Representation Of Density Matrix Research Articles

Quantum-inspired semantic matching based on neural networks with the duality of density matrices

Social media text can be semantically matched in different ways, viz paraphrase identification, answer selection, community question answering, and so on. The performance of the above semantic matching tasks depends largely on the ability of language modeling. Neural network based language models and probabilistic language models are two main streams of language modeling approaches. However, few prior work has managed to unify them in a single framework on the premise of preserving probabilistic features during the neural network learning process. Motivated by recent advances of quantum-inspired neural networks for text representation learning, we fill the gap by resorting to density matrices, a key concept describing a quantum state as well as a quantum probability distribution. The state and probability views of density matrices are mapped respectively to the neural and probabilistic aspects of language models. Concretizing this state-probability duality to the semantic matching task, we build a unified neural-probabilistic language model through a quantum-inspired neural network. Specifically, we take the state view to construct a density matrix representation of sentence, and exploit its probabilistic nature by extracting its main semantics, which form the basis of a legitimate quantum measurement. When matching two sentences, each sentence is measured against the main semantics of the other. Such a process is implemented in a neural structure, facilitating an end-to-end learning of parameters. The learned density matrix representation reflects an authentic probability distribution over the semantic space throughout the training process. Experiments show that our model significantly outperforms a wide range of prominent classical and quantum-inspired baselines.

Geometric visualizations of single and entangled qubits

The Bloch Sphere visualization of the possible states of a single qubit serves as a useful pedagogical and conceptual tool, offering a one-to-one map between qubit states and points in a 3D space. However, understanding many important concepts of quantum mechanics, such as entanglement, requires developing intuitions about states with a minimum of two qubits, which map one-to-one to unvisualizable spaces of six dimensions and higher. In this paper, we circumvent this visualization issue by creating maps of subspaces of 1- and 2-qubit systems that quantitatively and qualitatively encode properties of these states in their geometries. For the 1-qubit case, the subspace approach allows one to visualize how mixed states relate to different choices of measurement in a basis-independent way and how to read off the entries in a density matrix representation of these states from lengths in a simple diagram. For the 2-qubit case, a toroidal map of 2-qubit states illuminates the non-trivial topology of the state space while allowing one to simultaneously read off, in distances and angles, the level of entanglement in the 2-qubit state and the mixed-state properties of its constituent qubits. By encoding states and their evolutions through quantum logic gates with little to no need of mathematical formalism, these maps may prove particularly useful for understanding fundamental concepts of quantum mechanics and quantum information at the introductory level.

MPS-VQE: A variational quantum computational chemistry simulator with matrix product states

Quantum computing is attracting more and more attention in material and biological science. In the noisy intermediate-scale quantum (NISQ) computing era, the variational quantum eigensolver (VQE) is expected as an effective method to solve quantum chemistry problems which has potential quantum advantage. Classically simulating quantum algorithms plays a critical role in algorithmic development and the validation of noisy quantum devices before large-scale, fully fault-tolerant quantum computers are developed. However, most existing simulators are based on the state vector or density matrix representation of the quantum state which is faced with the memory bottleneck due to the exponential growth of memory requirement as the number of qubits increases. In this work, we present a VQE simulator based on the matrix product states (MPS) method from quantum many-body physics, and a detailed study of our MPS-VQE simulator in quantum chemistry applications. The memory requirement of MPS-VQE grows only polynomially and we demonstrate that accurate results can be obtained despite the truncation of the smallest singular values during the algorithm. A distributed parallelization scheme is also presented for massively parallel computer systems, and different implementations of the parallelization scheme based on the Julia language are benchmarked which demonstrate the efficiency and scalability of our method. Our method could open a new avenue towards large-scale classical simulation of quantum computational chemistry.

