Linear Unitary Transformation Research Articles

An Image Classification Algorithm Based on Hybrid Quantum Classical Convolutional Neural Network

Quantum machine learning is emerging as a strategy to solve real-world problems. As a quantum computing model, parameterized quantum circuits provide an approach for constructing quantum machine learning algorithms, which may either realize computational acceleration or achieve better algorithm performance than classical algorithms. Based on the parameterized quantum circuit, we propose a hybrid quantum-classical convolutional neural network (HQCCNN) model for image classification that comprises both quantum and classical components. The quantum convolutional layer is designed using a parameterized quantum circuit. It is used to perform linear unitary transformation on the quantum state to extract hidden information. In addition, the quantum pooling unit is used to perform pooling operations. After the evolution of the quantum system, we measure the quantum state and input the measurement results into a classical fully connected layer for further processing. We demonstrate its potential by applying HQCCNN to the MNIST dataset. Compared to a convolutional neural network in a similar architecture, the results reveal that HQCCNN has a faster training speed and higher testing set accuracy than a convolutional neural network.

Arbitrary optical wave evolution with Fourier transforms and phase masks.

A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. [Phys. Rev. Lett.73, 58 (1994)10.1103/PhysRevLett.73.58], it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a deterministic algorithm that can find an exact and efficient implementation of any unitary transformation, using only Fourier transforms and phase masks. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from some of the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.

Scalable low-rank tensor learning for spatiotemporal traffic data imputation

Missing value problem in spatiotemporal traffic data has long been a challenging topic, in particular for large-scale and high-dimensional data with complex missing mechanisms and diverse degrees of missingness. Recent studies based on tensor nuclear norm have demonstrated the superiority of tensor learning in imputation tasks by effectively characterizing the complex correlations/dependencies in spatiotemporal data. However, despite the promising results, these approaches do not scale well to large data tensors. In this paper, we focus on addressing the missing data imputation problem for large-scale spatiotemporal traffic data. To achieve both high accuracy and efficiency, we develop a scalable tensor learning model—Low-Tubal-Rank Smoothing Tensor Completion (LSTC-Tubal)—based on the existing framework of Low-Rank Tensor Completion, which is well-suited for spatiotemporal traffic data that is characterized by multidimensional structure of location × time of day × day. In particular, the proposed LSTC-Tubal model involves a scalable tensor nuclear norm minimization scheme by integrating linear unitary transformation. Therefore, tensor nuclear norm minimization can be solved by singular value thresholding on the transformed matrix of each day while the day-to-day correlation can be effectively preserved by the unitary transform matrix. Before setting up the experiment, we consider some real-world data sets, including two large-scale 5-min traffic speed data sets collected by the California PeMS system with 11160 sensors: 1) PeMS-4W covers the data over 4 weeks (i.e., 288×28 time points), and 2) PeMS-8W covers the data over 8 weeks (i.e., 288×56 time points). We compare LSTC-Tubal with some state-of-the-art baseline models, and find that LSTC-Tubal can achieve competitively accuracy with a significantly lower computational cost. In addition, the LSTC-Tubal will also benefit other tasks in modeling large-scale spatiotemporal traffic data, such as network-level traffic forecasting.

Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels

We consider the transmission of a Gaussian vector source over a multidimensional Gaussian channel where a random or a fixed subset of the channel outputs are erased. Within the setup where the only encoding operation allowed is a linear unitary transformation on the source, we investigate the minimum mean-square error (MMSE) performance, both in average, and also in terms of guarantees that hold with high probability as a function of the system parameters. Under the performance criterion of average MMSE, necessary conditions that should be satisfied by the optimal unitary encoders are established and explicit solutions for a class of settings are presented. For random sampling of signals that have a low number of degrees of freedom, we present MMSE bounds that hold with high probability. Our results illustrate how the spread of the eigenvalue distribution and the unitary transformation contribute to these performance guarantees. The performance of the discrete Fourier transform (DFT) is also investigated. As a benchmark, we investigate the equidistant sampling of circularly wide-sense stationary signals, and present the explicit error expression that quantifies the effects of the sampling rate and the eigenvalue distribution of the covariance matrix of the signal. These findings may be useful in understanding the geometric dependence of signal uncertainty in a stochastic process. In particular, unlike information theoretic measures such as entropy, we highlight the basis dependence of uncertainty in a signal with another perspective. The unitary encoding space restriction exhibits the most and least favorable signal bases for estimation.

