Arbitrary Operator Research Articles

A general automatic method for optimal construction of matrix product operators using bipartite graph theory.

Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.

Nanoparticle-based computing architecture for nanoparticle neural networks.

The lack of a scalable nanoparticle-based computing architecture severely limits the potential and use of nanoparticles for manipulating and processing information with molecular computing schemes. Inspired by the von Neumann architecture (VNA), in which multiple programs can be operated without restructuring the computer, we realized the nanoparticle-based VNA (NVNA) on a lipid chip for multiple executions of arbitrary molecular logic operations in the single chip without refabrication. In this system, nanoparticles on a lipid chip function as the hardware that features memory, processors, and output units, and DNA strands are used as the software to provide molecular instructions for the facile programming of logic circuits. NVNA enables a group of nanoparticles to form a feed-forward neural network, a perceptron, which implements functionally complete Boolean logic operations, and provides a programmable, resettable, scalable computing architecture and circuit board to form nanoparticle neural networks and make logical decisions.

On some inequalities for positive operators

In the present paper, our aim is to refine some operator inequalities for arbitrary operator means and positive maps. For example, If $$A,B\in {{\mathbb {B}}}({{\mathscr {H}}})$$ be two invertible positive operators with this condition that $$0< m^{'}\le B \le m < M\le A \le M^{'}$$ for some positive real numbers $$m,m^{'},M,M^{'},$$ $$\sigma _{1}$$ and $$\sigma _{2}$$ be two arbitrary means between geometric and arithmetic means and $$\Phi$$ be a positive linear map. Then $$\begin{aligned} \Phi ^{2}(A \sigma _{1} B)\le \left( \frac{ K(h^{'}) }{ \Theta _{\frac{1}{2}}(h)} \right) ^{2}\Phi ^{2}(A \sigma _{2} B), \end{aligned}$$ where $$K(h^{'})=\frac{(h^{'}+1)^{2}}{4h^{'}}$$ with $$h^{'}=\frac{m^{'}}{M^{'}}$$ is the Kantorovich constant and $$\Theta _{\frac{1}{2}}(h)=1+\frac{\sqrt{2}(h-1)^{2}}{4(h+1)^{\frac{3}{2}}}$$ with $$h=\frac{m}{M}$$ is decreasing.

Strictly Linear Light Cones in Long-Range Interacting Systems of Arbitrary Dimensions

In locally interacting quantum many-body systems, the velocity of information propagation is finitely bounded and a linear light cone can be defined. Outside the light cone, the amount of information rapidly decays with distance. When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists. Herein, we consider generic long-range interacting systems with decaying interactions, such as $R^{-\alpha}$ with distance $R$. We prove the existence of the linear light cone for $\alpha>2D+1$ ($D$: the spatial dimension), where we obtain the Lieb--Robinson bound as $\|[O_i(t),O_j]\|\lesssim{t}^{2D+1}(R-\bar{v}t)^{-\alpha}$ with $\bar{v}=\mathcal{O}(1)$ for two arbitrary operators $O_i$ and $O_j$ separated by a distance $R$. Moreover, we provide an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for $\alpha<2D+1$. In the regime of $\alpha>2D+1$, our result characterizes the best general constraints on the information spreading.

Choi–Davis–Jensen’s type trace inequalities for convex functions of self-adjoint operators in Hilbert spaces

Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among $$\mathrm{tr}(\Phi (C)A)$$ and $$\mathrm{tr}(\Phi (C)A^{-1})$$ . New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Holder type trace inequality for any arbitrary operator mean.

Approximation properties of multipoint boundary-value problems

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable functions $y:[a,b]\to\mathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $\mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.

An Extension of The First Eigen-type Ambarzumyan theorem

An extension of the first eigenvalue-type Ambarzumyan's theorem are provided for the arbitrary self-adjoint Sturm-Liouville differential operators. The result makes a contribution to the Poschel-Trubowitz inverse spectral theory as well.

Agile frequency transformations for dense wavelength-multiplexed communications.

