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ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.

This work considers the cooperative sequential hypothesis testing problem in a distributed network with quantized communication channels. The sensors observe independent sequences of samples and in the meantime, exchange their local information in the form of quantized statistics at every sampling interval. The communication links are represented as an undirected graph. In this distributed setup, every sensor performs its own sequential test based on the local samples and the messages from the neighbour sensors. Our goal is to devise the distributed sequential test that comprises the quantization scheme, the message-exchange protocol and the test procedure such that every sensor in the network fully exploits the network diversity and achieves the (asymptotically) optimal performance in terms of the stopping time. In particular, two distributed sequential tests are proposed based on different quantization schemes and a quantized message-exchange protocol that satisfies certain conditions. The first quantization scheme uniformly quantizes the local statistic at each sensor and at every sampling interval; the second one hinges on a modified level-triggered quantization technique, and resembles the Lebesgue sampling of the running local statistic. Our analyses show that the uniform quantization based distributed sequential test yields sub-optimal performance, while the one based on level-triggered quantization achieves the order-2 asymptotically optimal performance at every sensor for any fixed quantization step-size. Furthermore, we generalize the proposed sequential tests to the cluster-based network. Numerical results are provided to corroborate our analyses and demonstrate the effectiveness of the proposed sequential tests.

We have seen so far that particles can be represented by waves. The process by which particles are described as waves is called first quantization. Why is this called quantization? A “quantum” can be defined as a minimal amount of a physical entity involved in an interaction. In physics, the idea that physical entities may be quantized (discrete rather than continuous) is called the “quantization hypothesis”.

Quantum simulations of chemistry in first quantization offer important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside the Born-Oppenheimer approximation. However, as all prior work on quantum simulation in first quantization has been limited to asymptotic analysis, it has been impossible to compare the resources required for these approaches to those for more commonly studied algorithms in second quantization. Here, we analyze and optimize the resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al that realize block encodings for the qubitization and interaction picture frameworks of Low et al. The two algorithms we study enable simulation with gate complexities $\tilde{\cal O}(\eta^{8/3}N^{1/3}t+\eta^{4/3}N^{2/3}t)$ and $\tilde{\cal O}(\eta^{8/3} N^{1/3} t)$ where $\eta$ is the number of electrons, $N$ is the number of plane wave basis functions, and $t$ is the duration of time-evolution ($t$ is inverse to target precision when the goal is to estimate energies). In addition to providing the first explicit circuits and constant factors for any first quantized simulation and introducing improvements which reduce circuit complexity by about a thousandfold over naive implementations for modest sized systems, we also describe new algorithms that asymptotically achieve the same scaling in a real space representation. We assess the resources required to simulate various molecules and materials and conclude that the qubitized algorithm will often be more practical than the interaction picture algorithm. We demonstrate that our qubitized algorithm often requires much less surface code spacetime volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals., Comment: 96 pages, 9 figures, 8 tables

In this paper, we propose a novel robust visual classification framework that uses double quantization (dquant) to defend against adversarial examples in a specific attack scenario called “subsequent adversarial examples” where test images are injected with adversarial noise. The proposed system can remove the adversarial noise completely on this particular attack scenario. First, we analyze the potential sources of adversarial noise and classify adversarial examples into three classes. We then propose a novel effective solution, dquant, to target a specific class of adversarial examples. The first quantization is 1-bit dithering applied to both training and test images. The second one is linear quantization, which is applied to test images just before being inputted to a model to remove any adversarial noise. The linear quantizer guarantees that original 1-bit test images will be restored regardless of adversarial noise distance, and, therefore, dquant maintains identical accuracy whether or not the model is under attack. The results show that dquant achieves comparable accuracy, 85.28% on the CIFAR-10 and 94.99% on the Oxford-IIIT Pet datasets against three state-of-the-art adversaries with even a previously untested maximum adversarial distance of 64.

