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arXiv:2607.29054

Optimal complex conjugation of unknown isometry channels

Satoshi Yoshida, Mio Murao

quant-phSubmitted Jul 31, 2026• Announced Aug 3, 2026• 0 citations
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Abstract

Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry $\overline{V}$ from $n$ uses of an unknown isometry channel $V: \mathbb{C}^d\to\mathbb{C}^D$. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity $n=Θ(d[(D-d)/ε+1])$ for achieving infidelity $ε$. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity $O(\mathrm{poly}(D,1/ε))$. This task is extended to the multi-copy case $V^{\otimes n}\mapsto \overline{V}^{\otimes k}$. For fixed $d<D$ and $k$, we show that the optimal fidelity for the multi-copy case is $1-kd(D-d)/n+o(n^{-1})$, and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-$r$ quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank $r$ is constant.

Comments
25 pages, 8 figures

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Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry V‾\overline{V}V from nnn uses of an unknown isometry channel V:Cd→CDV: \mathbb{C}^d\to\mathbb{C}^DV:Cd→CD. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity n=Θ(d[(D−d)/ε+1])n=Θ(d[(D-d)/ε+1])n=Θ(d[(D−d)/ε+1]) for achieving infidelity εεε. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity O(poly(D,1/ε))O(\mathrm{poly}(D,1/ε))O(poly(D,1/ε)). This task is extended to the multi-copy case V⊗n↦V‾⊗kV^{\otimes n}\mapsto \overline{V}^{\otimes k}V⊗n↦V⊗k. For fixed d<Dd<Dd<D and kkk, we show that the optimal fidelity for the multi-copy case is 1−kd(D−d)/n+o(n−1)1-kd(D-d)/n+o(n^{-1})1−kd(D−d)/n+o(n−1), and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-rrr quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank rrr is constant.
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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 present a quantum circuit that implements the random dilation superchannel, transforming parallel queries of an unknown quantum channel into the same number of parallel queries of a randomly chosen dilation isometry of the input channel. This is a natural generalization of the random purification channel, that transforms copies of an unknown mixed state to copies of a randomly chosen purification state. The circuit complexity of our construction is O(poly(n,log⁡dI,log⁡dO))O(\mathrm{poly}(n, \log d_I, \log d_O))O(poly(n,logdI​,logdO​)), where nnn is the number of queries and dId_IdI​ and dOd_OdO​ are the input and output dimensions of the input channel, respectively. This random dilation superchannel is extended to the sequential queries approximately, by transforming the parallel random dilation isometry into sequential random dilation unitaries with O(poly(dI))O(\mathrm{poly}(d_I))O(poly(dI​)) overhead in the number of queries. We also show that our results can be further extended to the case of quantum superchannels. On the other hand, we show a no-go theorem on the exact random dilation of sequential queries with o(poly(min⁡{dI,dO}))o(\mathrm{poly}(\min\{d_I, d_O\}))o(poly(min{dI​,dO​})) query overhead, showcasing a fundamental difference between the parallel and sequential cases. As an application, we show an efficient storage-and-retrieval of an unknown quantum channel, which improves the program cost exponentially in the retrieval error ε\varepsilonε. For the case where the Kraus rank rrr is the least possible (i.e., r=dI/dOr = d_I/d_Or=dI​/dO​), we show quantum circuits that transform nnn parallel queries of an unknown quantum channel ΛΛΛ to Θ(nα)Θ(n^α)Θ(nα) parallel queries of ΛΛΛ for any α<2α<2α<2 approximately, and implement its Petz recovery map for the maximally mixed reference state probabilistically and exactly.
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Identification of possible transformations of quantum objects including quantum states and quantum operations is indispensable in developing quantum algorithms. Universal transformations, defined as input-independent transformations, appear in various quantum applications. Such is the case for universal transformations of unitary operations. However, extending these transformations to non-unitary operations is nontrivial and largely unresolved. Addressing this, we introduce isometry adjointation protocols that transform an input isometry operation into its adjoint operation, which include both unitary operation and quantum state transformations. The paper details the construction of parallel and sequential isometry adjointation protocols, derived from unitary inversion protocols using quantum combs and the (dual) Clebsch-Gordan transforms, and achieving optimal approximation error. This error is shown to be independent of the output dimension of the isometry operation. In particular, we explicitly obtain an asymptotically optimal parallel protocol achieving an approximation error ϵ=Θ(d2/n) , where d is the input dimension of the isometry operation and n is the number of calls of the isometry operation. The research also extends to isometry inversion and universal error detection, employing semidefinite programming to assess optimal performances. The findings suggest that the optimal performance of general protocols in isometry adjointation and universal error detection is not dependent on the output dimension, and that indefinite causal order protocols offer advantages over sequential ones in isometry inversion and universal error detection.
We consider the problem of noisy unitary purification. Given access to an unknown ddd-dimensional unitary channel followed by depolarizing noise of strength ppp, we aim to construct a superchannel that universally purifies the noisy unitary back to the original unknown unitary. We optimize over arbitrary adaptive sequential strategies and analytically derive the optimal fidelity to the leading order in the noise strength and number of channel uses, while also providing a concrete SU(d)\mathrm{SU}(d)SU(d)-covariant parallel strategy that attains the optimum. Our result implies the query complexity Θ(d2p/ε)Θ(d^2p/ε)Θ(d2p/ε) for achieving leading-order infidelity εεε in the low-noise regime, which scales better than the naive approach combining optimal state purification and storage-and-retrieval of quantum channels. We also consider the dual problem of noisy unitary conjugation, where the goal is to obtain the best approximation of the complex conjugate of the original unknown unitary from access to noisy queries. We show that the optimal fidelity for this task coincides with that of noisy unitary purification to the leading-order in the low-noise and large-query limit.