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The polar decomposition for a matrix $A$ is $A=UB$, where $B$ is a positive Hermitian matrix and $U$ is unitary (or, if $A$ is not square, an isometry). This paper shows that the ability to apply a Hamiltonian $pmatrix{ 0 & A^dagger cr A & 0 cr} $ translates into the ability to perform the transformations $e^{-iBt}$ and $U$ in a deterministic fashion. We show how to use the quantum polar decomposition algorithm to solve the quantum Procrustes problem, to perform pretty good measurements, to find the positive Hamiltonian closest to any Hamiltonian, and to perform a Hamiltonian version of the quantum singular value transformation.
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In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-search and phase-rotation factors. We discuss how the additive form can be utilized so that it effectively reduces the computational load of the iterative search, and we propose an effective shortcut gate that realizes the same outcome as the iterative search.
The quantum discord was introduced by Ollivier, Zurek, Henderson and Vedral as an indicator of the degree of quantumness of mixed states. In this paper, we give out the decomposition condition of quantum discords. Moreover, we show that under the condition, the quantum correlations between the quantum systems can be captured completely by the entanglement measure.
Currently available quantum hardware allows for small scale implementations of quantum machine learning algorithms. Such experiments aid the search for applications of quantum computers by benchmarking the near-term feasibility of candidate algorithms. Here we demonstrate the quantum learning of a two-qubit unitary by a sequence of three parameterized quantum circuits containing a total of 21 variational parameters. Moreover, we variationally diagonalize the unitary to learn its spectral decomposition, i.e., its eigenvalues and eigenvectors. We illustrate how this can be used as a subroutine to compress the depth of dynamical quantum simulations. One can view our implementation as a demonstration of entanglement-enhanced machine learning, as only a single (entangled) training data pair is required to learn a 4x4 unitary matrix.
Visual tracking (VT) is the process of locating a moving object of interest in a video. It is a fundamental problem in computer vision, with various applications in human-computer interaction, security and surveillance, robot perception, traffic control, etc. In this paper, we address this problem for the first time in the quantum setting, and present a quantum algorithm for VT based on the framework proposed by Henriques et al. [IEEE Trans. Pattern Anal. Mach. Intell., 7, 583 (2015)]. Our algorithm comprises two phases: training and detection. In the training phase, in order to discriminate the object and background, the algorithm trains a ridge regression classifier in the quantum state form where the optimal fitting parameters of ridge regression are encoded in the amplitudes. In the detection phase, the classifier is then employed to generate a quantum state whose amplitudes encode the responses of all the candidate image patches. The algorithm is shown to be polylogarithmic in scaling, when the image data matrices have low condition numbers, and therefore may achieve exponential speedup over the best classical counterpart. However, only quadratic speedup can be achieved when the algorithm is applied to implement the ultimate task of Henriquess framework, i.e., detecting the object position. We also discuss two other important applications related to VT: (1) object disappearance detection and (2) motion behavior matching, where much more significant speedup over the classical methods can be achieved. This work demonstrates the power of quantum computing in solving computer vision problems.
For two unknown quantum states $rho$ and $sigma$ in an $N$-dimensional Hilbert space, computing their fidelity $F(rho,sigma)$ is a basic problem with many important applications in quantum computing and quantum information, for example verification and characterization of the outputs of a quantum computer, and design and analysis of quantum algorithms. In this Letter, we propose a quantum algorithm that solves this problem in $text{poly}(log (N), r)$ time, where $r$ is the lower rank of $rho$ and $sigma$. This algorithm exhibits an exponential improvement over the best-known algorithm (based on quantum state tomography) in $text{poly}(N, r)$ time.
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