Universal adjointation of isometry operations using conversion of quantum supermaps

Satoshi Yoshida; Akihito Soeda; Mio Murao

doi:10.22331/q-2025-05-20-1750

Universal adjointation of isometry operations using conversion of quantum supermaps

Satoshi Yoshida¹, Akihito Soeda^2,3,1, and Mio Murao^1,4

¹Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
²Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
³Department of Informatics, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁴Trans-scale Quantum Science Institute, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

Published:	2025-05-20, volume 9, page 1750
Editor:	Borivoje Dakic
Eprint:	arXiv:2401.10137v3
Doi:	https://doi.org/10.22331/q-2025-05-20-1750
Citation:	Quantum 9, 1750 (2025).

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Abstract

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 $\textit{isometry adjointation}$ 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 $\epsilon = \Theta(d^2/n)$, 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.

Featured image: This work implements the "undo" operation $V_\mathrm{in}^\dagger$ of an unknown encoding operation $V_\mathrm{in}$ (right). The "undo" operation consists of two steps: first, a subspace check verifying whether the input state $\vert\phi_\mathrm{in}\rangle$ is in the image of $V_\mathrm{in}$, and second, decoding by applying $V_\mathrm{in}^{-1}$ on the input state. This construction is based on the conversion of the inversion protocol for unitary operations ($U_\mathrm{in}\to U_\mathrm{in}^{-1}$, left) to the broader class of isometry operations.

Popular summary

Controlling and transforming quantum systems is essential for advancing quantum algorithms. One important goal is to develop universal transformations—protocols that work regardless of the specific input. This transformation has been explored for unitary operations, reversible operations preserving the size of the system. However, extending it to more general operations has remained a significant challenge. Isometry operations form an important class of quantum operations since they represent the encoding of quantum information to a larger system and can represent general quantum operations by embedding the output system into a larger system. Isometry operation is an information-preserving operation, but it is reversible only within a certain subspace since it spreads the input information to a larger system.

This work introduces new protocols to "undo" an unknown isometry operation. These protocols first check whether the input state lies within the invertible subspace (subspace check) and, if it does, apply the inverse operation (decoding)—all by transforming the original "do" operation without learning what it is. These protocols are designed to be highly accurate and efficient, regardless of the output dimension of the isometry operation, which can be much larger than the input dimension. This construction is based on a quantum circuit that converts the corresponding protocol for the unitary operation to handle the broader class of isometry operations.

► BibTeX data

@article{Yoshida2025universal,
  doi = {10.22331/q-2025-05-20-1750},
  url = {https://doi.org/10.22331/q-2025-05-20-1750},
  title = {Universal adjointation of isometry operations using conversion of quantum supermaps},
  author = {Yoshida, Satoshi and Soeda, Akihito and Murao, Mio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1750},
  month = may,
  year = {2025}
}

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