Complex Wave Function Research Articles

Hybrid quantum-classical convolutional neural networks

Deep learning has been shown to be able to recognize data patterns better than humans in specific circumstances or contexts. In parallel, quantum computing has demonstrated to be able to output complex wave functions with a few number of gate operations, which could generate distributions that are hard for a classical computer to produce. Here we propose a hybrid quantum-classical convolutional neural network (QCCNN), inspired by convolutional neural networks (CNNs) but adapted to quantum computing to enhance the feature mapping process. QCCNN is friendly to currently noisy intermediate-scale quantum computers, in terms of both number of qubits as well as circuit’s depths, while retaining important features of classical CNN, such as nonlinearity and scalability. We also present a framework to automatically compute the gradients of hybrid quantum-classical loss functions which could be directly applied to other hybrid quantum-classical algorithms. We demonstrate the potential of this architecture by applying it to a Tetris dataset, and show that QCCNN can accomplish classification tasks with learning accuracy surpassing that of classical CNN with the same structure.

Direct measurement of ultrafast temporal wavefunctions.

The large capacity and robustness of information encoding in the temporal mode of photons is important in quantum information processing, in which characterizing temporal quantum states with high usability and time resolution is essential. We propose and demonstrate a direct measurement method of temporal complex wavefunctions for weak light at a single-photon level with subpicosecond time resolution. Our direct measurement is realized by ultrafast metrology of the interference between the light under test and self-generated monochromatic reference light; no external reference light or complicated post-processing algorithms are required. Hence, this method is versatile and potentially widely applicable for temporal state characterization.

Experimental entanglement characterization of two-rebit states

We characterize entanglement subject to its definition over real and complex, composite quantum systems. In particular, a method is established to assess quantum correlations with respect to a selected number system, illuminating the deeply rooted, yet rarely discussed question of why quantum states are described via complex numbers. With our experiment, we then realize two-photon polarization states that are entangled with respect to the notion of two rebits, comprising two two-level systems over real numbers. At the same time, the generated states are separable with respect to two complex qubits. Among other results, we reconstruct the best approximation of the generated states in terms of a real-valued, local expansion and show that this yields an incomplete description of our data. Conversely, the generated states are shown to be fully decomposable in terms of tensor-product states with complex wave functions. Thereby, we probe paradigms of quantum physics with modern theoretical tools and experimental platforms that are relevant for applications in quantum information science and technology and connected to the fundamentals of the quantum description of nature.

Auger recombination in narrow gap HgCdTe/CdHgTe quantum well heterostructures

We present detailed theoretical analysis of nonradiative Auger recombination in narrow-gap mercury-cadmium-telluride quantum wells (HgCdTe QWs). We suggest a microscopic model to calculate Auger recombination rates in the QWs with different Cd fraction as functions of non-equilibrium carrier density with account to the complex band dispersions and wavefunctions, degenerate carrier statistics, and screening effects. Our model is validated by the comparison of measured photoconductivity kinetics with the simulated curves. Furthermore, we use the developed calculation technique to evaluate different designs of HgCdTe/CdHgTe QWs for the far-IR emitters. In particular, we consider a series of QWs with the fixed bandgap of 40 meV (lasing wavelength about 30 μm) and find out that lasing may be favored in the QWs with moderate (6%–9%) cadmium content and not in the pure binary HgTe QWs, which is in contrast to intuitive expectations within threshold energy concept for Auger recombination. Though cadmium-free QWs do provide the highest possible Auger threshold energies, Cd-containing QWs feature much more efficient screening of Coulomb potential (and so Auger interaction) by free charge carriers. The latter effect contributes decisively into the suppression of Auger processes at low temperatures and high carrier concentrations.

Functional theory for Bose-Einstein condensates

One-particle reduced density matrix functional theory would potentially be the ideal approach for describing Bose-Einstein condensates. It namely replaces the macroscopically complex wavefunction by the simple one-particle reduced density matrix, therefore provides direct access to the degree of condensation and still recovers quantum correlations in an exact manner. We eventually initiate and establish this novel theory by deriving the respective universal functional $\mathcal{F}$ for general homogeneous Bose-Einstein condensates with arbitrary pair interaction. Most importantly, the successful derivation necessitates a particle-number conserving modification of Bogoliubov theory and a solution of the common phase dilemma of functional theories. We then illustrate this novel approach in several bosonic systems such as homogeneous Bose gases and the Bose-Hubbard model. Remarkably, the general form of $\mathcal{F}$ reveals the existence of a universal Bose-Einstein condensation force which provides an alternative and more fundamental explanation for quantum depletion.

