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Process tomography for unitary quantum channels

Gus Gutoski, Nathaniel Johnston

Published Jan 1, 2013• 10 citations
OpenAlex

Abstract

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.

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Quantum channel tomography: optimal bounds in the near-boundary regime

Kean Chen, Aadil Oufkir

Quantum channel tomography (also known as quantum process tomography) is a fundamental task in quantum information and quantum computing, aiming to learn the full description of an unknown quantum channel from black-box queries. For channels with input dimension $d_1$, output dimension $d_2$, and Kraus rank at most $r$, [CGO+26] identifies the dilation rate $τ=rd_2/d_1\geq 1$ as a key parameter governing a Heisenberg-to-classical transition in query complexity. While the optimal bounds in the boundary regime ($τ=1$) and the away-from-boundary regime ($τ\geq 1+Ω(1)$) are now known, the precise behavior in the near-boundary regime ($1<τ<1+o(1)$) has remained unresolved. In this paper, we close this gap, thereby fully settling the query complexity of quantum channel tomography. Specifically, we show that the query complexity is - $Ω(rd_1d_2/\varepsilon^2)$ for diamond norm error $\varepsilon$, matching the upper bound by [MB26, CGO+26], - $Θ(d_1^2/\varepsilon+\min\{(τ-1)d_1^2,(τ-1)^2d_1^2d_2^2\}/\varepsilon^2)$ for Choi trace norm error $\varepsilon$. Our results reveal two qualitatively different forms of the Heisenberg-to-classical transition. For diamond norm tomography, the Heisenberg scaling disappears suddenly as soon as $τ$ leaves the boundary regime. In contrast, Choi trace norm tomography exhibits a gradual transition from Heisenberg to classical scaling as $τ$ moves from the boundary regime to the away-from-boundary regime.

quant-pharXiv:2611.03639•Nov 3, 2026
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 ddd-level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only O(d2)O(d^2)O(d2) interactive observables, as opposed to the O(d4)O(d^4)O(d4) required for tomography of arbitrary channels. This result generalizes, so that channels with at most qqq Kraus operators can be identified with only O(qd2)O(qd^2)O(qd2) interactive observables. Slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with qqq 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.
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 din,doutd_{\mathrm{in}},d_{\mathrm{out}}din​,dout​ and Kraus rank at most kkk can be learned to accuracy ε\varepsilonε using O(dindoutk/ε2)O(d_{\mathrm{in}}d_{\mathrm{out}}k/\varepsilon^{2})O(din​dout​k/ε2) channel uses. Conversely, we prove that Ω(dindoutk)Ω(d_{\mathrm{in}}d_{\mathrm{out}}k)Ω(din​dout​k) uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate Ω(1/ε2)Ω(1/\varepsilon^{2})Ω(1/ε2) 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 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(d2) interactive observables, as opposed to the O(d4) required for tomography of arbitrary channels. This result generalizes to the problem of identifying channels with at most q Kraus operators, and 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 E\mathcal{E}E with input dimension d1d_1d1​, output dimension d2d_2d2​ and Kraus rank at most rrr. We establish a connection between the query complexities in two models: (i) access to E\mathcal{E}E, and (ii) access to a random dilation of E\mathcal{E}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 nnn queries to E\mathcal{E}E yet faithfully simulates the tester with nnn queries to a random dilation. As application, we show that: - O(rd1d2/ε2)O(rd_1d_2/\varepsilon^2)O(rd1​d2​/ε2) queries to E\mathcal{E}E suffice for channel tomography to within diamond norm error ε\varepsilonε. Moreover, when rd2=d1rd_2=d_1rd2​=d1​, we show that the Heisenberg scaling O(1/ε)O(1/\varepsilon)O(1/ε) can be achieved, even if E\mathcal{E}E is not a unitary channel: - O(min⁡{d12.5/ε,d12/ε2})O(\min\{d_1^{2.5}/\varepsilon,d_1^2/\varepsilon^2\})O(min{d12.5​/ε,d12​/ε2}) queries to E\mathcal{E}E suffice for channel tomography to within diamond norm error ε\varepsilonε, and O(d12/ε)O(d_1^2/\varepsilon)O(d12​/ε) queries suffice for the case of Choi state trace norm error ε\varepsilonε. - O(min⁡{d11.5/ε,d1/ε2})O(\min\{d_1^{1.5}/\varepsilon,d_1/\varepsilon^2\})O(min{d11.5​/ε,d1​/ε2}) queries to E\mathcal{E}E suffice for tomography of the mixed state E(∣0⟩⟨0∣)\mathcal{E}(|0\rangle\langle 0|)E(∣0⟩⟨0∣) to within trace norm error ε\varepsilonε.
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 ε\varepsilonε-close to the unknown unitary in diamond norm. We design an algorithm achieving error ε\varepsilonε using O( d2/ε)O\left(\mathrm{~d}^{2} / \varepsilon\right)O( d2/ε) applications of the unknown channel and only one qudit. This improves over prior results, which use O( d3/ε2)O\left(\mathrm{~d}^{3} / \varepsilon^{2}\right)O( d3/ε2) [via standard process tomography] or O( d2.5/ε)O\left(\mathrm{~d}^{2.5} / \varepsilon\right)O( d2.5/ε) [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 ε\varepsilonε-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires Ω(d2/ε)\Omega\left(\mathrm{d}^{2} / \varepsilon\right)Ω(d2/ε) 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.
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.
Quantum channel tomography (also known as quantum process tomography) is a fundamental task in quantum information and quantum computing, aiming to learn the full description of an unknown quantum channel from black-box queries. For channels with input dimension d1d_1d1​, output dimension d2d_2d2​, and Kraus rank at most rrr, [CGO+26] identifies the dilation rate τ=rd2/d1≥1τ=rd_2/d_1\geq 1τ=rd2​/d1​≥1 as a key parameter governing a Heisenberg-to-classical transition in query complexity. While the optimal bounds in the boundary regime (τ=1τ=1τ=1) and the away-from-boundary regime (τ≥1+Ω(1)τ\geq 1+Ω(1)τ≥1+Ω(1)) are now known, the precise behavior in the near-boundary regime (1<τ<1+o(1)1<τ<1+o(1)1<τ<1+o(1)) has remained unresolved. In this paper, we close this gap, thereby fully settling the query complexity of quantum channel tomography. Specifically, we show that the query complexity is - Ω(rd1d2/ε2)Ω(rd_1d_2/\varepsilon^2)Ω(rd1​d2​/ε2) for diamond norm error ε\varepsilonε, matching the upper bound by [MB26, CGO+26], - Θ(d12/ε+min⁡{(τ−1)d12,(τ−1)2d12d22}/ε2)Θ(d_1^2/\varepsilon+\min\{(τ-1)d_1^2,(τ-1)^2d_1^2d_2^2\}/\varepsilon^2)Θ(d12​/ε+min{(τ−1)d12​,(τ−1)2d12​d22​}/ε2) for Choi trace norm error ε\varepsilonε. Our results reveal two qualitatively different forms of the Heisenberg-to-classical transition. For diamond norm tomography, the Heisenberg scaling disappears suddenly as soon as τττ leaves the boundary regime. In contrast, Choi trace norm tomography exhibits a gradual transition from Heisenberg to classical scaling as τττ moves from the boundary regime to the away-from-boundary regime.