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arXiv:2605.15784

Quantum compressed sensing

Jianyong Hu, Wei Li, Shuxiao Wu, Liwen Zhang, Yongchuang Sun, Jiazhao Tian, Guosheng Feng, Zhixing Qiao, Jianqiang Liu, Changgang Yang, Ruiyun Chen, Chengbing Qin, Guofeng Zhang, Liantuan Xiao, Suotang Jia

quant-phphysics.opticsSubmitted May 15, 2026• Announced May 18, 2026• 0 citations
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Abstract

How many measurements are fundamentally required to capture a signal. Shannon's information theory established the bedrock of this question in 1948, the Nyquist Shannon theorem set the first answer, and compressed sensing (CS) rewrote it in 2006 by reducing the required measurement number to M = O(Klog(N/K)) for a K sparse signal. Here, we propose quantum compressed sensing (QCS), a paradigm that reframes signal acquisition as a unitary quantum evolution. By encoding high dimensional signal information into a single quantum probe state, then introducing domain-alignment evolution,a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis. QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M =O(K) and reducing reconstruction from ill posed optimization to linear estimation. We experimentally validate QCS using frequency and time domain sparse signals, confirming that the measurement number scales linearly with sparsity and decouples entirely from the signal dimension. Our work provides a physical pathway toward ultimate information acquisition efficiency, with broad implications for sensing, imaging, and communication.

