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arXiv:2405.07625 · Physical Review Letters

Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations

Tatsuki Odake, Satoshi Yoshida, Mio Murao

Published Nov 5, 2025• 0 citations
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Abstract

Recent developments have revealed deterministic and exact protocols for performing complex conjugation, inversion, and transposition of a general d-dimensional unknown unitary operation using a finite number of queries to a black-box unitary operation. In this Letter, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary. Specifically, our lower bound of d^{2} for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with O(d^{2}) queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions f: SU(d)→SU(d). As a corollary, we prove that a catalytic protocol-a new concept recently noted in the study of exact unitary inversion-is impossible for unitary complex conjugation. Furthermore, we extend our framework to the partially known setting, where the input unitary operation is promised to be within a subgroup of SU(d) and the probabilistic setting, where transformations succeed probabilistically.

DOI
10.1103/drp2-rzzw

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We construct exact deterministic circuits for transposing and inverting an unknown qutrit unitary U∈SU(3)U\in\mathrm{SU}(3)U∈SU(3) using only forward queries. Three queries prepare a catalyst independently of the input state. Four further queries implement transposition while returning the catalyst unchanged and uncorrelated with the output. Two uses within an antisymmetric encoding implement inversion with eight processing queries and return the same catalyst. Reusing the catalyst implements mmm transpositions or inversions with 3+4m3+4m3+4m or 3+8m3+8m3+8m total queries, respectively. More generally, we prove that exact transposition and inversion of an unknown U∈SU(d)U\in\mathrm{SU}(d)U∈SU(d), d≥2d\ge2d≥2, require at least d+1d+1d+1 and d2−1d^2-1d2−1 processing queries, respectively, in fixed causal order. These processing bounds hold for any program state prepared independently of the input state using finitely many forward queries, even if that state is consumed during processing. Both our catalytic qutrit algorithms attain their respective bounds. Moreover, exact qutrit transposition with four processing queries requires at least three preparation queries, and our construction attains this bound.
Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.
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.
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.
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.
Given oracle access to an unknown unitary U=eiHU=e^{iH}U=eiH , the fractional query problem asks how many queries are required to implement a noninteger power Ut=eitHU^t=e^{itH}Ut=eitH, 0<t<10<t<10<t<1, when the spectrum is separated from the branch cut by a gap δδδ. Quantum singular value transformation gives an upper bound of O ⁣(1δlog⁡1ε)O\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)O(δ1​logε1​) queries for approximation error ε\varepsilonε. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any NNN-query circuit to the approximation of eitθe^{itθ}eitθ by a trigonometric polynomial with degree bounded by O(N)O(N)O(N), together with Remez inequality. This allows us to establish the lower bound of Ωτ ⁣(1δlog⁡1ε)Ω_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)Ωτ​(δ1​logε1​). Consequently, the optimal query complexity for fractional query problem is Θτ ⁣(1δlog⁡1ε)Θ_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)Θτ​(δ1​logε1​), showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a Ωτ ⁣(log⁡1ε)Ω_τ\!\left(\log\frac{1}{\varepsilon}\right)Ωτ​(logε1​) bound uniform to δδδ.