Quantum canonical correlation analysis algorithm

Canonical correlation analysis (CCA) is a fundamental technique used to analyze data correlation in various fields, including video and medical data analysis. In this paper, we propose a quantum canonical correlation analysis (QCCA) algorithm. First, we introduce a combined density matrix representation method that transforms CCA into generalized eigenvalue decomposition. Moreover, to address the challenge of performing generalized eigenvalue decomposition in high-dimensional scenarios, we propose a quantum method for extracting the canonical principal axes. In this method, two sets of variables are transformed into a reduced density matrix, so that the product of variable matrices can be accelerated by phase estimation and controlled rotation. Complexity analysis shows that the QCCA algorithm achieves exponential acceleration in variable dimensions n, p and variable size m compared to classical algorithms. The QCCA algorithm serves as a foundation for the subsequent development of quantum algorithms for classification, regression, and other machine learning tasks.

On compression rate of quantum autoencoders: Control design, numerical and experimental realization

Quantum autoencoders which aim at compressing quantum information in a low-dimensional latent space lie in the heart of automatic data compression in the field of quantum information. In this paper, we establish an upper bound of the compression rate for a given quantum autoencoder and present a learning control approach for training the autoencoder to achieve the maximal compression rate. The upper bound of the compression rate is theoretically proven using eigen-decomposition and matrix differentiation, which is determined by the eigenvalues of the density matrix representation of the input states. Numerical results on 2-qubit and 3-qubit systems are presented to demonstrate how to train the quantum autoencoder to achieve the theoretically maximal compression, and the training performance using different machine learning algorithms is compared. Experimental results of a quantum autoencoder using quantum optical systems are illustrated for compressing two 2-qubit states into two 1-qubit states.

Finite temperature quantum dynamics of complex systems: Integrating thermo‐field theories and tensor‐train methods

AbstractThis review provides the fundamental theoretical tools for the development of a complete wave‐function formalism for the study of time‐evolution of chemico‐physical systems at finite temperature. The methodology is based on the non‐equilibrium thermo‐field dynamics (NE‐TFD) representation of quantum mechanics, which is alternative to the commonly used density matrix representation. TFD concepts are extended and integrated with the tensor‐train (TT) numerical tools leading to a novel and powerful theoretical and computational framework for the study of complex quantum dynamical problems. In addition, NE‐TFD techniques are extended to enable the study of dissipative open systems via a new formulation of the hierarchical equations of motion (HEOM) fully integrated with TT methodologies. We demonstrate that the combination of the TFD machinery with computational advantages of TTs results in a powerful theoretical and computational framework for scrutinizing dynamics of complex multidimensional electron‐vibrational systems. We illustrate the validity and the computational advantages of the developed methodologies by applying them to the study of quantum coherence effects in the energy‐transfer processes in antenna systems, to the analysis of fingerprints of vibrational modes in electron‐transfer and charge‐transfer processes in various model and realistic multidimensional molecular systems, as well as to simulation of other fundamental models of physical chemistry.This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics

Block generation in a two-dimensional space constructed by Hellinger metric and affinity for weather data fusion and learning inputs

The concise and reliable representation of a given source dataset is available for data based learning and decision-making, which can be obtained through a controllable data fusion process within a formatted space. Based on the density matrix representation of a given source weather dataset, this paper first calculates the Hellinger distances and affinities between different density matrices, and then takes them to construct a two-dimensional metric space using the longest Hellinger metric and affinity paths. Within this space, the given source data units are converted into the corresponding rectangle nodes which are determined by the minimum horizontal and vertical distances between different data units. According to a predefined detection size in this space, the basic blocks centred by different rectangle nodes are classified into different subset blocks for fusion. Each subset block is jointed by the basic blocks with overlapped areas. During the detection process, the detection size keeps increasing until a predefined reference variable, such as the relative density for all subset blocks, reaches a predefined turning point. The fusion of the rectangle nodes in a subset block depends on the distances between the included rectangle nodes’ positions and this subset block’s centre position which is calculated according to the included rectangle nodes’ weights and positions. The experimental analysis shows that the proposed weather data fusion method is controllable and stable and can obtain concise and reliable fusion results for learning inputs and decision-making.

Machine-Learning Coupled Cluster Properties through a Density Tensor Representation.

