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In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that $ell_2$-regularization with proper regularization gain provides even lower mean-square error under a cost in bias. The regularization parameter is tuned by two estimators in terms of a risk characterization. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed tuning estimators are asymptotically optimal as the number of samples grows to infinity under the risk metric. Additionally, numerical examples are provided to validate the established results.
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Linear regression is a basic and widely-used methodology in data analysis. It is known that some quantum algorithms efficiently perform least squares linear regression of an exponentially large data set. However, if we obtain values of the regression coefficients as classical data, the complexity of the existing quantum algorithms can be larger than the classical method. This is because it depends strongly on the tolerance error $epsilon$: the best one among the existing proposals is $O(epsilon^{-2})$. In this paper, we propose the new quantum algorithm for linear regression, which has the complexity of $O(epsilon^{-1})$ and keeps the logarithmic dependence on the number of data points $N_D$. In this method, we overcome bottleneck parts in the calculation, which take the form of the sum over data points and therefore have the complexity proportional to $N_D$, using quantum amplitude estimation, and other parts classically. Additionally, we generalize our method to some class of convex optimization problems.
In order to exploit quantum advantages, quantum algorithms are indispensable for operating machine learning with quantum computers. We here propose an intriguing hybrid approach of quantum information processing for quantum linear regression, which utilizes both discrete and continuous quantum variables, in contrast to existing wisdoms based solely upon discrete qubits. In our framework, data information is encoded in a qubit system, while information processing is tackled using auxiliary continuous qumodes via qubit-qumode interactions. Moreover, it is also elaborated that finite squeezing is quite helpful for efficiently running the quantum algorithms in realistic setup. Comparing with an all-qubit approach, the present hybrid approach is more efficient and feasible for implementing quantum algorithms, still retaining exponential quantum speed-up.
Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring $mathbb{D}$ and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-know scattering transformation in sd picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegles matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
Offline reinforcement learning (RL), also known as batch RL, offers the prospect of policy optimization from large pre-recorded datasets without online environment interaction. It addresses challenges with regard to the cost of data collection and safety, both of which are particularly pertinent to real-world applications of RL. Unfortunately, most off-policy algorithms perform poorly when learning from a fixed dataset. In this paper, we propose a novel offline RL algorithm to learn policies from data using a form of critic-regularized regression (CRR). We find that CRR performs surprisingly well and scales to tasks with high-dimensional state and action spaces -- outperforming several state-of-the-art offline RL algorithms by a significant margin on a wide range of benchmark tasks.
Ultrafast electron diffraction and time-resolved serial crystallography are the basis of the ongoing revolution in capturing at the atomic level of detail the structural dynamics of molecules. However, most experiments employ the classical ball-and-stick depictions, and the information of molecular quantum states, such as the density matrix, is missing. Here, we introduce a framework for the preparation and ultrafast coherent diffraction from rotational wave packets of molecules, and we establish a new variant of quantum state tomography for ultrafast electron diffraction to characterize the molecular quantum states. The ability to reconstruct the density matrix of molecules of arbitrary degrees of freedom will provide us with an unprecedentedly clear view of the quantum states of molecules, and enable the visualization of effects dictated by the quantum dynamics of molecules.
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