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arXiv:2306.08047 · npj Quantum Information

Information compression via hidden subgroup quantum autoencoders

Feiyang Liu, Kaiming Bian, Fei Meng, Wen Zhang, Oscar Dahlsten

Published Aug 8, 2024• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41534-024-00865-2

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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.
We experimentally demonstrate quantum data compression exploiting hidden subgroup symmetries using a photonic quantum processor. Classical databases containing generalized periodicities-symmetries that are in the worst cases inefficient for known classical algorithms to be detect-can efficiently compressed by quantum hidden subgroup algorithms. We implement a variational quantum autoencoder that autonomously learns both the symmetry type (e.g., Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ vs. Z4\mathbb{Z}_4Z4​) and the generalized period from structured data. The system uses single photons encoded in path, polarization, and time-bin degrees of freedom, with electronically controlled waveplates enabling tunable quantum gates. Training via gradient descent successfully identifies the hidden symmetry structure, achieving compression by eliminating redundant database entries. We demonstrate two circuit ansatzes: a parametrized generalized Fourier transform and a less-restricted architecture for Simon's symmetry. Both converge successfully, with the cost function approaching zero as training proceeds. These results provide experimental proof-of-principle that photonic quantum computers can compress classical databases by learning symmetries inaccessible to known efficient classical methods, opening pathways for quantum-enhanced information processing.
- Classical data are often compressed by using simple procedure which allows a string of data to take up less space during a computer’s memory. But, In Quantum Computing Data Compression is different because Quantum data are different and it's impossible to decide the frequencies of 1’s and 0’s in quantum information. The structure of the underlying autoencoder network is often chosen to represent the information on a smaller dimension, effectively compressing the input. Inspired by this concept, we introduce the model of a quantum autoencoder to perform similar tasks on quantum data. The quantum autoencoder is trained to compress a specific dataset of quantum states, where a classical compression algorithm cannot be employed. Using classical optimization algorithms, the parameters of the quantum autoencoder are trained. We show an example of an easy programmable circuit which will be trained as an efficient autoencoder with and without using dummy swap gates are implemented in IBMQ.
Abstract Classical autoencoders are neural networks that can learn efficient low-dimensional representations of data in higher-dimensional space. The task of an autoencoder is, given an input x , to map x to a lower dimensional point y such that x can likely be recovered from y . The structure of the underlying autoencoder network can be chosen to represent the data on a smaller dimension, effectively compressing the input. Inspired by this idea, we introduce the model of a quantum autoencoder to perform similar tasks on quantum data. The quantum autoencoder is trained to compress a particular data set of quantum states, where a classical compression algorithm cannot be employed. The parameters of the quantum autoencoder are trained using classical optimization algorithms. We show an example of a simple programmable circuit that can be trained as an efficient autoencoder. We apply our model in the context of quantum simulation to compress ground states of the Hubbard model and molecular Hamiltonians.
The search for practical problems that can be solved exponentially faster on a quantum computer is one of the primary goals in research into algorithms for quantum computers. It has long been known that such speedups are only possible for highly structured problems. We consider two problems with algebraic structure: (1) the hidden subgroup problem, where we have an unknown subgroup, a function that is constant on cosets of the subgroup, and the task is to find the unknown subgroup; (2) the hidden shift problem, where we have an unknown element of a group, a function that has been shifted by that group element, and the task is to find the unknown group element. We show that the Fourier transform is intimately connected to both these problems. This is because the Fourier transform respects group symmetries. For the hidden subgroup problem, we introduce the idea of using information theoretic techniques to design quantum algorithms. This consists of first finding an optimal set of measurements for distinguishing between different subgroups, followed by attempts to implement this set of measurements. We show that the standard algorithm for the abelian hidden subgroup problem is optimal. For the nonabelian hidden subgroup problem, the optimal measurements respect the Fourier transform in that they can be implemented by first performing a Fourier transform on each coset state, measuring the name of the irreducible representation for each state, followed by a joint measurement over the resulting lower dimensional space. This shows that the “strong standard method” cannot be used to implement the optimal set of measurements. For some instances of the hidden shift problem, we give a quantum algorithms that are exponentially faster than the best known classical algorithm. The algorithms make use of the Fourier transform in a way that may be interpreted as solving a deconvolution problem. As evidence of hardness, we give a reduction from algebraically homomorphic cryptosystems to the shifted Legendre symbol problem. This is an example of a problem which admits an exponential quantum speedup yet is not an instance of the abelian hidden subgroup problem.
As a ubiquitous aspect of modern information technology, data compression has a wide range of applications. Therefore, a quantum autoencoder which can compress quantum information into a low-dimensional space is fundamentally important to achieve automatic data compression in the field of quantum information. Such a quantum autoencoder can be implemented through training the parameters of a quantum device using classical optimization algorithms. In this paper, we demonstrate the condition of achieving a perfect quantum autoencoder and theoretically prove that a quantum autoencoder can losslessly compress high-dimensional quantum information into a low-dimensional space (also called latent space) if the number of maximum linearly independent vectors from input states is no more than the dimension of the latent space. Also, we experimentally realize a universal two-qubit unitary gate and design a quantum autoencoder device by applying a machine learning method. Experimental results demonstrate that our quantum autoencoder is able to compress two two-qubit states into two one-qubit states. Besides compressing quantum information, the quantum autoencoder is used to experimentally discriminate two groups of nonorthogonal states.
This paper explores the potential of quantum data compression through a novel algorithm predicated on the principles of quantum superposition and entanglement. Traditional data compression techniques rely on sequential processing, inherently limiting efficiency. We propose a quantum-based approach that leverages these principles to achieve exponential compression of data, exploiting inherent parallelism within quantum systems. The core mechanism involves quantum algorithms designed to systematically explore the data space, identifying and exploiting redundancies and correlations that classical algorithms struggle to detect. The resulting compression method offers a fundamentally different paradigm, potentially surpassing the capabilities of existing techniques in terms of compression ratios and computational speed. This research investigates the feasibility of implementing this quantum algorithm and analyzes its potential impact on various data types. The ultimate goal is to demonstrate a practical application of quantum data compression and explore its implications for future data storage and transmission.