In this paper, we propose a novel distributed digital transmission framework for two jointly sparse correlated signals. First, the non-zero coefficients of each signal are quantized by a standard quantizer or a novel distributed quantizer, as appropriate. Then, these quantized values are mapped to the elements of a finite field, except 0, while the zero coefficients are mapped to 0. Subsequently, compressed sensing over finite fields is applied to obtain measurements. We name such an order first quantization then compressed sensing. The two measurement signals are then converted to bit sequences, modulated, and transmitted through separate additive white Gaussian noise (AWGN) channels. At the central receiver, which has access to both channels, following demodulation, an innovative joint belief propagation (JBP) algorithm is performed for joint recovery. In this algorithm, we introduce a new type of constraint node, i.e., correlation constraint nodes, which connect two factor graphs that separately represent the CS encoding matrix of each signal. The experimental results prove that under the same framework the proposed scheme provides significant performance improvements compared to the scheme that ignores the correlated information between jointly sparse signals, especially when the correlation coefficient is high. Then, to answer the question of which order is better, we construct the first compressed sensing, then quantization framework, for fairness, two cutting edge jointly greedy pursuit algorithms are separately adopted at the joint decoder. Through simulations, we validate that the proposed framework provides more effective and robust transmissions.

Recent exploration of the commensurate structure in the turbostratic double layer graphene shows that the large angle twisting can be treated by the decrease of the effective velocity within the energy spectra of the single layer graphene. Within our work, we use this result as a starting point, aiming towards understanding the physics of by a large angle twisted double layer graphene (i.e. Moire) quantum dot systems. We show that within this simple approach using the language of the first quantization, yet another so far unrealized (not up to our knowledge), illustrative property of the commutation relation appears in the graphene physics. Intriguingly, large twisting angles show to be a suitable tunning knob of the position symmetry in the graphene systems. Complete overview of the large angle twisting on the considered dot systems is provided., Comment: 10 pages, 7 figures

Multimedia Forensics experts work consists in providing answers about integrity of a specific media content and from where it comes from. Exploitation of any traces from JPEG double compressed images is often one of the main investigative path to be used for these purposes. Thus it is fundamental to have tools and algorithms able to safely estimate the first quantization matrix to further proceed with camera model identification and related tasks. In this paper, a technique based on extensive simulation is proposed, with the aim to infer the first quantization for a certain numbers of Discrete Cosine Transform (DCT) coefficients exploiting local image statistics without using any a-priori knowledge. The method provides also a reliable confidence value for the estimation which is of great importance for forensic purposes. Experimental results w.r.t. the state-of-the-art demonstrate the effectiveness of the proposed technique both in terms of precision and overall reliability.

Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality, derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics. Quantum mechanics being just a generalization theory of the complex square root of classical statistical mechanics is both new physics and revolutionary discovery, which are affecting people’s deep philosophical thinking for modern physics development, solve all the crisises of quantum mechanics, quantum information and so on, and make quantum mechanics have scientific solid bases being checked and both no basic axiom presumption and no all the quantum strange incomprehensible properties, because classical statistical mechanics and the complex square root of classical statistical mechanics have the scientific solid bases being checked. In addition, this paper discovers the reason no taking the time derivative of space coordinates in Schrȍdinger equation. Therefore, this paper gives solution to the crisis of the first quantization origin, and mainly deduces quantum physics no all the quantum current strange incomprehensible properties.

The Ghirardi-Rimini-Weber (GRW) model of wave function collapse was successful in giving a solution to the measurement problem in quantum mechanics. Despite this, it lacked two desirable features—one mathematical and one physical—for a collapse model. The mathematical aspect of the GRW model that one would like to improve is its lacking of a unified description in terms of an equation for the wave function. On the physical side, the GRW model uses the first quantization language, and does not preserve the symmetry properties of indentical particles wave functions [1]. In 1990 Ghirardi, Rimini and Pearle took the desired step forward devising the Continuos Spontaneous Localisations (CSL) model [2]. They considered collapses to be driven by continuous stochastic processes (instead of having discrete collapses like in GRW), that allowed to describe the collapse dynamics by a stochastic Schrodinger equation. Moreover, the CSL model relies on the second quantization formalism, and correctly describes ensembles of identical particles.

The exploitation of traces in JPEG double compressed images is of utter importance for investigations. Properly exploiting such insights, First Quantization Estimation (FQE) could be performed in order to obtain source camera model identification (CMI) and therefore reconstruct the history of a digital image. In this paper, a method able to estimate the first quantization factors for JPEG double compressed images is presented, employing a mixed statistical and Machine Learning approach. The presented solution is demonstrated to work without any a-priori assumptions about the quantization matrices. Experimental results and comparisons with the state-of-the-art show the goodness of the proposed technique.

Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality (i.e., complementary principle), derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics, which will make people renew thinking modern physics development. In addition, this paper discovers the reason that Schrȍdinger equation doesn’t takes the time derivative of space coordinates. Therefore, this paper gives solution to the Crisis of the first quantization origin.

In this chapter we discuss most drastic non-perturbative effects of interactions. Namely, mutual interactions may give rise to new quasi-particles which are neither fermions nor bosons. We begin with the best-known example of such quasi-particles with a fractional charge in two-dimensional electron systems. Then we proceed to a simpler case in one dimension, where exact solution including such exotic quasi-particles is available. The bosonic description discussed in the previous chapter is regarded as the long-wavelength limit in the more general framework for non-perturbative effects. We emphasize that the fractional charge does not contradict with indivisible nature of electron, since such quasi-particle emerges as a collective phenomenon out of a non-perturbative ground state. All the treatment in this chapter use the first quantization formalism.

This paper presents a novel method for first quantization matrix estimation for non-aligned double JPEG compressed images. The non-aligned double JPEG scenario poses several challenges in the form of recompression artifacts which needs to be handled effectively in order to arrive at accurate first quantization estimate. One such recompression artifact which is prominent in the non-aligned scenario is the blocking artifacts arising due to the misalignment of the DCT grids in the successive compression cycles. The proposed method investigates the impact of the blocking artifacts induced errors on the DCT histogram and counters these errors via a novel DCT histogram filtering strategy. In addition, the residual noise which is also a recompression artifact is countered utilizing the local rank transform which adaptively filters the residual noise effects. The filtered histograms are compared with a synthetically created ideal second quantization matrix to estimate the degree of similarity between the two cases. Utilizing the maximum degree of similarity, the selection of first quantization value is performed. Experimental analysis utilizing several non-aligned double compressed JPEG images taken from UCID and RAISE datasets with Quality Factor (QF) of second compression being greater than the first compression ( Q F 2 > Q F 1 ) along with the comparative analysis with one of the state-of-the-art method shows accurate estimation of the first quantization values for the proposed method. The proposed method finds its application in image forensics as well as in steganalysis.

Unifying quantum and classical physics has proved difficult as their postulates are conflicting. Using the notion of counts of the fundamental measures—length, mass, and time—a unifying description is resolved. A theoretical framework is presented in a set of postulates by which a conversion between expressions from quantum and classical physics can be made. Conversions of well-known expressions from different areas of physics (quantum physics, gravitation, optics and cosmology) exemplify the approach and mathematical procedures. The postulated integer counts of fundamental measures change our understanding of length, suggesting that our current understanding of reality is distorted.

Transcorrelated coupled cluster and distinguishable cluster methods are presented. The Hamiltonian is similarity transformed with a Jastrow factor in the first quantization, which results in up to three-body integrals. The coupled cluster with singles and doubles equations on this transformed Hamiltonian are formulated and implemented. It is demonstrated that the resulting methods have a superior basis set convergence and accuracy to the corresponding conventional and explicitly correlated methods. Additionally, approximations for three-body integrals are suggested and tested.

A framework to calculate CC2 approximated coupled-cluster ground state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library. The CC2 working equations are formulated in first quantization which makes them suitable for real-space methods. The first quantized equations can be interpreted diagrammatically using the usual diagrams from second quantization with adjusted interpretation rules. Singularities arising from the nuclear and electronic potentials are regularized by explicitly taking the nuclear and electronic cusps into account. The regularized three- and six-dimensional cluster functions are represented directly on an adaptive grid. The resulting equations are free of singularities and virtual orbitals, which results in a low intrinsic scaling. Correlation energies close to the basis set limit are computed for small molecules. This work is the first step toward CC2 excitation energies in a multiresolution basis.

In this article, Scattering states of Klein–Gordon equation for three scatter potentials of single and double Dirac delta and a potential well in the q-deformed formalism of relativistic quantum mechanics have been derived. At first, we discussed how q-deformed formalism can be constructed and used. Postulates of this q-deformed quantum mechanics are noted. Then scattering problems for spin-zero bosons are studied.

