High-dimensional Hilbert Space Research Articles

Quantum structured light in high dimensions

Structured light has become topical of late, where controlling light in all its degrees of freedom has offered novel states of light long predicted, enhanced functionality in applications, and a modern toolbox for probing fundamental science. Structuring light as single photons and entangled states allows the spatial modes of light to be used to encode a large alphabet, accessing high dimensional Hilbert spaces for fundamental tests of quantum mechanics and improved quantum information processing tasks. In this tutorial, we outline the basic concepts of high dimensional quantum states expressed in a basis of spatial modes (structured light) and explain how to create, control, and detect such quantum states in the laboratory with a focus on transverse spatial modes such as the orbital angular momentum and pixel (position) modes. Finally, we highlight some example applications of such quantum structured light, from communications to imaging.

Native qudit entanglement in a trapped ion quantum processor

Quantum information carriers, just like most physical systems, naturally occupy high-dimensional Hilbert spaces. Instead of restricting them to a two-level subspace, these high-dimensional (qudit) quantum systems are emerging as a powerful resource for the next generation of quantum processors. Yet harnessing the potential of these systems requires efficient ways of generating the desired interaction between them. Here, we experimentally demonstrate an implementation of a native two-qudit entangling gate up to dimension 5 in a trapped-ion system. This is achieved by generalizing a recently proposed light-shift gate mechanism to generate genuine qudit entanglement in a single application of the gate. The gate seamlessly adapts to the local dimension of the system with a calibration overhead that is independent of the dimension.

Parametric t-stochastic neighbor embedding with quantum neural network

t-stochastic neighbor embedding (t-SNE) is a nonparametric data visualization method in classical machine learning. It maps the data from the high-dimensional space into a low-dimensional space, especially a two-dimensional plane, while maintaining the relationship or similarities between the surrounding points. In t-SNE, the initial position of the low-dimensional data is randomly determined, and the visualization is achieved by moving the low-dimensional data to minimize a cost function. Its variant called parametric t-SNE uses neural networks for this mapping. In this paper, we propose to use quantum neural networks for parametric t-SNE to reflect the characteristics of high-dimensional quantum data on low-dimensional data. We use fidelity-based metrics instead of Euclidean distance in calculating high-dimensional data similarity. To verify our proposed method, we visualize both classical (Iris dataset) and quantum (time-depending Hamiltonian dynamics) data for classification tasks. Since this method allows us to represent a quantum dataset in a higher dimensional Hilbert space by a quantum dataset in a lower dimension while keeping their similarity, the proposed method can also be used to compress quantum data for further quantum machine learning.

Imaging of Single-Photon Orbital-Angular-Momentum Bell States

Bell states are the most fundamental resource in realizing quantum information tasks and have a very unique position in quantum mechanics. Utilizing orbital angular momentum (OAM) to encode single-photon Bell states enables the realization of high-dimensional Hilbert space, which is crucial for the field of quantum information. In this paper, we design a single-photon OAM Bell-state evolution device based on the Sagnac interferometer, which can make one-to-one correspondence between the input Bell states and the output states. Moreover, we also develop a single-photon single-pixel imaging (SPI) technique, which improves spatial resolution and decreases acquisition time in the meantime, to capture interference images of output states. The results show that single-photon OAM Bell states can be fully recognized by comparing the differences of interference images. We innovatively utilize the SPI technique for single-photon OAM Bell-state recognition. This indicates that the SPI technique effectively promotes the study of quantum information based on OAM, while the quantum information based on OAM provides a clear application scenario for the SPI technique.

Entanglement Verification of Hyperentangled Photon Pairs

We experimentally investigate the properties of hyperentangled states displaying simultaneous entanglement in multiple degrees of freedom, and find that Bell tests beyond the standard Clauser, Horne, Shimony, Holt inequality can reveal a higher-dimensional nature in a device-independent way. Specifically, we show that hyperentangled states possess more than just simultaneous entanglement in separate degrees of freedom but also entanglement in a higher dimensional Hilbert space. We also verify the steerability of hyperentangled quantum states by steering different photonic degrees of freedom.

