In this work, we present a method to exponentiate non-sparse indefinite low-rank matrices on a quantum computer. Given an operation for accessing the elements of the matrix, our method allows singular values and associated singular vectors to be found quantum mechanically in a time exponentially faster in the dimension of the matrix than known classical algorithms. The method extends to non-Hermitian and non-square matrices via embedding matrices. In the context of the generic singular value decomposition of a matrix, we discuss the Procrustes problem of finding a closest isometry to a given matrix.Matrix computations are central to many algorithms in optimization and machine learning [1][2][3]. At the heart of these algorithms regularly lies an eigenvalue or a singular value decomposition of a matrix, or a matrix inversion. Such tasks could be performed efficiently via phase estimation on a universal quantum computer [4], as long as the matrix can be simulated (exponentiated) efficiently and controllably as a Hamiltonian acting on a quantum state. Almost exactly twenty years ago, Ref.[5] paved the way for such a simulation of quantum systems by introducing an efficient algorithm for exponentiating Hamiltonians with tensor product structure-enabling applications such as in quantum computing for quantum chemistry [6].Step by step, more general types of quantum systems were tackled and performance increased: Aharonov and Ta-Shma [7] showed a method for simulating quantum systems described by sparse Hamiltonians, while Childs et al. [8] demonstrated the simulation of a quantum walk on a sparse graph. Berry et al. [9] reduced the temporal scaling to approximately linear via higher-order Suzuki integrators. Further improvements in the sparsity scaling were presented in Ref. [10]. Beyond sparse Hamiltonians, quantum principal component analysis (qPCA) was shown to handle non-sparse positive semidefinite low-rank Hamiltonians [11] when given multiple copies of the Hamiltonian as a quantum density matrix. This method has applications in quantum process tomography and state discrimination [11], as well as in quantum machine learning [12][13][14][15][16][17][18], specifically in curve fitting [19] and support vector machines [20]. In an oracular setting, Ref. [10,21,22] showed the simulation of non-sparse Hamiltonians via discrete quantum walks. The scaling in terms of the simulated time t is t 3/2 or even linear in t.

In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross-validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.

We provide an argument to infer stationary entanglement between light and a mechanical oscillator based on continuous measurement of light only. We propose an experimentally realizable scheme involving an optomechanical cavity driven by a resonant, continuous-wave field operating in the non-sideband-resolved regime. This corresponds to the conventional configuration of an optomechanical position or force sensor. We show analytically that entanglement between the mechanical oscillator and the output field of the optomechanical cavity can be inferred from the measurement of squeezing in (generalized) Einstein-Podolski-Rosen quadratures of suitable temporal modes of the stationary light field. Squeezing can reach levels of up to 50% of noise reduction below shot noise in the limit of large quantum cooperativity. Remarkably, entanglement persists even in the opposite limit of small cooperativity. Viewing the optomechanical device as a position sensor, entanglement between mechanics and light is an instance of object-apparatus entanglement predicted by quantum measurement theory.

We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of unknown quantum fields based on data of correlation functions. At the basis of the analysis is the concept of continuous matrix product states (cMPS), a complete set of variational states grasping states in onedimensional quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used in the context of compressed sensing such as Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation functions. In the absence of a phase reference, we highlight how specific higher order correlation functions can still be predicted. We exemplify the functioning of the approach by reconstructing randomized cMPS from their correlation data and study the robustness of the reconstruction for different noise models. Furthermore, we apply the method to data generated by simulations based on cMPS and using the time-dependent variational principle. The presented approach is expected to open up a new window into experimentally studying continuous quantum systems, such as those encountered in experiments with ultra-cold atoms on top of atom chips. By virtue of the analogy with the input-output formalism in quantum optics, it also allows for studying open quantum systems.