The estimation of the first quantization step in double JPEG compressed images is still a challenging problem, especially when the first quantization step q 1 is smaller than the second quantization step q 2 . In this paper, we present a novel method to estimate q 1 . By introducing the mean square error (MSE) sequence of ratios among DCT coefficient histogram bins, we formulate the relationship between its periodic fluctuation and q 1 . And in order to enhance the periodic effect, we propose a strategy to adjust the histogram. Then, based on MSE sequence, several q 1 candidates can be obtained. Finally by histogram comparison, the estimated quantization step is selected from the candidates. Experimental results demonstrate that the proposed approach has better overall performance when compared with state of the art methods.

I consider a quantum system that possesses key features of quantum shape dynamics and show that the evolution of wave-packets will become increasingly classical at late times and tend to evolve more and more like an expanding classical system. At early times however, semiclassical effects become large and lead to an exponential mismatch of the apparent scale as compared to the expected classical evolution of the scale degree of freedom. This quantum inflation of an emergent and effectively classical system, occurs naturally in the quantum shape dynamics description of the system, while it is unclear whether and how it might arise in a constrained Hamiltonian quantization.

We look at some strategies for solving the macro-object problem for wave function realism. This is the problem of finding an account of the existence of macroscopic objects assuming a metaphysics in which objects in space-time are not fundamental; rather what is fundamental is the quantum wave function, a field characterized by an assignment of values to points in a much different kind of space, one adequate to realizing the full range of possible quantum pure states.

We generalize relativistic quantum mechanics and the Standard Model of elementary particle physics by considering a finite limit for the smallest measurable length. The resulting theory of Space-Time Quantization is logically consistent and accounts for all possible particle states by means of four new quantum numbers. They specify possible variations of wave functions at the smallest possible scale in space and time, while states of motion are defined by their large-scale variations. This theory also provides insight into the nature and properties of dark matter particles. It can facilitate their detection and identification because of a very strict conservation law.

Quantization has been an effective technology in ANN (approximate nearest neighbour) search due to its high accuracy and fast search speed. To meet the requirement of different applications, there is always a trade-off between retrieval accuracy and speed, reflected by variable code lengths. However, to encode the dataset into different code lengths, existing methods need to train several models, where each model can only produce a specific code length. This incurs a considerable training time cost, and largely reduces the flexibility of quantization methods to be deployed in real applications. To address this issue, we propose a Deep Recurrent Quantization (DRQ) architecture which can generate sequential binary codes. To the end, when the model is trained, a sequence of binary codes can be generated and the code length can be easily controlled by adjusting the number of recurrent iterations. A shared codebook and a scalar factor is designed to be the learnable weights in the deep recurrent quantization block, and the whole framework can be trained in an end-to-end manner. As far as we know, this is the first quantization method that can be trained once and generate sequential binary codes. Experimental results on the benchmark datasets show that our model achieves comparable or even better performance compared with the state-of-the-art for image retrieval. But it requires significantly less number of parameters and training times. Our code is published online: this https URL.

Quantization has been an effective technology in ANN (approximate nearest neighbour) search due to its high accuracy and fast search speed. To meet the requirement of different applications, there is always a trade-off between retrieval accuracy and speed, reflected by variable code lengths. However, to encode the dataset into different code lengths, existing methods need to train several models, where each model can only produce a specific code length. This incurs a considerable training time cost, and largely reduces the flexibility of quantization methods to be deployed in real applications. To address this issue, we propose a Deep Recurrent Quantization (DRQ) architecture which can generate sequential binary codes. To the end, when the model is trained, a sequence of binary codes can be generated and the code length can be easily controlled by adjusting the number of recurrent iterations. A shared codebook and a scalar factor is designed to be the learnable weights in the deep recurrent quantization block, and the whole framework can be trained in an end-to-end manner. As far as we know, this is the first quantization method that can be trained once and generate sequential binary codes. Experimental results on the benchmark datasets show that our model achieves comparable or even better performance compared with the state-of-the-art for image retrieval. But it requires significantly less number of parameters and training times. Our code is published online: https://github.com/cfm-uestc/DRQ.