PQKLP: Projected Quantum Kernel based Link Prediction in Dynamic Networks

Link prediction in dynamic networks finds new or future links based on the previously seen structure of the network. Its study is crucial to comprehending network evolution and its effects on individual nodes. Accuracy and efficiency of link prediction on dynamic networks are the two aspects research. We present Projected Quantum Kernel-based Link Prediction (PQKLP), a quantum-enhanced feature-based framework for solving link prediction problems in dynamic networks. According to our study, the Projected Quantum Kernel has not been utilized in the field of link prediction. Thus, we propose this method that combines the disciplines of social networks and quantum computing. We employed high-dimensional Hilbert spaces to enhance the prediction data in this model, which otherwise we only have access to via inner products provided by measurements. Such enhancement leads to better prediction results from machine learning-based link prediction techniques. We trained six classical machine learning models and their quantum-enhanced counterparts based on the enhanced features generated by the Projected Quantum Kernel (PQK) technique. The proposed model outperforms traditional link prediction methods, classical machine learning approaches, and current state-of-the-art methods on five well-known dynamic network datasets, as per the results of four performance metrics.

A universal qudit quantum processor with trapped ions

Most quantum computers use binary encoding to store information in qubits—the quantum analogue of classical bits. Yet, the underlying physical hardware consists of information carriers that are not necessarily binary, but typically exhibit a rich multilevel structure. Operating them as qubits artificially restricts their degrees of freedom to two energy levels1. Meanwhile, a wide range of applications—from quantum chemistry2 to quantum simulation3—would benefit from access to higher-dimensional Hilbert spaces, which qubit-based quantum computers can only emulate4. Here we demonstrate a universal quantum processor using trapped ions that act as qudits with a local Hilbert-space dimension of up to seven. With a performance similar to qubit quantum processors5, this approach enables the native simulation of high-dimensional quantum systems3, as well as more efficient implementation of qubit-based algorithms6,7. Qudits are generalizations of qubits that have more than two states, which gives them a performance advantage in some quantum algorithms. The operations needed for a universal qudit processor have now been demonstrated using trapped ions.

Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space

Informationally complete (IC) positive operator-valued measures (POVMs) are generalized quantum measurements that offer advantages over the standard computational basis readout of qubits. For instance, IC-POVMs enable efficient extraction of operator expectation values, a crucial step in many quantum algorithms. POVM measurements are typically implemented by coupling one additional ancilla qubit to each logical qubit, thus imposing high demands on the device size and connectivity. Here, we show how to implement a general class of IC-POVMs without ancilla qubits. We exploit the higher-dimensional Hilbert space of a qudit in which qubits are often encoded. POVMs can then be realized by coupling each qubit to two of the available qudit states, followed by a projective measurement. We develop the required control pulse sequences and numerically establish their feasibility for superconducting transmon qubits through pulse-level simulations. Finally, we present an experimental demonstration of a qudit-space POVM measurement on IBM Quantum hardware. This paves the way to making POVM measurements broadly available to quantum computing applications.

A nonlinear correlation measure with applications to gene expression data.