Simulation of Anharmonic Bloch Oscillations: Numerical Problems and Nonlinear Effects

A stable and fast numerical method for calculating complex energies and wave functions of carriers in one-dimensional electrically biased periodic structures is presented. Optical transitions in a superlattice (Bloch oscillations) have been investigated using this method. It is shown that the transition probabilities depend nonlinearly on the applied (sufficiently strong) field. In this case, the transitions at the double and triple Bloch frequencies can be more intense than that at the fundamental frequency. A similar situation is observed in the superlattice with a split miniband (the nonlinearity is more pronounced at strong splitting).

Simulating Periodic Systems on a Quantum Computer Using Molecular Orbitals.

The variational quantum eigensolver (VQE) is one of the most appealing quantum algorithms to simulate electronic structure properties of molecules on near-term noisy intermediate-scale quantum devices. In this work, we generalize the VQE algorithm for simulating periodic systems. However, the numerical study of a one-dimensional (1D) infinite hydrogen chain using existing VQE algorithms shows a remarkable deviation of the ground-state energy with respect to the exact full configuration interaction (FCI) result. Here, we present two schemes to improve the accuracy of quantum simulations for periodic systems. The first one is a modified VQE algorithm, which introduces a unitary transformation of Hartree-Fock orbitals to avoid the complex wave function. The second one is combining VQE with the quantum subspace expansion approach (VQE/QSE). Numerical benchmark calculations demonstrate that both of the two schemes provide an accurate description of the potential energy curve of the 1D hydrogen chain. In addition, excited states computed with the VQE/QSE approach also agree very well with FCI results.

ИСПОЛЬЗОВАНИЕ УРАВНЕНИЯ ШРЁДИНГЕРА ДЛЯ ИССЛЕДОВАНИЯ ЯВЛЕНИЙ, ПРОТЕКАЮЩИХ В ПЛАЗМОГЕНЕРАТОРЕ ТЛЕЮЩЕГО РАЗРЯДА

The work purpose is development of the theory ensuring creation of an efficient system of fast process control in the plasma generator of a glow discharge and contributing to the development of new techniques and equipment for their realization under conditions of controllable automated tool production. 
 There are used regulations of quantum mechanics lying in that any system can be described by setting in a general case a complex wave function of the kind of . A possibility to find out a charged particle at the time t in some point of the near-cathode area of the closed volume of the plasma generator with the radius-vector was defined by probability density which is presented by a module square of the wave function of .
 In the course of the investigation carried out there are obtained the following results. The formation of charged particle flows in the plasma generator of a glow discharge has a probabilistic character. The Schroedinger equation use to obtain analytical dependences describing the processes of charged particle flows formation in the plasma generator of a glow charge is the most corresponding as it allows defining a value of their energy depending on the gas technological environment used. The rate change of gas technological environment pumping allows forming a corresponding volume of ions having specified energy and frequency, in the flow taking into account their mass and energy according to the adopted exponential distribution of ion mass in the flow. 
 In automated technological environment having changed a kind of gas technological environment and a rate of its pumping it is possible to obtain predictable results of the impact of glow discharge plasma upon a surface of products worked explaining the effect of defect generation. As a result of high-energy ion bombardment of the surface of products under processing in the plasma generator of the glow discharge there is discovered the presence of a dissipative process with the elements of self-organization. The low-energy ion presence in the flow ensures an ion current transfer which results in the change of chemical and phase structure in the surface volume of material, its modification and reduction of a crystalline structure and also in amorphism on the surface.

