Efficient compression of unknown quantum information

Yun Kai Deng, Ming‐Xing Luo, S. Y.

Feiyang Liu, Kaiming Bian, Fei Meng +2 more

We design a quantum method for classical information compression that exploits the hidden subgroup quantum algorithm. We consider sequence data in a database with a priori unknown symmetries of the hidden subgroup type. We prove that data with a given group structure can be compressed with the same query complexity as the hidden subgroup problem, which is exponentially faster than the best-known classical algorithms. We moreover design a quantum algorithm that variationally finds the group structure and uses it to compress the data. There is an encoder and a decoder, along the paradigm of quantum autoencoders. After the training, the encoder outputs a compressed data string and a description of the hidden subgroup symmetry, from which the input data can be recovered by the decoder. In illustrative examples, our algorithm outperforms the classical autoencoder on the mean squared value of test data. This classical-quantum separation in information compression capability has thermodynamical significance: the free energy assigned by a quantum agent to a system can be much higher than that of a classical agent. Taken together, our results show that a possible application of quantum computers is to efficiently compress certain types of data that cannot be efficiently compressed by current methods using classical computers.

François Chapeau‐Blondeau

Amir Kalev, Carlos A. Riofrío, Robert L. Kosut +1 more

Compressed sensing (CS) is a technique to faithfully estimate an unknown signal from relatively few data points when the measurement samples satisfy a restricted isometry property (RIP). Recently, this technique has been ported to quantum information science to perform tomography with a substantially reduced number of measurement settings. In this work we show that the constraint that a physical density matrix is positive semidefinite provides a rigorous connection between the RIP and the informational completeness of a POVM used for state tomography. This enables us to construct informationally complete measurements that are robust to noise using tools provided by the CS methodology. The exact recovery no longer hinges on a particular convex optimization program; solving any optimization, constrained to the cone of positive semidefinite matrices, effectively results in a CS estimation of the state. From a practical point of view, we can therefore employ fast algorithms developed to handle large dimensional matrices for efficient tomography of quantum states of a large dimensional Hilbert space.

Matej Pivoluska, Martin Plesch

Advances in development of quantum computing processors brought ample opportunities to test the performance of various quantum algorithms with practical implementations. In this paper we report on implementations of quantum compression algorithm that can efficiently compress unknown quantum information. We restricted ourselves to compression of three pure qubits into two qubits, as the complexity of even such a simple implementation is barely within the reach of today's quantum processors. We implemented the algorithm on IBM quantum processors with two different topological layouts-a fully connected triangle processor and a partially connected line processor. It turns out that the incomplete connectivity of the line processor affects the performance only minimally. On the other hand, it turns out that the transpilation, i.e. compilation of the circuit into gates physically available to the quantum processor, crucially influences the result. We also have seen that the compression followed by immediate decompression is, even for such a simple case, on the edge or even beyond the capabilities of currently available quantum processors.

Yuxiang Yang