The broad bandwidth and spectral efficiency of photonics has facilitated unparalleled speeds in long-distance lightwave communication. Yet efficient routing and control of photonic information without optical-to-electrical conversion remains an ongoing research challenge. Here, we demonstrate a practical approach for dynamically transforming the carrier frequencies of dense wavelength-division-multiplexed data. Combining phase modulators and pulse shapers into an all-optical frequency processor, we realize both cyclic channel hopping and 1-to-N broadcasting of input data streams for systems with N = 2 and N = 3 users. Our method involves no optical-to-electrical conversion and enables low-noise, reconfigurable routing of fiber-optic signals with in principle arbitrary wavelength operations in a single platform, offering new potential for low-latency all-optical networking.

Alternative scheme of universal optical programmable multi-qubit gates for polarization qubits

We propose an alternative scheme for programmable quantum gates specifically designed for polarization qubits of single photons. Unlike the conventional approach based on the scheme of Reck et al., we adopt the Hilbert-space expansion technique to enable a novel feature in the design of arbitrary quantum operations. We present a scheme that independently programs each matrix element of an operator. The proposed scheme can support the realization of many kinds of quantum operations. In anticipation of advances in the current quantum photonics technology, we assume the manipulation and detection of many photon qubits and propose a theoretical scheme to achieve n-polarization-qubit optical reconfigurable quantum gates by linear optical elements.

Convergence in Measure and τ-Compactness of τ-Measurable Operators, Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

Lithium Plating: How Mathematics Can Help to Identify Fast-Charging Limits

With their growing importance in the market of electric vehicles and energy storage, a more comprehensive evaluation of Li-ion batteries behavior has become necessary for the power sources industry. Mathematical modelling and numerical simulation have become standard techniques in Li-ion battery research and development, with the purpose of studying the issues of batteries, including performance and ageing, and consequently increasing the predictability of model-based life expectancy. The efficient and fast charging of Li-ion batteries remains at the moment one of the most delicate challenges for the automotive and energy industries, being seriously affected by the formation of lithium metal on the surface of the anode during charge. This degradation process is called lithium plating and happens to be very damaging for the mechanical and chemical integrity of the battery, which not only sees its capacity lowered but could also incur serious damage and the risk of explosion. It is very difficult to detect lithium plating in situ without a direct observation of the open cell, but it is possible to deduce its presence by analyzing the cell behavior during cycles of charge/discharge in critical conditions and detecting some peculiarities which have been shown to indicate plating. The most common hints are a voltage plateau due to lithium oxidation during discharge at constant temperature and a voltage drop due to re-intercalation of metallic lithium during heating of the cell. On the other hand, the absence of any evidence of changes in voltage should not be considered as proof of evidence of a complete absence of lithium plating.Following our development of a comprehensive modelling and simulation framework for a commercial 0.35 Ah high-power lithium-ion pouch cell with LCO/NCA blend cathode1, here we present an extended pseudo-3D (P3D) model in which a lithium plating reaction has been integrated and parameterized. An extended bibliographic research helped us in choosing suitable parameters for modelling the plating kinetics. The model is now able to describe and predict both the equilibrium potentials and the non-equilibrium kinetics of the competing intercalation and plating reactions for arbitrary macroscopic operating conditions (C-rate, temperature, SOC). A relatively simple and common way to assess plating risk with P2D models is to compare the simulated local anode potential ΔΦ an with the thermodynamic plating condition of ΔΦ Li eq = 0 V, but this approach shows several pitfalls that have not been well discussed in literature. These issues have been included and discussed in detail in the present model, which has carefully been built to consider the effects of temperature, pressure and ion concentration on the thermodynamics and kinetics of the plating reaction. Consequently, this model also allows the creation of operation maps and an accurate spatiotemporal analysis of the competing reactions and lithium plating formation at the electrode-pair scale (1D, mesoscale) and intraparticle scale (1D, microscale) over a wide range of conditions. The governing equations for this model are implemented in the in-house multiphysics software package DENIS. The electrochemistry model is based on the use of the open-source chemical kinetics code CANTERA, enabling the thermodynamically consistent description of the main and side reactions, which is coupled to the DENIS transport model via the chemistry source terms.To validate our extended model, we also simulated and successfully reproduced the experimental data from Ecker et al. 2 for 1C charge-discharge at different temperatures. The unavoidable differences are due to the different size and characteristics of Ecker’s battery, measured experimentally (40 Ah high-power lithium-ion pouch cell with NMC cathode) and our virtual battery (0.35 Ah high-power lithium-ion pouch cell with LCO/NCA blend cathode).