ИСПОЛЬЗОВАНИЕ УРАВНЕНИЯ ШРЁДИНГЕРА ДЛЯ ИССЛЕДОВАНИЯ ЯВЛЕНИЙ, ПРОТЕКАЮЩИХ В ПЛАЗМОГЕНЕРАТОРЕ ТЛЕЮЩЕГО РАЗРЯДА

The work purpose is development of the theory ensuring creation of an efficient system of fast process control in the plasma generator of a glow discharge and contributing to the development of new techniques and equipment for their realization under conditions of controllable automated tool production. 
 There are used regulations of quantum mechanics lying in that any system can be described by setting in a general case a complex wave function of the kind of . A possibility to find out a charged particle at the time t in some point of the near-cathode area of the closed volume of the plasma generator with the radius-vector was defined by probability density which is presented by a module square of the wave function of .
 In the course of the investigation carried out there are obtained the following results. The formation of charged particle flows in the plasma generator of a glow discharge has a probabilistic character. The Schroedinger equation use to obtain analytical dependences describing the processes of charged particle flows formation in the plasma generator of a glow charge is the most corresponding as it allows defining a value of their energy depending on the gas technological environment used. The rate change of gas technological environment pumping allows forming a corresponding volume of ions having specified energy and frequency, in the flow taking into account their mass and energy according to the adopted exponential distribution of ion mass in the flow. 
 In automated technological environment having changed a kind of gas technological environment and a rate of its pumping it is possible to obtain predictable results of the impact of glow discharge plasma upon a surface of products worked explaining the effect of defect generation. As a result of high-energy ion bombardment of the surface of products under processing in the plasma generator of the glow discharge there is discovered the presence of a dissipative process with the elements of self-organization. The low-energy ion presence in the flow ensures an ion current transfer which results in the change of chemical and phase structure in the surface volume of material, its modification and reduction of a crystalline structure and also in amorphism on the surface.

Efficient update of determinants for many-electron wave function overlaps

The calculation of overlaps between many-electron wave functions at different nuclear geometries during nonadiabatic dynamics simulations requires the evaluation of a large number of determinants of matrices that differ only in a few rows/columns. While this calculation is fast for small systems, its cost grows faster than the alternative electronic structure calculation used to obtain the wave functions. For wave functions that can be written as a CIS expansion, all determinants can be computed using the set of level-2 minors of the reference matrix. However, this is still a costly computation for large systems.In this paper, we provide an algorithm for efficiently calculating all level-2 minors of a matrix by re-utilizing and updating the LU factorization for the determinants of the minors. This approach results in a parallel version of the algorithm that is up to an order of magnitude faster then the current best parallel implementation. The algorithm thus allows the computation of exact wave function overlaps for relatively large systems, with a high density of states, at virtually no cost compared with the electronic structure calculations. Furthermore, the new algorithm opens the path to further investigations in efficient computing of the exact wave function overlaps for complex wave functions such as MR-CIS and MR-CISD. Program summaryProgram Title: CIS OverlapLicensing provisions: MITProgramming language: FORTRAN 2008, CNature of problem: Calculation of overlaps between CIS type wave functions at different nuclear geometries during nonadiabatic dynamics simulation requires calculating a large number of connected determinants and scales with the seventh power of the size of the system being studied. Without additional approximations, for large systems this computation becomes more costly than the electronic structure calculation used to obtain the wave functions.Solution method: All of the determinants required for a CIS wave function overlap calculation can be derived from the set of all level-2 minors of the matrix of overlaps between the reference Slater determinants. We developed an algorithm for efficiently computing all level-2 minors of a matrix. This part of the computational process was the bottleneck in previous solutions to the problem and is now an order of magnitude faster resulting in significantly faster overall calculation of the wave function overlaps.

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

For the parametrically driven nonlinear Schrödinger equation, two criteria for the dynamical stability of solitons are presented, using the driving force f(x, t) = a exp[2 i K(t) x]. Both criteria are based on the results of a collective coordinate theory. In this theory, the parameters of the 1-soliton solution are allowed to depend on time and a variational approach yields a set of coupled nonlinear ODEs for the collective coordinates. The solutions of these ODEs are used to compute the soliton’s normalized momentum p(t), velocity v(t), and a parametric plot p(v). The first criterion states that a sufficient condition for instability is fulfilled if the curve p(v), or a piece of it, has a negative slope. If the slope is positive then a necessary condition for stability is fulfilled. For constant K, a second criterion is presented. Here a complex wavefunction is calculated and a parametric plot of its imaginary part vs its real part is made. This yields a closed orbit in the complex plane. If this orbit has a negative sense of rotation with respect to a fixed point, then a sufficient condition for instability is fulfilled, whereas a positive sense is a necessary condition for stability. The ensemble of orbits and fixed points is a so-called phase portrait, which is very helpful for the classification of the different types of solutions of the ODEs. Finally, the validity of the two criteria is tested by comparing with simulations for the driven nonlinear Schrödinger equation. Both criteria make correct predictions about the stability and instability of the solitons, with the exception of a few cases at the border between stability and instability regions.