Integrated multimode interferometers with arbitrary designs for photonic boson sampling

The evolution of bosons undergoing arbitrary linear unitary transformations quickly becomes hard to predict using classical computers as we increase the number of particles and modes. Photons propagating in a multiport interferometer naturally solve this so-called boson sampling problem1, thereby motivating the development of technologies that enable precise control of multiphoton interference in large interferometers2,3,4. Here, we use novel three-dimensional manufacturing techniques to achieve simultaneous control of all the parameters describing an arbitrary interferometer. We implement a small instance of the boson sampling problem by studying three-photon interference in a five-mode integrated interferometer, confirming the quantum-mechanical predictions. Scaled-up versions of this set-up are a promising way to demonstrate the computational advantage of quantum systems over classical computers. The possibility of implementing arbitrary linear-optical interferometers may also find applications in high-precision measurements and quantum communication5.

Time evolution of a time-dependent harmonic oscillator in a static magnetic field

The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su(1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators (a, a†), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.

Linear extension of the Robinson–Schensted algorithm

AbstractThe Robinson–Schensted (RS) algorithm demonstrates a bijection between the set of magnetic configurations f and the set of pairs of tableaux: a semistandard Weyl tableau P(f) accompanied by a standard Young tableau Q(f). We show that it is not only a bijection between sets, but it can be extended to a linear unitary transformation within the space of all quantum states of the magnet. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

A time-dependent closed-form formulation of the linear unitary transformation for harmonic-oscillator annihilation and creation operators is presented in the Schrödinger picture using the Lie algebraic approach. The time evolution of the quantum mechanical system described by a general time-dependent quadratic Hamiltonian is investigated by combining this formulation with the time evolution equation of the system. The analytic expressions of the evolution operator and propagator are found. The motion of a charged particle with variable mass in the time-dependent electric field is considered as an illustrative example of the formalism. The exact time evolution wave function starting from a Gaussian wave packet and the operator expectation values with respect to the complicated evolution wave function are obtained readily.

On linear unitary transformations of two canonical variables

A useful parametrization of the groups SO(0,3) and SO↑(2,1) is presented that has simple, rational composition laws, albeit that it has a (quasigraded) structure. This ‘‘tangential’’ parametrization is also advantageous in providing a rather simple 1‐1 ‘‘picture’’ of the elements of the groups in question. As well, the parametrization allows an explicitly finite result for the composition law needed for the (abstract) Baker–Campbell–Hausdorff formula for exponentials of objects formed from the corresponding Lie algebras. This approach, in turn, allows a useful beginning to the problem of the determination of flows on the group manifold, i.e., the determination of analytic curves given the initial direction of the curve.

Quantum mechanics in phase space: III. Linear transformations

The correspondence of a linear canonical transformation in phase space with a linear unitary transformation in the Hilbert space L 2( R ) yields in a unique way the Wigner kernel. With this kernel function the unitary transformations corresponding to finite linear transformations are calculated in different representations. The results are applied to propagators derived from quadratic hamiltonians.

A new formulation of relativistic quantum field theory

Quantum field theory is reformulated in sucha manner that a complete set of ocillators for modes with both positive and negative energies are introduced. The theory leads to the proper connection between spin and statistics as in the standard formulation, but it implements the time reversal transformation and the TCP transformation as linear unitary transformations. Negative energy particles in the initial states are identified with antiparticles in the final state with reversed motion (andvice versa) as far as scattering amplitudes are concerned. A covariant perturbation theory is developed which yields scattering amplitudes which are essentially the same as in the usual theory.