of the Hamiltonian depends on the choice of these points. Sufficient conditions on the timing to ensure complete identifiability of the Hamiltonian are presented under the assumption that unitary matrices obtained from QPT are exact without numerical errors.

Chinese Control Conference (CCC)Jul 25, 20120 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

Query-optimal estimation of unitary channels in diamond distance

Jeongwan Haah, Robin Kothari, Ryan O’Donnell +1 more

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$ -close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$ -error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O\left(\mathrm{~d}^{2} / \varepsilon\right)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)$ [via standard process tomography] or $O\left(\mathrm{~d}^{2.5} / \varepsilon\right)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $\Omega\left(\mathrm{d}^{2} / \varepsilon\right)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

arXiv:2302.14066Annual Symposium on Foundations of Computer ScienceNov 6, 202323 citations

Efficient non-destructive direct characterization of arbitrary many-body quantum channels

Zhiyuan Wang, Qing He, Zijing Zhang

Quantum process tomography (QPT) is a crucial technique for characterizing unknown quantum channels. However, traditional QPT methods encounter scalability problems as the particle numbers increase, requiring exponentially more state preparations and measurement operators. The characteristics of sparse target channels (e.g., multiqubit phase-shift gates) can be obtained by measuring only a few specific matrix elements without requiring global QPT. Therefore, direct quantum channel characterization is vital. This paper proposes a direct protocol for both qubit and qudit systems that extracts specific process matrix elements without full reconstruction. The measurement operator requirements remain independent of system size and dimensionality. Notably, the proposed protocol uses nondestructive measurements and preserves qubits after evolution through unknown processes for potential reuse, making it uniquely promising for applications such as real-time monitoring of noise processes in quantum error correction and real-time feedback control requiring quantum state preservation. To validate the theoretical correctness of our approach, we conducted experimental demonstrations of both the unitary and non-unitary processes of single qubits using a nuclear magnetic resonance system on the SpinQ quantum cloud platform. The experimental results confirmed the correctness and effectiveness of the proposed method.Quantum process tomography (QPT) is a crucial technique for characterizing unknown quantum channels. However, traditional QPT methods encounter scalability problems as the particle numbers increase, requiring exponentially more state preparations and measurement operators. The characteristics of sparse target channels (e.g., multiqubit phase-shift gates) can be obtained by measuring only a few specific matrix elements without requiring global QPT. Therefore, direct quantum channel characterization is vital. This paper proposes a direct protocol for both qubit and qudit systems that extracts specific process matrix elements without full reconstruction. The measurement operator requirements remain independent of system size and dimensionality. Notably, the proposed protocol uses nondestructive measurements and preserves qubits after evolution through unknown processes for potential reuse, making it uniquely promising for applications such as real-time monitoring of noise processes in quantum error correction and real-time feedback control requiring quantum state preservation. To validate the theoretical correctness of our approach, we conducted experimental demonstrations of both the unitary and non-unitary processes of single qubits using a nuclear magnetic resonance system on the SpinQ quantum cloud platform. The experimental results confirmed the correctness and effectiveness of the proposed method.

npj Quantum InformationDec 14, 20251 citation

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.

d

-level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only

O(d^2)

interactive observables, as opposed to the

O(d^4)

required for tomography of arbitrary channels. This result generalizes, so that channels with at most

q

Kraus operators can be identified with only

O(qd^2)

interactive observables. Slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with

q

Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.

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 consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a d-dimensional qudit, we aim to output a classical description of a unitary that is

\varepsilon

-close to the unknown unitary in diamond norm. We design an algorithm achieving error

\varepsilon

using

O\left(\mathrm{~d}^{2} / \varepsilon\right)

applications of the unknown channel and only one qudit. This improves over prior results, which use

O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)

[via standard process tomography] or

O\left(\mathrm{~d}^{2.5} / \varepsilon\right)

[Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to “bootstrap” an algorithm that can produce constant-error estimates to one that can produce

\varepsilon

-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires

\Omega\left(\mathrm{d}^{2} / \varepsilon\right)

applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.