The worldline formalism is a useful scheme in quantum field theory which has also become a powerful tool for numerical computations. The key ingredient in this formalism is the first quantization of an auxiliary point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field which is confined within some boundaries one needs to restrict the path integration domain of the auxiliary point-particle to a specific subset of worldlines enclosed by those boundaries. We show how to implement this restriction for the case of a scalar field confined to the D-dimensional ball under Dirichlet and Neumann boundary conditions, and compute the first few heat-kernel coefficients as a verification of our construction. We argue that this approach could admit different generalizations., Facultad de Ciencias Exactas, Instituto de Física La Plata

We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode splitting proposed by Killoran et al. (2014)., 11 pages, 5 figures, Revtex 4.1; (v2) minor corrections, some references added; (v3) minor corrections and clarifications; (v4) Accepted version (PRA)

Relativistic particles with higher spin can be described in first quantization using actions with local supersymmetry on the worldline. First, we present a brief review of these actions and their use in first quantization. In a Dirac quantization scheme the field equations emerge as Dirac constraints on the Hilbert space, and we outline how they lead to the description of higher spin fields in terms of the more standard Fronsdal-Labastida equations. Then, we describe how these actions can be extended so that the propagating particle is allowed to take different values of the spin, i.e. carry a reducible representation of the Poincar\'e group. This way one may identify a four dimensional model that carries the same degrees of freedom of the minimal Vasiliev's interacting higher spin field theory. Extensions to massive particles and to propagation on (A)dS spaces are also briefly commented upon., Comment: 10 pages, updated references. To appear in Proceedings of Victor Villanueva Memorial Workshop

Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure., Comment: 16 pages, 11 figures

Nonlinear field theories which are often studied perturbatively can be interpreted also in terms of their soliton solutions. In addition to states related to quantization of free-field modes, a set of various particles will be present. A quantum expansion of the scalar theory is given in terms of a coupling constant. There occur states associated with quantization of the free field modes but also heavy particles. The heavy particles have a new quantum number and will be stable. An interpretation of the stability of the particles as related to a topological property is also described.

We review the fundamental concepts of quantum chaos in Hamiltonian systems. The quantum evolution of bound systems does not possess the sensitive dependence on initial conditions, and thus no chaotic behaviour occurs, whereas the study of the stationary solutions of the Schrodinger equation in the quantum phase space (Wigner functions) reveals precise analogy of the structure of the classical phase portrait. We analyze the regular eigenstates associated with invariant tori in the classical phase space, and the chaotic eigenstates associated with the classically chaotic regions, and the corresponding energy spectra. The effects of quantum localization of the chaotic eigenstates are treated phenomenologically, resulting in Brody-like level statistics, which can be found also at very high-lying levels, while the coupling between the regular and the irregular eigenstates due to tunneling, and of the corresponding levels, manifests itself only in low-lying levels.

The ordering problem has been one of the long standing and much discussed questions in quantum mechanics from its very beginning. Nowadays, there is more or less a consensus among physicists that the right prescription is Weyl’s rule, which is closely related to the Moyal–Wigner phase space formalism. We propose in this report an alternative approach by replacing Weyl quantization with the less well-known Born–Jordan quantization. This choice is actually natural if we want the Heisenberg and Schrodinger pictures of quantum mechanics to be mathematically equivalent. It turns out that, in addition, Born–Jordan quantization can be recovered from Feynman’s path integral approach provided that one used short-time propagators arising from correct formulas for the short-time action, as observed by Makri and Miller. These observations lead to a slightly different quantum mechanics, exhibiting some unexpected features, and this without affecting the main existing theory; for instance quantizations of physical Hamiltonian functions are the same as in the Weyl correspondence. The differences are in fact of a more subtle nature; for instance, the quantum observables will not correspond in a one-to-one fashion to classical ones, and the dequantization of a Born–Jordan quantum operator is less straightforward than that of the corresponding Weyl operator. The use of Born–Jordan quantization moreover solves the “angular momentum dilemma”, which already puzzled L. Pauling. Born–Jordan quantization has been known for some time (but not fully exploited) by mathematicians working in time–frequency analysis and signal analysis, but ignored by physicists. One of the aims of this report is to collect and synthesize these sporadic discussions, while analyzing the conceptual differences with Weyl quantization, which is also reviewed in detail. Another striking feature is that the Born–Jordan formalism leads to a redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution reducing interference effects.