Sorting of Fully Homomorphic Encrypted Cloud Data: Can Partitioning be Effective?

The challenge of maintaining confidentiality of stored data in cloud is of utmost importance to realize the potential of cloud computing as an emerging storage solution service. Storing data in encrypted form may solve the problem, but exposes data to an adversary for each required computation. This repeated encryption decryption also diminishes the essence of cloud for storing encrypted database and huge computation power of cloud remains unused. Fully homomorphic encryption (FHE) is an effective scheme to support arbitrary operations directly on encrypted data, but has serious performance issues. In this paper, we have considered sorting on encrypted data, which is a frequently required database operation. We have investigated the feasibility of performing comparison as well as partition based sort on CPA resistant FHE data and highlight an important observation that time requirement of partition based sort on FHE data is no better than comparison based sort owing to the underlying security of the cryptosystem. We identify the recrypt operation, which is the denoising step of FHE as the main reason of costly timing requirement of such operations. We propose a FHE specific two stage sorting technique termed as $\sf {Lazy sort}$ Lazy sort with reduced recrypt operation, which proves to be better in terms of performance on FHE data in comparison to partition as well as comparison sort. Finally, we provide some multi-core implementation results to show that with proper implementation tricks performance of FHE computations can be improved further.

Universal programmable photonic architecture for quantum information processing

We present a photonic integrated circuit architecture for a quantum programmable gate array (QPGA) capable of preparing arbitrary quantum states and operators. The architecture consists of a lattice of phase-modulated Mach-Zehnder interferometers, which perform rotations on path-encoded photonic qubits, and embedded quantum emitters, which use a two-photon scattering process to implement a deterministic controlled-$\sigma_z$ operation between adjacent qubits. By appropriately setting phase shifts within the lattice, the device can be programmed to implement any quantum circuit without hardware modifications. We provide algorithms for exactly preparing arbitrary quantum states and operators on the device and we show that gradient-based optimization can train a simulated QPGA to automatically implement highly compact approximations to important quantum circuits with near-unity fidelity.

Aggregate in my way: Privacy-preserving data aggregation without trusted authority in ICN

Information-Centric Networking (ICN) is a novel future network architecture which in contrast to IP-based networks relies on content and its name. It separates the physical location of data from the discovery and forwarding process and solely relies on the content itself. For the Internet of Thing (IoT) networks, stripping the location information may provide privacy, but this does not translate to operational privacy. Attackers can infer user behavior patterns through operational privacy, by eavesdropping on meaningful information. The content name and designed signature in ICN can associate content with the identity of the provider, even if the IP address is hidden. Although several research efforts focus on the privacy protection of ICN, they do not consider the privacy implications of data aggregation. Most existing privacy-preserving data aggregation protocols are designed for special operations, such as sum, average, variance, max or min, and they cannot support arbitrary aggregation operations in the ciphertexts domain. Besides, the need for trusted authority (TA) restricts the use of existing protocols in the real world. In this paper, we propose a practical and privacy-preserving data aggregation scheme that can compute arbitrary aggregation functions without a TA. On one hand, our scheme can ensure users’ anonymity and privacy protection, while on the other, the scheme is efficient in enabling participants to join or leave the system dynamically. Security analysis shows that the proposed scheme can achieve the desired security properties, while experimental results demonstrate its effectiveness and efficiency.

Engineering cross resonance interaction in multi-modal quantum circuits

The existing scalable superconducting quantum processors have only nearest-neighbor coupling. This leads to a reduced circuit depth, requiring a large series of gates to perform an arbitrary unitary operation in such systems. Recently, multi-modal devices have been demonstrated as a promising candidate for small quantum processor units. Always on longitudinal coupling in such circuits leads to implementation of native high fidelity multi-qubit gates. We propose an architecture using such devices as building blocks for a highly connected larger quantum circuit. To demonstrate a quantum operation between such blocks, a standard transmon is coupled to the multi-modal circuit using a 3D bus cavity giving rise to small exchange interaction between the transmon and one of the modes. We study the cross-resonance interaction in such systems and characterize the entangling operation and the unitary imperfections and crosstalk as a function of device parameters. Finally, we tune up the cross-resonance drive to implement multi-qubit gates in this architecture.