The introduction of machine-learning (ML) algorithms to quantum mechanics enables rapid evaluation of otherwise intractable expressions at the cost of prior training on appropriate benchmarks. Many computational bottlenecks in the evaluation of accurate electronic structure theory could potentially benefit from the application of such models, from reducing the complexity of the underlying wave function parameter space to circumventing the complications of solving the electronic Schrödinger equation entirely. Applications of ML to electronic structure have thus far been focused on learning molecular properties (mainly the energy) from geometric representations. While this line of study has been quite successful, highly accurate models typically require a "big data" approach with thousands of training data points. Herein, we propose a general, systematically improvable scheme for wave function-based ML of arbitrary molecular properties, inspired by the underlying equations that govern the canonical approach to computing the properties. To this end, we combine the established ML machinery of the t-amplitude tensor representation with a new reduced density matrix representation. The resulting model provides quantitative accuracy in both the electronic energy and dipoles of small molecules using only a few dozen training points per system.

Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field

We report a combined theoretical and experimental investigation on the unambiguous role of electronic coherence underlying the generation of laserlike coherent emission from ${{\mathrm{N}}_{2}}^{+}$ ions launched in a strong laser field. The coherence manifests itself by giving rise to a dramatic enhancement of ${{\mathrm{N}}_{2}}^{+}$ laserlike radiation with two temporally separated pump laser pulses centered at 800- and 1580-nm wavelengths, respectively. We further show that coherent control of ${{\mathrm{N}}_{2}}^{+}$ lasing in a subfemtosecond timescale can be achieved by replacing the single 800-nm laser pulse with a pair of 800-nm laser pulses of adjustable time delay. The experimental results are well reproduced in theory using a three-level quantum model of ${{\mathrm{N}}_{2}}^{+}$ ions under the density matrix representation.

Condition for zero and nonzero discord in graph Laplacian quantum states

This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.

Preparation and measurement in quantum memory models

Quantum Cognition has delivered a number of models for semantic memory, but to date these have tended to assume pure states and projective measurement. Here we relax these assumptions. A quantum inspired model of human word association experiments will be extended using a density matrix representation of human memory and a POVM based upon non-ideal measurements. Our formulation allows for a consideration of key terms like measurement and contextuality within a rigorous modern approach. This approach both provides new conceptual advances and suggests new experimental protocols.

Geometric multiaxial representation of N -qubit mixed symmetric separable states

Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use geometric multiaxial representation(MAR) of density matrix to characterize the mixed symmetric separable states. Since separability problem is NP hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P-distribution function \lambda(\theta; \phi). We show that the N-qubit mixed symmetric separable state can be visualized as a uniaxial system if the distribution function is independent of \theta and \phi. We further choose distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two qubit mixed symmetric separable state. We also observe that there exists a correspondence between separability and classicality of states.

Stimulated Emission Enhancement Using Shaped Pulses.

Controlling stimulated emission is of importance because it competes with absorption and fluorescence under intense laser excitation. We performed resonant nonlinear optical spectroscopy measurements using femtosecond pulses shaped by π- or π/2-step phase functions and carried out calculations based on density matrix representation to elucidate the experimental results. In addition, we compared enhancements obtained when using other pulse shaping functions (chirp, third-order dispersion, and a time-delayed probe). The light transmitted through the high optical density solution was dominated by an intense stimulated emission feature that was 14 times greater for shaped pulses than for transform limited pulses. Coherent enhancement depending on the frequency, temporal, and phase characteristics of the shaped pulse is responsible for the experimental observations.

Comment on “Concrete Representation and Separability Criteria for Symmetric Quantum State”

Recently, Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) presented the concrete representation of density matrix of symmetric quantum states . Moreover , according to this concrete representation of the density matrix for symmetric quantum states, Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) have established Theorem 4.1. In this Comment, we would like to point out that Theorem 4.1 given by Li et al. (Int. J. Theor. Phys. 53(9), 2923–2930 (2014)) is incorrect in general.

