Quantized Form Research Articles

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form in which they can act on multi-qubit states. 
Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian.
They are, however,
masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits.
To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations.
They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices.
We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations
for many-body systems.

The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices

Let Λ be an n-dimensional lattice. For any n-dimensional vector c and positive real number s, let Ds,c and DΛ,s,c denote the continuous Gaussian distribution and the discrete Gaussian distribution over Λ, respectively. In this paper, we establish the exact relationship between the second and fourth moments centered around c of the discrete Gaussian distribution DΛ,s,c and those of the continuous Gaussian distribution Ds,c, respectively. This provides a quantization form of the result obtained by Micciancio and Regev on the second and fourth moments of discrete Gaussian distribution. Using the relationship, we also derive an uncertainty principle for Gaussian functions, which extend the result of Zheng, Zhao, and Xu. Our proof is based on combination of the idea of Micciancio and Regev and the idea of Zheng, Zhao, and Xu, where the main tool is high-dimensional Fourier transform.

Efficient simulation of potential energy operators on quantum hardware: a study on sodium iodide (NaI)

This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O(2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(2^n)$$\\end{document} to O∑i=1rnCr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\sum _{i=1}^{r} {} ^nC_r \\right)$$\\end{document} (for some r≪n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ll n$$\\end{document}). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.

Local double quantitative fuzzy rough sets over two universes

As an important expanded quantification fuzzy rough set model, the local fuzzy rough set model is used to measure relative quantitative information between the fuzzy similarity classes and the basic concept. Although many studies have taken on this topic, the existing local fuzzy rough sets ignore the absolute quantitative information in the fuzzy information system over two universes. Therefore, based on this observation, we propose a local double quantitative fuzzy rough set model over two universes, which considers the problems from the double quantization form of the local fuzzy rough sets over two universes. Furthermore, the corresponding properties and decision rules of the local double quantitative “logical and” fuzzy rough set model over two universes are studied. Then, in order to improve the applicability of the model, an effective reduction method is introduced. Finally, we conduct experimental comparisons showing the computational efficiency and approximate accuracy of the model in concept approximation and reduction.

Time-symmetric quantization of relativistic fields. 1. Complex fields, massless gauge fields and gravitons

In the standard formulation of quantum field theory (QFT), where there are only positive energy particles and antiparticles, the energy and charge of the vacuum diverge, which, due to the existence of gravity, leads to the inconsistency of the theory (cosmological constant problem). In the article, it is shown that in the Stueckelberg-Feynman (SF) interpretation, where antiparticles are described as negative energy particles moving backward in time, the zero-point energy and zero-point charge of vacuum of complex fields are absent and there is no cosmological constant problem. However, until now it was believed that the SF interpretation leads to negative probabilities and incompatible with QFT. In the article, it is presented a new formulation of QFT on the basis of the SF interpretation in the form of time-symmetric quantization (TSQ), where the probability of states is positive. In TSQ, the consequences of the SF interpretation are taken into account consecutively and it is shown that: a) the negative sign of the norm of states only changes the sign of the wave function, and not the probabilities; b) the expression of backward in time integrals through the forward in time integrals changes sign; c) the time ordering of the operators is symmetric in time and writing them through the usual ordering leads to the standard diagram technique. For this reason, TSQ correctly describes the known observable effects. In TSQ, the results of unification models change, in particular, a) there is no zero-point energy even with broken supersymmetry between complex fields; b) there is no zero-point energy of modes in string theories, which allows to include gravity, but there is no a conformal anomaly and the dimension of space can be arbitrary.

Privacy-preserving federated learning on lattice quantization

Federated learning (FL) is an important approach to cooperate with multiple devices for learning without exchanging data between devices and central server. However, due to bandwidth and other reasons, the communication efficiency should be considered when the volume of information transmitted is limited. In this paper, we utilize the tool of lattice quantization form quantization theory and the variable intercommunication interval to improve communication efficiency. Meanwhile, to make strong privacy guarantee, we incorporate the notion of differential privacy (DP) to the FL framework with local SGD algorithm. By adding calibrated noises, we propose a universal lattice quantization for differentially private federated averaging algorithm (ULQ-DP-FedAvg). We provide tight privacy bound by using some privacy techniques. We also analyze the convergence bound of ULQ-DP-FedAvg based on bits rate constraints and the growing inter-communication interval as well as the data are non-independent identically distribution (Non-IID). It turns out that the algorithm converges and preserves that the privacy has scarcely influenced on the convergence rate. The effectiveness of our algorithm is demonstrated by synthetic and real datasets.

