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arXiv:2610.12664

A Catalytic Algorithm for Transposing Unknown Qutrit Unitaries with Tight Query Bounds

Yu-Ao Chen

quant-phcs.CCcs.DSSubmitted Oct 8, 2026• Announced Oct 12, 2026• 0 citations
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Abstract

We construct exact deterministic circuits for transposing and inverting an unknown qutrit unitary $U\in\mathrm{SU}(3)$ 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 $m$ transpositions or inversions with $3+4m$ or $3+8m$ total queries, respectively. More generally, we prove that exact transposition and inversion of an unknown $U\in\mathrm{SU}(d)$, $d\ge2$, require at least $d+1$ and $d^2-1$ 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.

Comments
31 pages, including supplementary material

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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.
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Symmetry plays a crucial role in the design and analysis of quantum protocols. This result shows a canonical circuit decomposition of a (G×H)(G\times H)(G×H)-invariant quantum comb for compact groups GGG and HHH using the corresponding Clebsch--Gordan transforms, which naturally extends to the GGG-covariant quantum comb. By using this circuit decomposition, we propose a parametrized quantum comb with group symmetry, and derive the optimal quantum comb which transforms an unknown unitary operation U∈SU(d)U\in \mathrm{SU}(d)U∈SU(d) into its inverse U†U^\daggerU† or transpose U⊤U^\topU⊤. From numerics, we find a deterministic and exact unitary transposition protocol for d=3d=3d=3 with 777 queries to UUU. This protocol improves upon the protocol shown in the previous work, which requires 131313 queries to UUU.
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