Nonlinear correlation exists in many types of biomedical data. Several types of pairwise gene expression in humans and other organisms show nonlinear correlation across time, e.g., genes involved in human T helper (Th17) cells differentiation, which motivated this study. The proposed procedure, called Kernelized correlation (Kc), first transforms nonlinear data on the plane via a function (kernel, usually nonlinear) to a high-dimensional (Hilbert) space. Next, we plug the transformed data into a classical correlation coefficient, e.g., Pearson's correlation coefficient (r), to yield a nonlinear correlation measure. The algorithm to compute Kc is developed and the R code is provided online. In three simulated nonlinear cases, when noise in data is moderate, Kc with the RBF kernel (Kc-RBF) outperforms Pearson's r and the well-known distance correlation (dCor). However, when noise in data is low, Pearson's r and dCor perform slightly better than (equivalently to) Kc-RBF in Case 1 and 3 (in Case 2); Kendall's tau performs worse than the aforementioned measures in all cases. In Application 1 to discover genes involved in the early Th17 cell differentiation, Kc is shown to detect the nonlinear correlations of four genes with IL17A (a known marker gene), while dCor detects nonlinear correlations of two pairs, and DESeq fails in all these pairs. Next, Kc outperforms Pearson's and dCor, in estimating the nonlinear correlation of negatively correlated gene pairs in yeast cell cycle regulation. In conclusion, Kc is a simple and competent procedure to measure pairwise nonlinear correlations.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N): Mathematical Framework

This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N), which enables the most efficient computation of exactly determined expressions of arbitrarily high-order sensitivities of generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in the model’s phase space. The mathematical framework underlying the nth-CASAM-N is proven to be correct by using mathematical induction. The nth-CASAM-N is formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems.

Multiple Riemannian Manifold-Valued Descriptors Based Image Set Classification With Multi-Kernel Metric Learning

The importance of wild video based image set recognition is monotonically increasing due to the large amount of video data being collected by various devices including surveillance cameras, drive recorders, smart phones, and internet. The content of these videos is often complex, and it raises the question of how to perform image set modeling and feature extraction for image set-based classification. In recent years, image set classification methods have advanced considerably by modeling the image set in terms of a covariance matrix, linear subspace, or Gaussian distribution. Moreover, the distinctive geometry spanned by them include Symmetric Positive Definite (SPD) manifold, Grassmannian manifold, and Gaussian embedded Riemannian manifold, respectively. As a matter of fact, most of the approaches just adopt a single geometric model to describe each given image set, which may lose information useful for classification. To tackle this problem, we propose a novel algorithm to model each image set from a multi-geometric perspective. Specifically, the covariance matrix, linear subspace, and Gaussian distribution are applied to set representation simultaneously. In order to fuse these multiple heterogeneous Riemannian manifold-valued features, the well-equipped Riemannian kernel functions are first employed to map them into high dimensional Hilbert spaces. Then, a multi-kernel metric learning framework is devised to embed the learned hybrid kernels into a common lower dimensional subspace to facilitate classification. We conduct experiments on six widely used datasets each representing a different classification task: video-based face recognition, set-based object categorization, video-based emotion recognition, dynamic scene classification, set-based cell identification, and 3D hand pose estimation, to evaluate the classification performance of the proposed algorithm. The extensive experimental results confirm its superiority over the state-of-the-art methods.

Majorana’s stellar representation for the quantum geometric tensor of symmetric states

Majorana’s stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana’s stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana’s stars.

Disorder averaging and its UV discontents

We present a stringy realization of quantum field theory ensembles in $D\ensuremath{\le}4$ spacetime dimensions, thus realizing a disorder averaging over coupling constants. When each member of the ensemble is a conformal field theory with a standard semiclassical holographic dual of the same radius, the resulting bulk can be interpreted as a single asymptotically anti--de Sitter space geometry with a distribution of boundary components joined by wormhole configurations, as dictated by the Hartle--Hawking wave function. This provides a UV completion of a recent proposal by Marolf and Maxfield that there is a high-dimensional Hilbert space for baby universes, but one that is compatible with the proposed swampland constraints of McNamara and Vafa. This is possible because our construction is really an approximation that breaks down both at short distances, but also at low energies for objects with a large number of microstates. The construction thus provides an explicit set of counterexamples to various claims in the literature that holographic and effective field theory considerations can be reliably developed without reference to any UV completion.