Theoretical research on the transverse spin of structured optical fields inside a waveguide

Structured optical fields inside a waveguide possess the transverse spin, i.e., the spin angular momentum perpendicular to the direction of the waveguide. The physical origin of the transverse spin can be attributed to the presence of an effective rest mass of photons in guided waves, or equivalently, to the existence of a longitudinal field component, such that the transverse and longitudinal fields together form an elliptical polarization plane. In contrary to the traditional viewpoint, the transverse spin of photons in guided waves is also quantized, and its quantization form is related to the ellipticity of the polarization ellipse. The direction of the transverse spin depends on the propagation direction of electromagnetic waves along the waveguide, such a spin-momentum locking may have important applications in spin-dependent unidirectional optical interfaces. By means of a coupling between the transverse spin of guided waves and some physical degrees of freedom, one can develop an optical analogy of spintronics, i.e., spinoptics.

A Survey of Schematic of Vacuum Quantum Structure: General Equation for EBSF (III)

Researchers have studied the tiny effects on fundamental particles with quantum vacuum about electroweak baryogenesis mechanism, and so we can further confirm the existence of the energy basic state field of the universe (EBSF). In this paper, we propose that since EBSF is the basic energy field of the universe, the quantum superfluid state must be the next level of fundamental particle (quantum state). Then, the evolution-developed equation of quantum superfluid state should be nonlinear, which can be expressed by the fraction fractal KdV equation temporarily. This paper reviews the experimental observation of superfluid in the existing physics and puts forward the possible quantized form of KdV equation. One possible form of quantization for vortex line of quantum superfluid is the string theory, so the basic physical scenario and foundation of physics in string theory can be revealed. At the same time, through the analysis of the KdV equation and general solution, the quantum superfluid state as the lowest level physical state in physics can only be described by the topological invariant in topological theory, and its image is not uniform; there are common phenomena such as energy gap, non-closure and wrinkle (folding) in Euclidean space. Based on the physical view of the formation of the universe system, the idea of nonlinear evolution development, and the idea of level emergence theory in condensed matter physics, the authors preliminarily put forward the framework of the hierarchy theory of physics. The paper also analyzes the broad physical prospect of quantum superfluid physics (or new physics). The essence of some physics concepts is further clear, some present physics puzzles that are expected to be solved, but their mathematical physical processing has considerable complexity, and some rigorous proofs need further breakthroughs.

A straight forward path to a path integration of Einstein’s gravity

Path integration is a respected form of quantization that all theoretical quantum physicists should welcome. This elaboration begins with simple examples of three different versions of path integration. After an important clarification of how gravity can be properly quantized, an appropriate path integral, that also incorporates necessary constraint issues, becomes a proper path integral for gravity that can effectively be obtained. How to evaluate such path integrals is another aspect, but most likely best done by computational efforts including Monte Carlo-like procedures.

Realistic magnetic thermodynamics by local quantization of a semiclassical Heisenberg model

Classical Monte Carlo simulation of the Heisenberg model poorly describes many thermodynamic phenomena due to its neglect of the quantum nature of spins. Alternatively, we discuss how to semiclassically approach the quantum problem and demonstrate a simple method for introducing a locally approximate form of spin quantization. While the procedure underestimates magnetic short-range order, our results suggest a simple correction for recovering realistic spin–spin correlations above the critical temperature. Moreover, ensemble fluctuations are found to provide reasonably accurate thermodynamics, largely reproducing quantum mechanically calculated heat capacities and experimental magnetometry for ferromagnetic Fe and antiferromagnetic RbMnF3. Extensions of the method are proposed to address remaining inaccuracies.

Quantum Simulation of Molecular Electronic States with a Transcorrelated Hamiltonian: Higher Accuracy with Fewer Qubits.

Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.

Characterization of Limit Cycle Oscillations Induced by Fixed Threshold Samplers

In this work, a generalized study of the conditions for the appearance of limit cycle oscillations induced by any kind of sampler with multilevel fixed thresholds is presented. These kinds of samplers, which will be referred to as Fixed Threshold Samplers (FTS), are characterized by a series of parameters, which, when selected properly, allow obtaining some of the most used forms of quantization in Event-Based Control (EBC). Because of some sampler characteristics, the obtained limit cycle oscillations can present a bias, therefore, to characterize them the Dual Input Describing Function (DIDF) method is used. The obtained DIDF is analyzed revealing some interesting properties allowing to simplify the robustness analysis. The analysis takes into account the effect of the disturbance and reference signal influence on the system, generally overlooked in DF analysis. Guidelines about how to perform the robustness analysis are given, showing their application through some study cases.