Comment on “Criteria for Separability of Multipartite Quantum System”

Recently, Tao et al. (Int. J. Theor. Phys. 52 (6), 1970–1978, 2013) established a representation of density matrix for multipartite quantum system. Moreover, according to this representation of the density matrix for multipartite quantum system, Tao et al. (Int. J. Theor. Phys. 52 (6), 1970–1978, 2013) have presented two necessary criteria for separability of multipartite quantum system of arbitrary dimensions, for example, see Theorem 2.1 and Theorem 2.2. In this Comment, we would like to point out that Theorem 2.2 given by Tao et al. (Int. J. Theor. Phys. 52 (6), 1970–1978, 2013) is incorrect in general.

Bethe free-energy approximations for disordered quantum systems.

Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

Concrete Representation and Separability Criteria for Symmetric Quantum State

Using the typical generators of the special unitary groups SU(2), the concrete representation of symmetric quantum state is established, then the relations satisfied by those coefficients in the representation are presented. Based on the representation of density matrix, the PPT criterion and CCNR criterion are proved to be equivalent on judging the separability of symmetric quantum states. Moreover, it is showed that the matrix Γρ of symmetric quantum state only has five efficient entries, thus the calculation of ∥Γρ∥ is simplified. Finally, the quantitative expressions of real symmetric quantum state under the ∥Γρ∥ separability criterion are obtained.

CUGatesDensity—Quantum circuit analyser extended to density matrices

CUGatesDensity is an extension of the original quantum circuit analyser CUGates (Loke and Wang, 2011) [7] to provide explicit support for the use of density matrices. The new package enables simulation of quantum circuits involving statistical ensemble of mixed quantum states. Such analysis is of vital importance in dealing with quantum decoherence, measurements, noise and error correction, and fault tolerant computation. Several examples involving mixed state quantum computation are presented to illustrate the use of this package. Program summaryProgram title: CUGatesDensity.mCatalogue identifier: AEPY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5368No. of bytes in distributed program, including test data, etc.: 143994Distribution format: tar.gzProgramming language: Mathematica.Computer: Any computer installed with a copy of Mathematica 6.0 or higher.Operating system: Any system with a copy of Mathematica 6.0 or higher installed.Classification: 4.15.Nature of problem:To simulate arbitrarily complex quantum circuits comprised of single/multiple qubit and qudit quantum gates with mixed state registers.Solution method:A density matrix representation for mixed states and a state vector representation for pure states are used. The construct is based on an irreducible form of matrix decomposition, which allows a highly efficient implementation of general controlled gates with multiple conditionals.Running time:The examples provided in the notebook CUGatesDensity.nb take approximately 30 s to run on a laptop PC.

Criteria for Separability of Multipartite Quantum System

We first present the Hamel base of the density operator space for multipartite quantum system, and thus establish a representation of density matrix. Moreover, according to the structure of the density matrix for multipartite quantum system, we present two necessary criteria for separability of multipartite quantum system of arbitrary dimensions.

Quantum correlations in different density-matrix representations of spin-1/2 open chain

We consider quantum correlations in a spin-1/2 open chain of $N$ nodes with the XY Hamiltonian using different bases for the density matrix representation and the initial state with a single polarized node. These bases of our choice are following: (i) the basis of eigenvectors of the fermion operators; this basis appears naturally through the Jordan-Wigner transformation (this representation of the density matrix is referred to as the $\beta$-representation), (ii) its Fourier representation ($c$-representation of the density matrix) and (iii) the basis of eigenvectors of the operators $I_{jz}$ (the $z$-projection of the $j$th spin, $j=1,...,N$). Although for the short chains (a few nodes) the qualitative behavior of the entanglement and the discord are very similar (the difference is quantitative), this is not valid for longer chains ($N\gtrsim 10$). In this case, there are qualitative and quantitative distinctions between the entanglement and the discord in all three cases. We underline three most important features: (i) the quantum discord is static in the $\beta$-representation, where the entanglement is identical to zero; (ii) in the $c$-representation, the concurrence may be non-zero only between the nearest neighbors (with a single exception), while the discord is nonzero between any two nodes; (iii) there is so-called "echo" in the evolution of the discord, which is not observed in the evolution of the concurrence. Using different bases, we may choose the preferable behavior of quantum correlations which allows a given quantum system to be more flexible in applications.