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.

我们设计了一种利用隐藏子群量子算法的经典信息压缩量子方法。我们考虑数据库中具有隐藏子组类型的先验未知对称性的序列数据。我们证明，具有给定组结构的数据可以以与隐藏子组问题相同的查询复杂度进行压缩，这比最著名的经典算法要快得多。此外，我们还设计了一种量子算法，可以通过变体方式找到群结构并使用它来压缩数据。按照量子自动编码器的范式，有一个编码器和一个解码器。训练后，编码器输出压缩数据串和隐藏子组对称性的描述，解码器可以从中恢复输入数据。在说明性示例中，我们的算法在测试数据的均方值上优于经典自动编码器。这种信息压缩能力中的经典量子分离具有热力学意义：量子智能体分配给系统的自由能可以远高于经典智能体的自由能。总而言之，我们的结果表明，量子计算机的一个可能应用是有效压缩某些类型的数据，而这些数据无法通过使用经典计算机的当前方法有效压缩。