Intermediate symmetric construction of transformation between anyon and Gentile statistics

Gentile statistics describes fractional statistical systems in the occupation number representation. Anyon statistics researches those systems in the winding number representation. Both of them are intermediate statistics between Bose–Einstein and Fermi–Dirac statistics. The second quantization of Gentile statistics shows a lot of advantages. According to the symmetry requirement of the wave function and the property of braiding, we give the general construction of transformation between anyon and Gentile statistics. In other words, we introduce the second quantization form of anyons in an easier way. This construction is a correspondence between two fractional statistics and gives a new description of anyon. Basic relations of second quantization operators, the coherent state and Berry phase are also discussed.

Entanglement Hamiltonian of interacting systems: Local temperature approximation and beyond

We investigate the second quantization form of the entanglement Hamiltonian (EH) of various subregions for the ground-state of several interacting lattice fermions and spin models. The relation between the EH and the model Hamiltonian itself is an unsolved problem for the ground-state of generic local Hamiltonians. In this letter, we demonstrate that the EH is practically local and its dominant components are related to the terms present in the model Hamiltonian up to a smooth spatially varying temperature even for (a) discrete lattice systems, (b) systems with no emergent conformal or Lorentz symmetry, and (c) for subsystems with non-flat boundaries, up to relatively strong interactions. We show that the mentioned local temperature at a given point decays inversely proportional to its distance from the boundary between the subsystem and the environment. We find the subdominant terms in the EH as well and show that they are severely suppressed away from the boundaries of subsystem and are relatively small near them.

Information-Distilling Quantizers

Let X and Y be dependent random variables. This paper considers the problem of designing a scalar quantizer for Y to maximize the mutual information between the quantizer's output and X, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low I(X;Y), where it is shown that, if X is binary, a constant fraction of the mutual information can always be preserved using <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (log(1/I(X;Y))) quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets 2 <; | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> | <; ∞, it is established that an η-fraction of the mutual information can be preserved using roughly (log(| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> | /I(X;Y))) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η·(|<i>X</i>| - 1)</sup> quantization levels.

The electrical properties of poly(dA)-poly(dT) DNA molecule: electrical current influenced by magnetic field and electric field

This theoretical study aims to determine the electrical properties of poly(dA)-poly(dT) DNA molecule under influence of electric and magnetic fields as external disturbances. The electrical properties of the DNA is investigated using a tight-binding model in the second quantization form. Then, the Landauer-Buttiker formalism is used in calculating electric current from the transmission probability which is calculate using the green function approach. While the external interference is considered by modelling the magnetic dependent hopping phase and the electric field dependent hopping amplitude in the tight binding Hamiltonian. The calculation results shows that the electric field applied along the DNA decreases sharply the peak of the transmission band. This means that the electric field significantly weakening the quality of transmission in poly(dA)-poly(dT) DNA molecule. On the other hand, the increase in magnetic field results in the increment of maximum current and the decrement of threshold voltage.

Quantization of the 1-D Forced Harmonic Oscillator in the Space (&amp;lt;i&amp;gt;x&amp;lt;/i&amp;gt;, &amp;lt;i&amp;gt;v&amp;lt;/i&amp;gt;)

The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrodinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrodinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.

The beauty of quantized form and the pleasures of constructing with equality – euclid’s platonic solids

The beauty of quantized form and the pleasures of constructing with equality – euclid’s platonic solids

Diagonalization of High-order Second Quantization Form

In this letter, by introducing high order tensor representation, the method to diagonalizing second quantization quadratic form is generalized to any orders operators. By some simple calculation and comparison to classic exact diagonalization, this new technique can be verified since has the same result. Moreover, this procedure could also be applied to some analytical computation, which may give some useful information in many-body interacting systems.

MELASTOMA MALABATHRICUM L. EXTRACTS-BASED INDICATOR FOR MONITORING SHRIMP FRESHNESS INTEGRATED WITH CLASSIFICATION TECHNOLOGY USING NEAREST NEIGHBOURS ALGORITHM

As a maritime country, shrimp commodity production in Indonesia is very high and continues to increase. However, because shrimp is a perishable food, we need a detection device. This is because conventional methods that are widely used by the community in detecting freshness of shrimp are only based on the smell. Of course, this is a problem when shrimp are packed in closed containers. In this paper, a method for detecting shrimp is proposed using the Melastoma malabathricum L. - based label indicator. The high content of flavonoids in the extracts allows the changing the colour of the label from red to grey due to the interaction between the label with the OH- group that arises from the shrimp spoilage process. The colour that appears on the label indicator will correlate with the level of shrimp freshness. By increasing detection effectiveness, the classification is performed using the nearest-neighbours algorithm, which is equipped with an image processing mechanism in the form of colour quantization. There are four classifications used to express the quality of shrimp, namely "acceptable," "just acceptable," "unacceptable," and "more unacceptable." The accuracy of applying this method is 71.9%, with the majority of detection errors occurring in the "acceptable" class. Based on these results, it can be stated that the label indicators prepared in this study are very promising to be developed into intelligent packaging components.