The dependence of the time delay on the shape of the wave packet tunneling through the potential barrier

The problem of quasi-monochromatic wave packet tunneling through a potential rectangular barrier is studied. It is shown that the approach to calculating the time delay of a wave packet called ``phase time’’ is not universally correct though well known in the literature. In this paper we discuss the mathematical reasons for this discrepancy, and substantiate another, more correct analytical method for calculating the time delay in the approximation of wide barriers. This time depends on the shape of the envelope complex wave function falling on the barrier. Analytical results for the Gaussian wave packet are presented.

Schrödinger's original struggles with a complex wave function

In this paper, some of Schrödinger's initial struggles with the complex nature of his wave function are outlined. It is shown that he first attached physical meaning only to its real component and even tried to avoid the explicit appearance of the imaginary unit i in his fundamental (time-dependent) equation. This attitude is quite understandable, since he was committed to the classical framework of wave theory, not only ontologically, but also formally. Around one year after the publication of his famous papers on wave mechanics, Schrödinger seems to have accepted that the wave function must be complex and that the physical interpretation is to be related to its absolute square. This is justified by him both due to equivalence considerations with matrix mechanics and in analogy with Maxwell's equations. It is argued that an awareness of the struggles Schrödinger went through to accept a complex wave function can be a relief to students who themselves wonder why ψ needs to be complex when they encounter the topic for the first time.

Scattering of SH Waves by a Partially Debonded Cylindrical Inclusion in the Covering Layer

This article presents analytical solutions to the problem of dynamic stress concentration and the surface displacement of a partially debonded cylindrical inclusion in the covering layer under the action of a steady-state horizontally polarized shear wave (SH wave); these solutions are using the complex function method and wave function expansion method. By applying the large-arc assumption method, the straight line boundary of the half-space covering layer is transformed into a curved boundary. The wave field of the debonded inclusion is constructed utilizing a Fourier series and boundary conditions of continuity. The impact of debonding upon the dynamic stress concentration and surface displacement around the cylindrical concrete or steel inclusion is analyzed through numerical examples of the SH waves that are incident at normal angles, from a harder medium to a softer medium and from a softer medium to a harder medium. The examples show that various factors (including the medium parameters of the soil layers and the inclusion, the frequency of the incident waves, and the debonding situations) jointly affect the dynamic stress concentration factor and the surface displacement around the structure.

Three-dimensional fluorescence imaging using the transport of intensity equation.

.We propose a nonscanning three-dimensional (3-D) fluorescence imaging technique using the transport of intensity equation (TIE) and free-space Fresnel propagation. In this imaging technique, a phase distribution corresponding to defocused fluorescence images with a point-light-source-like shape is retrieved by a TIE-based phase retrieval algorithm. From the obtained phase distribution, and its corresponding amplitude distribution, of the defocused fluorescence image, various images at different distances can be reconstructed at the desired plane after Fresnel propagation of the complex wave function. Through the proposed imaging approach, the 3-D fluorescence imaging can be performed in multiple planes. The fluorescence intensity images are captured with the help of an electrically tunable lens; hence, the imaging technique is free from motion artifacts. We present experimental results corresponding to microbeads and a biological sample to demonstrate the proposed 3-D fluorescence imaging technique.

A novel view on successive quantizations, leading to increasingly more “miraculous” states

A series of successive quantizations is considered, starting with the quantization of a non-relativistic or relativistic point particle: (1) quantization of a particle’s position, (2) quantization of wave function, (3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first-order equation in time derivative for complex wave function (or functional), the equivalent second-order equation for its real part was used. In such a way, at each level of quantization, the equation a quantum state satisfies is just like that of a harmonic oscillator, and wave function(al) is composed in terms of the pair of its canonically conjugated variables.