Higher-order topological biphoton corner states in two-dimensional photonic lattices

Higher-order topological insulators have attracted great interest for their unconventional bulk-boundary correspondence especially gapless corner states. Here, we realize topological biphoton corner states in two-dimensional lattices. In addition to the striking state localization, the correlations in topologically nontrivial cases are more than 60 times higher than those in trivial cases and the visibility of Hong-Ou-Mandel interference for output topological corner-entangled states is higher than 90.3%. We confirm that the second-order photonic topological insulators support stable biphoton states that can maintain spatial distributions in high-dimensional Hilbert space, which have potential applications in quantum algorithms, simulation, and computation.

Creation of high-dimensional entanglement of polar molecules via optimal control fields

Quantum entanglement in high-dimensional Hilbert space offers broad prospects for quantum information science. Polar molecules have a rich internal structure and long coherence time, serving as an attractive candidate for quantum information processing. In this paper, we propose a theoretical scheme for creating high-dimensional entangled states based on ultracold SrO molecules. Qudits are encoded in the pendular states induced by an external electric field and coupled by the dipole-dipole interaction. With the assistance of optimal control theory, a series of optimal microwave fields is designed for the generation of an entangled qutrit-qubit state, qutrit-qutrit state, and ququart-ququart state through numerical iterations. We detail the relation of the fidelity and entanglement of the systemic final states with the number of iteration steps after the control fields are applied and analyze the population dynamics of the field-driven wave functions during time evolution. Moreover, three coupled SrO molecules arranged in an equilateral triangle configuration are employed as pendular qubits, and the trimolecular $W$ and Greenberger-Horne-Zeilinger states are realized with high fidelities by optimal control. The specific molecule and experimental parameters used in our theoretical studies are chosen for computational convenience, but we include a discussion of how our results can be applied to realistic situations as well. In principle, our results could be extended to a situation with more pendular qudits, providing a significant step toward the achievement of high-dimensional quantum information processing with arrays of polar molecules.

Polarization-path-frequency entanglement using interferometry and frequency shifters

Higher dimensional Hilbert space along with ability to control multiple degrees of freedom of photon and entangle them has enabled new quantum protocols for various quantum information processing applications. Here, we propose a scheme to generate and control polarization-path-frequency entanglement using the operative elements required to implement a polarization-controlled quantum walk in the path (position) space and frequency domain. Hyperentangled states manifests in the controlled dynamics using an interferometric setup where half-wave plates, beam-splitters and frequency shifters such as those based on the electro-optic effect are used to manipulate the polarization, path and frequency degrees of freedom respectively. The emphasis is on utilizing the polarization to influence the movement to a specific value in the frequency and position space. Negativity between the subspaces is calculated to demonstrate the controllability of the entanglement between the three degrees of freedom and the effect of noise on the entanglement is modelled using the depolarizing channel. Progress reported with experimental demonstration of realization of quantum walk using quantum states of light makes quantum walks a practical approach to generate hyperentangled states.

Maximum margin Riemannian manifold-based hyperdisk for fault diagnosis of roller bearing with multi-channel fusion covariance matrix

For rotating machinery, the sudden failure of roller bearing would lead to the downtime of the whole system and even catastrophic accidents. Therefore, multiple accelerometers are usually arranged to comprehensively evaluate the health of roller bearing, enhancing the stability and reliability of monitoring results. This paper proposes a novel fault diagnosis framework by utilizing a multi-channel fusion covariance matrix (MFCM) and Riemannian manifold-based hyperdisk. First, 22 statistical features are acquired from each channel data. Then, MFCM is calculated as the fault feature representation of roller bearing to achieve multi-channel feature fusion, where the element of MFCM represents the correlation information between different channels. Finally, since MFCM is a symmetric positive definite (SPD) matrix, lying on a Riemannian manifold, we design a maximum margin Riemannian manifold-based hyperdisk (MMRMHD) classifier to conduct fault classification, where Log-Euclidean metric (LEM) is introduced to calibrate the distribution of MFCMs. Moreover, to further improve the classification ability of nonlinear SPD data, we map MFCMs into a high-dimensional Hilbert space with the LEM-based kernel function and construct a novel kernelized MMRMHD model. The experimental results on two bearing datasets with multi-channel vibration signals demonstrate the effectiveness and superiority of the proposed fault diagnosis framework.