The standard time-of-arrival distribution cannot reproduce both the temporal and the spatial profile of the modulus squared of the time-evolved wave function for an arbitrary initial state. In particular, the time-of-arrival distribution gives a non-vanishing probability even if the wave function is zero at a given point for all values of time. This poses a problem in the standard formulation of quantum mechanics where one quantizes a classical observable and uses its spectral resolution to calculate the corresponding distribution. In this work, we show that the modulus squared of the time-evolved wave function is in fact contained in one of the degenerate eigenfunctions of the quantized time-of-arrival operator. This generalizes our understanding of quantum arrival phenomenon where particle detection is not a necessary requirement, thereby providing a direct link between time-of-arrival quantization and the outcomes of the two-slit experiment.

In this work, quantization of electromagnetic fields in a general anisotropic inhomogeneous media is considered. First, the classical Hamiltonian is derived. Once the Hamiltonian is derived, with it the classical equations of motion follow. Next, the fields are elevated to become quantum operators, for which conjugate pairs are endowed with commutators. Since as previously shown, such commutators induce functional derivatives, the quantum equations of motion can be easily derived. Next, the mode decomposition of the fields is demonstrated. Upon substitution into the classical Hamiltonian, each mode then behaves like a simple harmonic oscillator. These oscillators can then be elevated to become quantum harmonic oscillators. Both the traveling wave modes and the standing wave modes are discussed. Furthermore, the longitudinal modes are described. Finally, the case of impressed sources are considered, with the corresponding Hamiltonian and field quantization. The Green's functions for the linear system are presented. Finally, the quantum Maxwell's equations are derived. A short discussion on field-atom interaction is given.

Estimation of the first quantization matrix of a nonaligned double compressed JPEG (NA-DJPG) image remains a challenging problem. To solve this problem, a novel method for estimating the original quantization steps of NA-DJPG images is proposed in this paper. Our method is separated into two phases: estimation of the misalignment parameters and estimation of the quantization steps. In the first phase, since the texture of the image has great influence on the traces of the first compression grid, an appropriate block selection strategy is proposed that takes the interference of texture into consideration, and this is followed by a differential boundary analysis to estimate the misalignment parameters between the first and second compression grids. In the second phase, a DCT coefficient histogram rearrangement strategy is first proposed to make the periodicity of the histogram more apparent. This is followed by a histogram frequency spectrum analysis to estimate the period of the histogram, which is close to the first quantization step. To further suppress the interference of the second compression, several quantization step candidates around the rounded period value are selected, and finally, the quantization step for each band is estimated via a comparison strategy of periodic bin accumulation. The experimental results and a comparison with existing methods demonstrate the effectiveness and superiority of the proposed method.

Choosing the right first quantization basis in quantum optics is critical for the interpretation of experimental results. The usual frequency basis is, for instance, inappropriate for short, subcycle waveforms. Deriving first quantization in time domain shows that the electromagnetic field is not directly proportional, nor even causally related, to the photonic field (the amplitude probability of a photon detection). We derive the relation between the two and calculate the statistics of the electromagnetic field for specific states in time domain, such as the single photon Fock state. We introduce the dual of the Hamiltonian in time domain and extend the concept of quadratures to all first quantization bases., 4 pages, 3 figures; supplementary material: 6 pages, 1 figure; changes from version 1: discussion of results largely extended

