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arXiv:2510.24880 · Communications Physics

Structure, optimality, and symmetry in shadow unitary inversion

Guocheng Zhen, Yu-Ao Chen, Mingrui Jing, Jingu Xie, Xin Wang, Ranyiliu Chen

Published May 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Reversing unknown quantum dynamics is vital for quantum control and learning, yet full unitary inversion requires a resource-intensive O(d2) queries. Because many applications only require the reversed evolution for a given observable, a critical gap remains in understanding the minimal resources for such targeted reversal. Here, we address this by introducing shadow unitary inversion. We establish a lower bound showing the query complexity must scale at least linearly with system dimension for spectrally biased observables, with the constant determined by the observable’s spectral properties. For qubit case, we construct an explicit, deterministic three-query sequential protocol achieving exact shadow inversion and completely characterize all admissible channels, with numerical evidence suggesting optimality. For higher dimensions, we develop a semidefinite-programming formulation and introduce a representation-theoretic symmetry reduction that decomposes the optimization into invariant blocks, substantially reducing the problem size. Shadow unitary inversion thus offers a resource-efficient path to inverse-dynamics estimation for future quantum control, diagnostics, and learning tasks. Reversing unitary operations is crucial in quantum computing and control, yet remains a challenging task. The authors introduce shadow unitary inversion, achieving efficient inversion at the expectation value level with reduced query complexity, offering a promising approach for quantum error correction and information recovery.

DOI
10.1038/s42005-026-02690-9

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Reversing unknown quantum dynamics is vital for quantum control and learning, yet full unitary inversion requires a resource-intensive O(d2) queries. Because many applications only require the reversed evolution for a given observable, a critical gap remains in understanding the minimal resources for such targeted reversal. Here, we address this by introducing shadow unitary inversion. We establish a lower bound showing the query complexity must scale at least linearly with system dimension for spectrally biased observables, with the constant determined by the observable’s spectral properties. For qubit case, we construct an explicit, deterministic three-query sequential protocol achieving exact shadow inversion and completely characterize all admissible channels, with numerical evidence suggesting optimality. For higher dimensions, we develop a semidefinite-programming formulation and introduce a representation-theoretic symmetry reduction that decomposes the optimization into invariant blocks, substantially reducing the problem size. Shadow unitary inversion thus offers a resource-efficient path to inverse-dynamics estimation for future quantum control, diagnostics, and learning tasks. Reversing unitary operations is crucial in quantum computing and control, yet remains a challenging task. The authors introduce shadow unitary inversion, achieving efficient inversion at the expectation value level with reduced query complexity, offering a promising approach for quantum error correction and information recovery.
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Classical shadow estimation (CSE) is a powerful tool for learning the properties of quantum states and quantum processes. Here we consider the CSE task for quantum unitary channels. By querying an unknown unitary channel U multiple times in quantum experiments, the goal is to learn a classical description from which one can accurately predict many different linear properties of the channel, i.e., the expectation values of arbitrary observables measured on the output of U upon arbitrary input states. Based on collective measurements on multiple systems, we propose a query-efficient protocol for this task, whose query complexity has a quadratic advantage over the previous best approach for this problem, and almost saturates the information-theoretic lower bound. To further enhance practicality, we also present a variant protocol using only single-copy measurements, which still offers much better query performance than previous protocols that do not use quantum memory, and can serve as a key subroutine for learning an arbitrary unknown Hamiltonian from dynamics. In addition to linear properties of unitary channels, our protocol can also be applied to simultaneously predict many nonlinear properties, such as out-of-time-ordered correlators.
Deterministic exact inversion of an arbitrary ddd-dimensional unitary requires {Θ(d2)Θ(d^2)Θ(d2)} coherent forward calls in the worst case. We ask how this cost changes for Hamiltonian evolution U(x)=exp⁡(i∑jxjHj)U(x)=\exp(i\sum_j x_jH_j)U(x)=exp(i∑j​xj​Hj​) when the generators are known but the parameters are hidden. For one-parameter families with a fixed eigenbasis, we show that additive relations among the distinct eigenvalues determine the optimal query number exactly, and we construct the corresponding inversion protocol. For general families, we prove that repeated symmetry sectors do not affect the exact query complexity and give an automatic construction for combining inverses from inequivalent active sectors. We also give a sufficient phase-alignment condition under which family-specific structure can reduce the query number. These results establish structure-dependent bounds for reversing the unknown dynamics arising in Tavis-Cummings out-of-time-order correlator protocols, collective-spin echo verification, and passive multimode links, without requiring prior knowledge or explicit estimation of the underlying coupling strengths.
Unknown unitary inversion is a fundamental primitive in quantum computing and physics. Although recent work has demonstrated that quantum algorithms can invert arbitrary unknown unitaries without accessing their classical descriptions, improving the efficiency of such protocols remains an open question. In this work, we present efficient quantum algorithms for inverting unitaries with specific Hamiltonian structures, achieving significant reductions in both ancilla qubit requirements and unitary query complexity. We identify cases where unitaries encoding exponentially many parameters can be inverted using only a single query. We further extend our framework to implement unitary complex conjugation and transposition operations, and develop modified protocols capable of inverting more general classes of Hamiltonians. We have also demonstrated the efficacy and robustness of our algorithms via numerical simulations under realistic noise conditions of superconducting quantum hardware. Our results establish more efficient protocols that improve the resources required for quantum unitary inversion when prior information about the quantum system is available, and provide practical methods for implementing these operations on near-term quantum devices.
We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state ∣ψ⟩|ψ\rangle∣ψ⟩, a Hermitian matrix HHH, a polynomial fff, and a set of observables {O1,…,Om}\{O_1,\dots,O_m\}{O1​,…,Om​}, the goal is to estimate ⟨ψ∣f(H)†Ojf(H)∣ψ⟩\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle⟨ψ∣f(H)†Oj​f(H)∣ψ⟩ for all j∈{1,…,m}j\in\{1,\dots,m\}j∈{1,…,m}. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of f(H)f(H)f(H). It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with HHH and ∣ψ⟩|ψ\rangle∣ψ⟩ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from HHH, fff, and ∣ψ⟩|ψ\rangle∣ψ⟩ without prior knowledge of the observables or explicit preparation of the target state proportional to f(H)∣ψ⟩f(H)|ψ\ranglef(H)∣ψ⟩. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.
We report a deterministic and exact protocol to reverse any unknown qubit-unitary operation, which simulates the time inversion of a closed qubit system. To avoid known no-go results on universal deterministic exact unitary inversion, we consider the most general class of protocols transforming unknown unitary operations within the quantum circuit model, where the input unitary operation is called multiple times in sequence and fixed quantum circuits are inserted between the calls. In the proposed protocol, the input qubit-unitary operation is called 4 times to achieve the inverse operation, and the output state in an auxiliary system can be reused as a catalyst state in another run of the unitary inversion. We also present the simplification of the semidefinite programming for searching the optimal deterministic unitary inversion protocol for an arbitrary dimension presented by M. T. Quintino and D. Ebler [Quantum 6, 679 (2022)2521-327X10.22331/q-2022-03-31-679]. We show a method to reduce the large search space representing all possible protocols, which provides a useful tool for analyzing higher-order quantum transformations for unitary operations.