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How many measurements are fundamentally required to capture a signal. Shannon's information theory established the bedrock of this question in 1948, the Nyquist Shannon theorem set the first answer, and compressed sensing (CS) rewrote it in 2006 by reducing the required measurement number to M = O(Klog(N/K)) for a K sparse signal. Here, we propose quantum compressed sensing (QCS), a paradigm that reframes signal acquisition as a unitary quantum evolution. By encoding high dimensional signal information into a single quantum probe state, then introducing domain-alignment evolution,a physically realizable unitary transformation that maps the sparse basis directly onto the measurement basis. QCS executes the support-set search at the quantum level without consuming measurement trials. The logarithmic penalty vanishes, compressing the required measurement number from the classical bound to M =O(K) and reducing reconstruction from ill posed optimization to linear estimation. We experimentally validate QCS using frequency and time domain sparse signals, confirming that the measurement number scales linearly with sparsity and decouples entirely from the signal dimension. Our work provides a physical pathway toward ultimate information acquisition efficiency, with broad implications for sensing, imaging, and communication.
Conventional compressed sensing (CS) has a measurement lower bound of M = Ω(K log(N/K)) under non-adaptive measurements. Recent experiments on quantum compressed sensing (QCS) have reported numbers of measurements below this classical lower bound. In this work, we establish lower bounds on the measurement complexity of QCS from an information-theoretic and quantum-physical perspective. QCS exploits quantum parallelism, enabling a unitary domain-alignment evolution to act on a superposition of all N basis states and map the sparse basis onto the measurement basis in a one-to-one manner. Consequently, the locations of nonzero components are explicitly carried by measurement-index labels. If only basic statistical information about the K nonzero components is required, the total number of effective index samples satisfies the coarse necessary condition Ms = Ω(K). Under independent single-index projective measurements, when the support is unknown, the nonzero probabilities satisfy pn = Θ(1/K), and exact support recovery is required with a fixed success probability, the sampling-coverage requirement leads to Ms = Θ(K ln K). If each nonzero amplitude coefficient is further required to achieve a coordinate-wise relative root-mean-square error ε, the optimal effective index sample complexity is Ms = Θ(K ln K + K/ε2), which reduces to Θ(K ln K) for fixed ε. Multimode photon-number-resolving detection can reduce the number of experimental rounds but not the total effective index sample complexity. These results show that quantum parallelism is not only a core resource for quantum computing but can also play a transformative role in measurement science.
Quantum state tomography (QST) is a crucial technique for characterizing the state of quantum systems. However, traditional QST requires a large number of measurements, leading to significant experimental overhead in terms of time, resources, and data acquisition. This work proposes a novel approach to QST by leveraging compressed sensing (CS) techniques. CS allows for accurate reconstruction of the quantum state from a significantly reduced set of measurements, thereby mitigating the experimental burden. We employ sparsity regularization, a key component of CS, to minimize the number of measurements needed. The theoretical framework and the underlying principles are presented, focusing on the mathematical formulation of the problem and the optimization strategies involved. This method offers a pathway to more efficient and cost-effective QST, particularly for complex quantum states or when limited experimental resources are available. We demonstrate the potential of this approach through a detailed mathematical analysis.
Compressive sensing is a signal processing technique that enables the reconstruction of sparse signals from a limited number of measurements, leveraging the signal's inherent sparsity to facilitate efficient recovery. Recent works on the Quantum Compressive Sensing (QCS) architecture, a quantum data-driven approach to compressive sensing where the state of the tensor network is represented by a quantum state over a set of entangled qubits, have shown promise in advancing quantum data-driven methods for compressive sensing. However, the QCS framework has remained largely untested on quantum computing resources or in the presence of quantum noise. In this work, we present a practical implementation of QCS on Amazon Braket, utilizing the Quantum Imaginary Time Evolution (QITE) projection technique to assess the framework's capabilities under quantum noise. We outline the necessary modifications to the QCS framework for deployment on Amazon Braket, followed by results under four types of quantum noise. Finally, we discuss potential long-term directions aimed at unlocking the full potential of quantum compressive sensing for applications such as signal recovery and image processing.
Compressed Sensing (CS) is a new method of signal and image processing which allows for exact recovery of an image from a number of samples much smaller than that required by the Nyquist/Shannon theorem. Compressed Sensing uses a priori information about the object called “sparsity”, which means that only a small number of image samples are nonzero. We have analyzed the superresolution behavior of CS taking into account the quantum fluctuations in the image. Our analysis allows to characterize the ultimate capabilities of CS imposed by the quantum nature of the light.
Quantum state tomography (QST) is a crucial technique for characterizing quantum states, forming the bedrock of quantum information processing and quantum metrology. However, traditional QST relies on a large number of measurements, making it computationally expensive and experimentally demanding. This work addresses this challenge by proposing a novel algorithm that integrates compressed sensing (CS) and Bayesian inference to achieve accurate quantum state reconstruction from limited measurement data. The core idea is to exploit the inherent sparsity of quantum states and leverage CS principles to significantly reduce the required measurement count. The algorithm utilizes a sparse dictionary learned from prior knowledge of the quantum state and employs Bayesian inference to estimate the state parameters, providing a robust and efficient reconstruction process. The results demonstrate the feasibility and effectiveness of this approach, offering a pathway towards more practical and accessible QST methods. Key mathematical formulations are presented, including the Bayesian inference framework and the CS reconstruction process.
Compressive sensing is a sensing protocol that facilitates the reconstruction of large signals from relatively few measurements by exploiting known structures of signals of interest, typically manifested as signal sparsity. Compressive sensing’s vast repertoire of applications in areas such as communications and image reconstruction stems from the traditional approach of utilizing non-linear optimization to exploit the sparsity assumption by selecting the lowest-weight (i.e., maximum sparsity) signal consistent with all acquired measurements. Recent efforts in the literature consider instead a data-driven approach, training tensor networks to learn the structure of signals of interest. The trained tensor network is updated to “project” its state onto one consistent with the measurements taken, and is then sampled site by site to “guess” the original signal. In this paper, we take advantage of this computing protocol by formulating an alternative “quantum” protocol, in which the state of the tensor network is a quantum state over a set of entangled qubits. Accordingly, we present the associated algorithms and quantum circuits required to implement the training, projection, and sampling steps on a quantum computer. We supplement our theoretical results by simulating the proposed circuits with a small, qualitative model of LIDAR imaging of earth forests. Our results indicate that a quantum, data-driven approach to compressive sensing may have significant promise as quantum technology continues to make new leaps.