In dieser Arbeit werden Algorithmen für die Berechnung elektronischer Korrelations- und Anregungsenergien mittels der Coupled-Cluster Methode auf adaptiven Gittern entwickelt und implementiert. Die jeweiligen Funktionen und Operatoren werden adaptiv durch Multiskalenanalyse dargestellt, was eine Basissatz unabängige Beschreibung mit kontrollierter numerischer Genauigkeit ermöglicht. Gleichungen für die Coupled-Cluster Methode werden in einem verallgemeinerten Rahmen, unabhängig von virtuellen Orbitalen und globalen Basissätzen, neu formuliert. Hierzu werden die amplitudengewichteten Anregungen in virtuelle Orbitale ersetzt durch Anregungen in n-Elektronenfunktionen, welche durch Gleichungen im n-Elektronen Ortsraum bestimmt sind. Die erhaltenen Gleichungen können, analog zur Basissatz abh¨angigen Form, mit leicht angepasster Interpretation diagrammatisch dargestellt werden. Aufgrund des singulären Coulomb Potentials werden die Arbeitsgleichungen mit einem explizit korrelierten Ansatz regularisiert. Coupled-Cluster singles mit genäherten doubles (CC2) und ähnliche Modelle werden, für geschlossenschalige Systeme und in regularisierter Form, in die MADNESS Bibliothek (eine allgemeine Bibliothek zur Darstellung von Funktionen und Operatoren mittels Multiskalenanalyse) implementiert. Mit der vorgestellten Methode können elektronische CC2 Paarkorrelationsenergien und Anregungsenergien mit bestimmter numerischer Genauigkeit unabhängig von globalen Basissätzen berechnet werden, was anhand von kleinen Molekülen verifiziert wird In this work algorithms for the computation of electronic correlation and excitation energies with the Coupled-Cluster method on adaptive grids are developed and implemented. The corresponding functions and operators are adaptively represented with multiresolution analysis allowing a basis-set independent description with controlled numerical accuracy. Equations for the coupled-cluster model are reformulated in a generalized framework independent of virtual orbitals and global basis-sets. For this, the amplitude weighted excitations into virtuals are replaced by excitations into n-electron functions which are determined by projected equations in the n-electron position space. The resulting equations can be represented diagrammatically analogous to basis-set dependent approaches with slightly adjusted rules of interpretation. Due to the singular Coulomb potential, the working equations are regularized with an explicitly correlated ansatz. Coupled-cluster singles with approximate doubles (CC2) and similar models are implemented for closed-shell systems and in regularized form into the MADNESS library (a general library for the representation of functions and operators with multiresolution analysis). With the presented approach electronic CC2 pair-correlation energies and excitation energies can be computed with definite numerical accuracy and without dependence on global basis sets, which is verified on small molecules.

We present a quantum algorithm for simulating quantum chemistry with gate complexity $\tilde{O}(N^{1/3} \eta^{8/3})$ where $\eta$ is the number of electrons and $N$ is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity $\tilde{O}(N^{8/3} /\eta^{2/3})$. We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when $N \gg \eta$, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation., Comment: 8 pages, 1 figure

Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational "supremacy". There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates bosonic sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantisation. The techniques are quite general and have application beyond boson sampling., Comment: 25 pages, 4 figures, 2 algorithms, comments welcome

Based on the particle-wave duality principle of de Broglie, the state of a system can be described by a wave function. The treatment of systems using the methods of quantum mechanics involves finding the wave function. A set of postulates describes the formal approach for applying quantum mechanics to determine the appropriate wave functions. In general, wave functions have certain properties, such as finite, single-valued, and continuous. In one interpretation of quantum mechanics, the square of the wave function is related to the probability of some event (e.g., position of a particle, etc.). The wave function, in principle, makes it possible to determine other properties of the system. This chapter illustrates the application of the postulates of quantum mechanics and the application of a wave function for the 1s state of a hydrogen atom.

We study strong compactly aligned product systems of $\mathbb{Z}_+^N$ over a C*-algebra $A$. We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of $A$. This approach encompasses product systems where the left action is given by compacts, as well as a wide class of higher rank graphs (beyond row-finite). Moreover we analyze higher rank factorial languages and their C*-algebras. Many of the rank one results in the literature find here their higher rank analogues. In particular, we show that the Cuntz-Nica-Pimsner algebra of a higher rank sofic language coincides with the Cuntz-Krieger algebra of its unlabeled follower set higher rank graph. However there are also differences. For example, the Cuntz-Nica-Pimsner can lie in-between the first quantization and its quotient by the compactly supported operators., Comment: 45 pages, Section 4 and 5 merged with the GIUT appearing now as Theorem 4.2, minor corrections of the text

We propose a new world-line Lagrangian model of the D=4 massless relativistic particle with continuous spin and develop its twistorial formulation. The description uses two Penrose twistors subjected to four first class constraints. After the first quantization of the world-line twistorial model, the wave function is defined by an unconstrained function on the two-dimensional complex affine plane. We find the twistor transform that determines the space-time field of the continuous spin particle through the corresponding twistor one, which plays the role of a prepotential. It is shown that this space-time field is an exact solution of the space-time constraints defining the irreducible massless representation of the Poincare group with continuous spin., Comment: 1 + 19 pages, v3: minor corrections in the published version