Restricted Boltzmann machines in quantum physics

A type of stochastic neural network called a restricted Boltzmann machine has been widely used in artificial intelligence applications for decades. They are now finding new life in the simulation of complex wavefunctions in quantum many-body physics.

Ligand Field Theory and Angular Overlap Model Based Analysis of the Electronic Structure of Homovalent Iron-Sulfur Dimers.

The electronic structure of multinuclear transition metal complexes is a highly challenging problem for quantum chemical methods. The problems to be solved for a successful analysis include the following: (1) many unpaired electrons leading to “highly entangled” wave functions that cannot be calculated by standard electronic structure methods, (2) drastic differences between the one-particle and many-particle spectra and a high density of low-lying states, and (3) the interpretation of such highly complex wave functions in chemical terms. In this work, we continue our research on oligonuclear clusters by presenting an in-depth analysis of the electronic structure of a prototypical iron–sulfur (Fe2S2) dimer. Accurate wave functions are obtained from a variety of advanced wave function based methods. The wave function results are interpreted in terms of an effective Hamiltonian that in turn is parametrized in terms of the angular overlap model (AOM) that provides the chemical insights that we are striving for. A hierarchical analysis allows us to interpret the local electronic structure in terms of the thiolate, sulfide ligands, and metal–metal interaction strengths. The many-particle spectrum is analyzed in terms of configurations involving ligand and metal centers. Finally, we are able to derive simple yet effective interpretations of ligand interaction strengths, the metal–metal interaction strength, and the low-lying many-particle spectrum of the Fe2S2 dimer.

Transition-Dipole Moments for Electronic Excitations in Strong Magnetic Fields Using Equation-of-Motion and Linear Response Coupled-Cluster Theory.

An implementation of transition-dipole moments at the equation-of-motion coupled-cluster singles-doubles (EOM-CCSD) and CCSD linear response (LR) levels of theory for the treatment of atoms and molecules in strong magnetic fields is presented. The presence of a finite magnetic field leads, in general, to a complex wave function and a gauge-origin dependence, necessitating a complex computer code together with the use of gauge-including atomic orbitals. As in the field-free case, for EOM-CC, the evaluation of transition-dipole moments consists of setting up the one-electron transition-density matrix (TDM) which is then contracted with dipole-moment integrals. In the case of CC-LR, the evaluation proceeds with a modified TDM but additionally requires a second contribution accounting for the amplitude response which is missing in EOM-CC theory for properties. We present a selected set of transitions for the sodium atom and investigate the LiH molecule in both a parallel as well as a perpendicular magnetic field. The dependence of excited-state energies and transition moments on the magnetic-field strength is discussed with a focus on magnetic-field-induced avoided crossings. Additionally, the differences between field-dependent EOM-CCSD and CCSD-LR transition moments are investigated.

Resultant gradient information, kinetic energy and molecular virial theorem

Resultant gradient-information is introduced and applied to problems in chemical reactivity theory. This local measure of the structural information contained in (complex) wavefunctions of electronic states is related to the system overall kinetic energy combining the modulus (probability) and phase (current) contributions. The grand-ensemble representation of thermodynamic equilibria in open systems demonstrates the physical equivalence of the variational energetic and information principles. It is used and to relate the populational derivatives of ensemble-average functionals in both these representations, which represent reactivity criteria for diagnosing the charge-transfer (CT) phenomena. Their equivalence is demonstrated by using the in situ potential and hardness descriptors to predict the direction and optimum amount of CT. The virial theorem is generalized into thermodynamic quantities and used to extract the kinetic energy component from qualitative energy profiles in the bond-formation and (exo/endo)-ergic reactions. The role of electronic kinetic energy in such chemical processes is reexamined, the virial theorem implications for the Hammond postulate of reactivity theory are explored, and variations of the structural-information in chemical processes are addressed. The maximum thermodynamic information rule is formulated and “production” of the gradient- information in chemical reactions is addressed. The Hammond postulate is shown to be indexed by the geometric derivative of resultant gradient-information at transition-state complex.