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.
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.
Compressive sensing utilizes sparsity to realize efficient image reconstruction. It is a valuable processing technique when cost, power, technology or computational overhead are limited or high. In the quantum domain technology usually limits efficient acquisition of weak or fragile signals. I will discuss the basics of information theory, compression, and compressive sensing. I will then discuss our recent work in compressive sensing. The topics of discussion include low-flux laser Radar, photonic phase transitions, high resolution biphoton ghost imaging, Ghost object tracking, 3D object tracking and high dimensional entanglement characterization. I will touch lightly on our current work of rapid wavefunction reconstruction and wavefront sensing. As an example (shown below), we were able efficiently and rapidly reconstruct high dimensional joint probability functions of biphotons in momentum and position. With conventional raster scanning this process would take approximately a year, but using double-pixel compressive sensing, the pictures were acquired in a few hours with modest flux.
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.
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.
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.
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.