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 E with input dimension d1, output dimension d2 and Kraus rank at most r. We establish a connection between the query complexities in two models: (i) access to E, and (ii) access to a random dilation of 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 E yet faithfully simulates the tester with n queries to a random dilation. As application, we show that:
O(rd1d2/ε2) queries to E suffice for channel tomography to within diamond norm error ε.
Moreover, when rd2=d1, we show that the Heisenberg scaling O(1/ε) can be achieved, even if E is not a unitary channel:
O(min{d12.5/ε,d12/ε2}) queries to E suffice for channel tomography to within diamond norm error ε, and O(d12/ε) queries suffice for the case of Choi state trace norm error ε.
O(min{d11.5/ε,d1/ε2}) queries to E suffice for tomography of the mixed state E(∣0⟩⟨0∣) to within trace norm error ε.
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 d1, output dimension d2, and Kraus rank at most r, to within error ε. We identify the dilation rate τ=rd2/d1 (which always satisfies τ≥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(rd1d2/ε) for Choi trace norm error ε, and is upper bounded by O(min{rd11.5d2/ε,rd1d2/ε2}) and lower bounded by Ω(rd1d2/ε) for diamond norm error ε.
- In the away-from-boundary regime (τ≥1+Ω(1)): we show that the query complexity is Θ(rd1d2/ε2) for both Choi trace norm and diamond norm errors ε.
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/ε, whereas for τ≥1+Ω(1), it exhibits classical scaling 1/ε2. 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.
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,dout and Kraus rank at most k can be learned to accuracy ε using O(dindoutk/ε2) channel uses. Conversely, we prove that Ω(dindoutk) uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate 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.
Consider quantum channels with input dimension d1, output dimension d2 and Kraus rank at most r. Any such channel must satisfy the constraint rd2≥d1, and the parameter regime rd2=d1 is called the boundary regime. In this paper, we show an optimal query lower bound Ω(rd1d2/ε2) for quantum channel tomography to within diamond norm error ε in the away-from-boundary regime rd2≥2d1, matching the existing upper bound O(rd1d2/ε2). In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions d1=d2=d with r≥2, in sharp contrast to the unitary case r=1 where Heisenberg scaling Θ(d2/ε) is achievable.
We study tomography for quantum channels with input dimension d1, output dimension d2, and Kraus rank at most r, to within diamond norm error ε, using adaptive experiments that retain no quantum memory between channel queries.
- For quantum channels whose non-zero Choi eigenvalues are bounded below by Ω(d1/r), we establish optimal query upper and lower bounds Θ(d1d2r2/ε2). 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(d1d2r2log(2d1)/ε2), nearly matching the above lower bound Ω(d1d2r2/ε2). To achieve this, we generalize the above nonadaptive algorithm by adapting the input state over O(log(2d1)) rounds with the Matrix Multiplicative Weight Update algorithm.
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 O(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)).
In the study of quantum process tomography, it has remained open how to learn a unitary channel in diamond distance with strictly parallel queries as efficiently as with sequential queries. Sequential queries allow one to exploit adaptivity to refine the estimate and adjust the learning strategy accordingly, whereas such adjustments are impossible for parallel queries. From the perspective of higher-order quantum operations, any parallel learning strategy can be simulated by a sequential one, whereas the converse does not hold in general, suggesting that sequential learning strategies are potentially more powerful. Contrary to this intuition, we present a new protocol for learning an unknown d-dimensional unitary channel to within ε in diamond distance using O(d2/ε) 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.