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arXiv:2608.26061

Asymptotically optimal purification of noisy unitary channels in any dimension

Ryotaro Niwa, Satoshi Yoshida, Mio Murao

quant-phSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

We consider the problem of noisy unitary purification. Given access to an unknown $d$-dimensional unitary channel followed by depolarizing noise of strength $p$, 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 $\mathrm{SU}(d)$-covariant parallel strategy that attains the optimum. Our result implies the query complexity $Θ(d^2p/ε)$ for achieving leading-order infidelity $ε$ 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.

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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.
We consider the problem of purifying noisy qubit unitary channels. Given the ability to apply an unknown qubit unitary channel followed by depolarizing noise, we aim to construct a superchannel that purifies the noisy unitary back to the original unknown unitary. We first provide numerical evidence that sequential strategies can strictly outperform parallel strategies when the number of channel uses is finite, highlighting the fundamental distinction from state purification. We then provide a concrete U(2)\mathrm{U}(2)U(2)-covariant parallel protocol based on a novel entanglement-assisted quantum error-correcting code that suppresses the first-order noise strength as O(1/n)O(1/n)O(1/n) with nnn channel uses and show this scaling is asymptotically optimal in the low-noise regime, even when sequential strategies are allowed.
Quantum gates executed on physical hardware are inevitably degraded by environmental noise. While state purification effectively distills static quantum resources, the dynamic execution of quantum algorithms requires a higher-order approach to mitigate errors on the operations themselves. In this work, we investigate universal unitary purification: the task of utilizing a quantum higher-order operation to partially restore the ideal action of an unknown unitary corrupted by a known noise model. We consider deterministic purification of unknown single-qubit unitaries under independent depolarizing noise with ideal auxiliary operations. We prove that within the indefinite causal order (ICO) framework, no nontrivial 2-slot higher-order operation can universally purify the set of single-qubit unitaries. Overcoming this strict limitation, we establish that a 3-slot parallel architecture provides the minimal realization for nontrivial purification. We analytically derive the optimal average fidelity over all 3-slot ICO strategies and show that a parallel strategy attains it, strictly surpassing trivial strategies through input encoding and joint decoding of the channel outputs and a quantum memory. Furthermore, we provide a concrete quantum circuit construction attaining this optimum. Our results clarify the possibilities and limitations of purifying unknown qubit unitaries under depolarizing noise, offering insights for robust gate design.
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.
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.
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.
We present fast algorithms $\unicode{x2013}$ under the depolarizing noise assumption, often made in fault-tolerant quantum computations $\unicode{x2013}$ to test its strength. Our optimal algorithms answer the following question: is the quantum channel implemented by a given black box identical to a target unitary or ε\varepsilonε-far from it in the diamond distance, assuming that the deviation is a depolarizing channel with unknown parameter? Our algorithm has a query complexity of Θ(1/ε).Θ(1/\varepsilon).Θ(1/ε). The query complexity of the relaxed problem of testing whether the black-box channel is ε1\varepsilon_1ε1​-close to a target unitary or ε2\varepsilon_2ε2​-far in the diamond distance is Θ(ε2/(ε2−ε1)2).Θ\bigl(\varepsilon_2/(\varepsilon_2 - \varepsilon_1)^2\bigr).Θ(ε2​/(ε2​−ε1​)2). In both cases, we provide matching lower bounds that hold even for adaptive, ancilla-assisted protocols with multi-outcome incoherent measurements.