Historically, quantization has meant turning the dynamical variables of classical mechanics that are represented by numbers into their corresponding operators. Thus the relationships between classical variables determine the relationships between the corresponding quantum mechanical operators. Here, we take a radically different approach to this conventional quantization procedure. Our approach does not rely on any relations based on classical Hamiltonian or Lagrangian mechanics nor on any canonical quantization relations, nor even on any preconceptions of particle trajectories in space and time. Instead we examine the symmetry properties of certain Hermitian operators with respect to phase changes. This introduces harmonic operators that can be identified with a variety of cyclic systems, from clocks to quantum fields. These operators are shown to have the characteristics of creation and annihilation operators that constitute the primitive fields of quantum field theory. Such an approach not only allows us to recover the Hamiltonian equations of classical mechanics and the Schrodinger wave equation from the fundamental quantization relations, but also, by freeing the quantum formalism from any physical connotation, makes it more directly applicable to non-physical, so-called quantum-like systems. Over the past decade or so, there has been a rapid growth of interest in such applications. These include, the use of the Schrodinger equation in finance, second quantization and the number operator in social interactions, population dynamics and financial trading, and quantum probability models in cognitive processes and decision-making. In this paper we try to look beyond physical analogies to provide a foundational underpinning of such applications.

A fully numerical method for the time-dependent Hartree-Fock and density functional theory (TD-HF/DFT) with the Tamm-Dancoff (TD) approximation is presented in a multiresolution analysis (MRA) approach. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. The integral equation is efficiently and adaptively solved using a numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H2, Be, N2, H2O, and C2H4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. We introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.

The quantum Hall effect (QHE) has captivated the attention of theorists and experimentalists following its discovery. First came the astounding integer quantum Hall effect (IQHE) discovered by von Klitzing, Dorda, and Pepper in 1980 [1]. Then came the even more mysterious discovery of the fractional quantum Hall effect (FQHE) by Tsui, Stormer, and Gossard in 1982 [2]. Obviously I cannot provide even an overview of this vast subject. Instead, I will select two techniques that come into play in the theoretical description of the FQHE. Along the way I will cover some aspects of IQHE. However, of necessity, I will be forced to leave out many related developments, too numerous to mention. The books in [3–7] and online notes in [8] may help you with further reading. The first technique is due to Bohm and Pines (BP) [9], and was used to describe an excitation of the electron gas called the plasmon . Since the introduction by BP of this technique in first quantization, it has been refined and reformulated in the diagrammatic framework. I will stick to the wavefunction-based approach because it is very beautiful, and because two of the great problems in recent times – the theory of superconductivity and the theory of the FQHE – were first cracked open by ingenious trial wavefunctions that captured all the essentials. I will introduce the BP approach in terms of the electron gas. The second technique is Chern–Simons field theory. Originally a product of the imaginations of the mathematicians S. S. Chern and J. Simons, it first entered particle physics in the work of Deser, Jackiw, and Templeton [10], and then condensed matter [11–14]. I will describe its role in the FQHE after introducing the problem to you. The Bohm–Pines Theory of Plasmons: The Goal Consider a system of N spinless fermions experiencing the Coulomb interaction Invoking the Fourier transformation (in unit spatial volume) we find that is the density operator (in first quantization), and the q =0 component is presumed to have been neutralized by some background charge.

In this paper, a complexity-quality analysis with transcoding architectures is proposed for reduced inverse quantization numbers. This architecture is different from conventional transcoding scheme, which neglects the relation between first and second quantized step size. However, our proposed transcoding architecture depends on the modulus of the ratio of second quantization and first quantization parameter. By analyzing quantized area of first quantization and second quantization parameter, we concluded the part of undoing first inverse quantization, to reduce computing complexity. From computer simulation we verify the merits of the proposed scheme over the conventional transcoding approaches, in terms of achieving better performance based on the computing complexity and objective (e.g., the peak signal-to-noise ratio) analysis.

In conformal quantum mechanics with the vacuum of a real scaling dimension and with a complete orthonormal set of energy eigenstates which is preferable under the unitary evolution, the dilatation expectation value between energy eigenstates monotonically decreases along the flow from the UV to the IR. In such conformal quantum mechanics there exist bounds on scaling dimensions of the physical states and the gauge operators., 25 pages, 2 figures; v3: published version in Physical Review D