Uniqueness results for higher order Lane-Emden systems

In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet’s: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier’s boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

Sensitivity-driven adaptive sparse stochastic approximations in plasma microinstability analysis

Quantifying uncertainty in predictive simulations for real-world problems is of paramount importance – and far from trivial, mainly due to the large number of stochastic parameters and significant computational requirements. Adaptive sparse grid approximations are an established approach to overcome these challenges. However, standard adaptivity is based on global information, thus properties such as lower intrinsic stochastic dimensionality or anisotropic coupling of the input directions, which are common in practical applications, are not fully exploited. We propose a novel structure-exploiting dimension-adaptive sparse grid approximation methodology using Sobol' decompositions in each subspace to introduce a sensitivity scoring system to drive the adaptive process. By employing sensitivity information, we explore and exploit the anisotropic coupling of the stochastic inputs as well as the lower intrinsic stochastic dimensionality. The proposed approach is generic, i.e., it can be formulated in terms of arbitrary approximation operators and point sets. In particular, we consider sparse grid interpolation and pseudo-spectral projection constructed on (L)-Leja sequences. The power and usefulness of the proposed method is demonstrated by applying it to the analysis of gyrokinetic microinstabilities in fusion plasmas, one of the key scientific problems in plasma physics and fusion research. In this context, it is shown that a 12D parameter space can be scanned very efficiently, gaining more than an order of magnitude in computational cost over the standard adaptive approach. Moreover, it allows for the uncertainty propagation and sensitivity analysis in higher-dimensional plasma microturbulence problems, which would be almost impossible to tackle with standard screening approaches.

Quasi-distributions for arbitrary non-commuting operators

We present a new approach for obtaining quantum quasi-probability distributions, P(α,β), for two arbitrary operators, a and b, where α and β are the corresponding c-variables. We show that the quantum expectation value of an arbitrary operator can always be expressed as a phase space integral over α and β, where the integrand is a product of two terms: One dependent only on the quantum state, and the other only on the operator. In this formulation, the concepts of quasi-probability and correspondence rule arise naturally in that simultaneously with the derivation of the quasi-distribution, one obtains the generalization of the concept of correspondence rule for arbitrary operators.

Operations and poly-operations in algebraic cobordism

In the case of a field of characteristic zero, we describe all operations (including non-additive ones) from a theory A⁎ obtained from Algebraic Cobordism Ω⁎ of M. Levine-F. Morel by change of coefficients to any oriented cohomology theory B⁎ (in the sense of Definition 2.1). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done in [24], as well as the topological one obtained by T. Kashiwabara - see [6]. The key new ingredients which permit us to treat the non-additive operations are: the use of poly-operations and the “Discrete Taylor expansion”. As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p - see [23], which permits to sharpen results on the structure of Algebraic Cobordism - see [22]. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the case of multiplicative operations proved by I. Panin in [13].

Reinforcement learning for semi-autonomous approximate quantum eigensolver

The characterization of an operator by its eigenvectors and eigenvalues allows us to know its action over any quantum state. Here, we propose a protocol to obtain an approximation of the eigenvectors of an arbitrary Hermitian quantum operator. This protocol is based on measurement and feedback processes, which characterize a reinforcement learning protocol. Our proposal is composed of two systems, a black box named environment and a quantum state named agent. The role of the environment is to change any quantum state by a unitary matrix where is a Hermitian operator, and τ is a real parameter. The agent is a quantum state which adapts to some eigenvector of by repeated interactions with the environment, feedback process, and semi-random rotations. With this proposal, we can obtain an approximation of the eigenvectors of a random qubit operator with average fidelity over 90% in less than 10 iterations, and surpass 98% in less than 300 iterations. Moreover, for the two-qubit cases, the four eigenvectors are obtained with fidelities above 89% in 8000 iterations for a random operator, and fidelities of 99% for an operator with the Bell states as eigenvectors. This protocol can be useful to implement semi-autonomous quantum devices which should be capable of extracting information and deciding with minimal resources and without human intervention.