Universal construction of decoders from encoding black boxes

Satoshi Yoshida1, Akihito Soeda1,2,3, and Mio Murao1,4

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

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Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a $d$-dimensional system to a $D$-dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of $D$. For a qubit ($d=2$) encoded in $n$ qubits, our protocol achieves an exponential improvement over any tomography-based or unitary-embedding method, which cannot avoid $D$-dependence. We present a quantum operation that converts multiple parallel calls of any given isometry operation to random parallelized unitary operations, each of dimension $d$. Applied to our setup, it universally compresses the encoded quantum information to a $D$-independent space, while keeping the initial quantum information intact. This compressing operation is combined with a unitary inversion protocol to complete the isometry inversion. We also discover a fundamental difference between our isometry inversion protocol and the known unitary inversion protocols by analyzing isometry complex conjugation and isometry transposition. General protocols including indefinite causal order are searched using semidefinite programming for any improvement in the success probability over the parallel protocols. We find a sequential "success-or-draw" protocol of universal isometry inversion for $d = 2$ and $D = 3$, thus whose success probability exponentially improves over parallel protocols in the number of calls of the input isometry operation for the said case.

Encoding quantum information to a larger system and its inverse, decoding back to the original system, are essential operations utilized in various quantum information processing protocols for spreading and refocusing quantum information. This work explores a universal protocol to convert an encoder to its decoder as a higher-order quantum transformation without assuming classical descriptions of the encoder, given as a black box. This protocol allows to “undo'' encoding by executing the encoding operation multiple times, but does not require a full knowledge of the encoding operation. We call this task “isometry inversion,'' as encoding is mathematically represented by an isometry operation.

Remarkably, the success probability of our protocol does not depend on the output dimension of the isometry operation. The straightforward strategy for isometry inversion using known protocols is inefficient because its success probability depends on the output dimension, which is typically much larger than the input dimension. Therefore, the protocol proposed in this work outperforms the aforementioned protocol. We also compare isometry inversion with unitary inversion and show a crucial difference between them. Any isometry inversion protocol cannot be composed of complex conjugation and transposition of the input operations, while the known unitary inversion protocol can.

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