Infrared Small Object Tracking Based on Att-Siam Network

Aiming at the practical problems such as background clutter interference and occlusion during the tracking of small air targets by infrared imaging guided missiles, we propose an improved SiamFC infrared small target tracking algorithm. By mapping the original spatial data to a high-dimensional Hilbert space, the nonlinear mapping is implicit in the linear learner. This method has the advantages of learnability, efficient calculation, linearizability, and good generalization performance. The target tracking problem provides a new and effective approach with the actual needs of the application. We propose a Siamese web tracker (Att-Siam). Att-Siam fuses convolutional channel attention mechanism, stacked channel attention mechanism and spatial attention mechanism to improve tracking performance. Feature extraction of target objects is enhanced by fusing the channel attention mechanism and two convolution blocks at low convolutional layers. Compared with the benchmark algorithm SiamFC, the improved algorithm is tested on the infrared air weak and small target data set, and the tracking success rate and accuracy are increased by 32.3% and 20.9% respectively. The experimental results show that the proposed algorithm can adapt to complex and diverse infrared air scenes, and achieve effective and stable real-time tracking of small infrared air targets.

Spacetime Path Integrals for Entangled States

Although the path-integral formalism is known to be equivalent to conventional quantum mechanics, it is not generally obvious how to implement path-based calculations for multi-qubit entangled states. Whether one takes the formal view of entangled states as entities in a high-dimensional Hilbert space, or the intuitive view of these states as a connection between distant spatial configurations, it may not even be obvious that a path-based calculation can be achieved using only paths in ordinary space and time. Previous work has shown how to do this for certain special states; this paper extends those results to all pure two-qubit states, where each qubit can be measured in an arbitrary basis. Certain three-qubit states are also developed, and path integrals again reproduce the usual correlations. These results should allow for a substantial amount of conventional quantum analysis to be translated over into a path-integral perspective, simplifying certain calculations, and more generally informing research in quantum foundations.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): I. Mathematical Framework

This work presents the mathematical framework of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “nth-CASAM-L”), which is conceived for obtaining the exact expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Since many of the most important responses for linear systems involve the solutions of both the forward and the adjoint linear models that correspond to the respective physical system, the sensitivity analysis of such responses makes it necessary to treat linear systems in their own right, rather than treating them as particular cases of nonlinear systems. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward functions, since nonlinear operators do not admit bona-fide adjoint operators (only a linearized form of a nonlinear operator admits an adjoint operator). The nth-CASAM-L determines the exact expression of arbitrarily-high order sensitivities of responses to the parameters underlying both the forward and adjoint models of a nonlinear system, thus enable the most efficient and accurate computation of such sensitivities. The mathematical framework underlying the nth-CASAM is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods, thus providing the basis for overcoming the “curse of dimensionality” in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. In particular, for a scalar-valued valued response associated with a nonlinear model comprising TP parameters, the 1st-−CASAM-L requires 1 additional large-scale adjoint computation (as opposed to TP large-scale computations, as required by other methods) for computing exactly all of the 1st-−order response sensitivities. All of the (mixed) 2nd-order sensitivities are computed exactly by the 2nd-CASAM-L in at most TP computations, as opposed to TP(TP + 1)/2 computations required by all other methods, and so on. For every lower-order sensitivity of interest, the nth-CASAM-L computes the “TP next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space. Very importantly, the nth-CASAM-L computes the higher-level adjoint functions using the same forward and adjoint solvers (i.e., computer codes) as used for solving the original forward and adjoint systems, thus requiring relatively minor additional software development for computing the various-order sensitivities.