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arXiv:2609.38145

Query-optimal unitary channel tomography in diamond distance with parallel access

Entong He, Zihao Li, Yuxiang Yang

quant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

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 $\varepsilon$ 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.

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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 ddd-dimensional unitary channel to within ε\varepsilonε in diamond distance using O(d2/ε)\mathcal{O}(d^2/\varepsilon)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.
Unitary process tomography is the act of learning how an unknown unitary gate acts on a quantum system. It is an essential procedure for realizing a functioning quantum computer. As quantum technologies develop, it is important to understand the computational resources needed to conduct unitary process tomography using a variety of types of circuits. In particular, when measuring error in diamond distance, it is known how to achieve optimal query complexity using a sequential circuit architecture, but it had not been known whether a parallel circuit architecture could achieve the same optimal query complexity. In this work, we prove that unitary process tomography with near-optimal query complexity is indeed possible using a parallel circuit by introducing an explicit, novel parallel procedure. This closes the gap between sequential and parallel circuits up to a logarithmic factor, showing that parallel and sequential strategies are nearly equally capable for this task.
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.
The diamond distance between two quantum channels determines the optimal probability of distinguishing them from a single use. We give a quantum algorithm that estimates the diamond distance between two unknown ddd-dimensional unitary channels to within additive error ε\varepsilonε, using O(d/ε)O(\sqrt{d}/\varepsilon)O(d​/ε) queries to the unitary channels and their inverses. The query complexity is optimal, as it matches the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).
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.
Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown ddd-dimensional unitary UUU and stores classical data that can later be used to predict expectation values tr[OUρU†]\mathrm{tr}[O UρU^\dagger]tr[OUρU†] for arbitrary input states ρρρ and observables OOO. Here we propose a parallel and non-adaptive CSEU protocol that uses O(min⁡{dε−2,d3/2ε−1})\mathcal{O}(\min\{d\varepsilon^{-2},d^{3/2}\varepsilon^{-1}\})O(min{dε−2,d3/2ε−1}) queries to UUU when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error ε\varepsilonε and is query-optimal, as we prove a matching lower bound Ω(min⁡{dε−2,d3/2ε−1})Ω(\min\{d\varepsilon^{-2},d^{3/2}\varepsilon^{-1}\})Ω(min{dε−2,d3/2ε−1}) for the query complexity, which remains valid even when stronger access to the unknown unitary is available. Our CSEU protocol is a versatile primitive with applications to several quantum learning tasks. In particular, it yields a near-optimal algorithm for learning an arbitrary Hamiltonian from its real-time evolution in the high-precision regime and recovers the best-known bounds for several tasks in a unified way.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2)Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Θ(din3​dout3​/ε2), where dind_{\mathrm{in}}din​ and doutd_{\mathrm{out}}dout​ are the channel input and output dimensions, respectively, and ε\varepsilonε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2)Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Ω(din3​dout3​/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2)O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)O(din3​dout3​/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1d_{\mathrm{in}}=1din​=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2)Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)Θ(din2​dout2​/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.