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The first excited state of conjugated donor–acceptor molecules of C3 symmetry (octupolar molecules) is doubly degenerate. Such a doublet is known to be isomorphic to a spin 1/2. It is shown that a large electric dipole moment is associated with this spin. Since the mean value of the electric dipole moment of an octupolar molecule is a measure of the symmetry breaking charge transfer, a dimensionless dipole moment called the dissymmetry vector is introduced. The dissymmetry vector operator is constructed. A linear tensor connection between this operator and the Pauli matrices is found. The tensor character is due to the two-dimensionality of the dipole moment. The dipole moment can rotate freely in the plane of the molecule as long as the C3 symmetry is maintained. The rotation is associated only with the rearrangement of the electronic subsystem of the molecule and does not affect the spatial position of the nuclei. This opens up the possibility of changing the dipole moment state on a subpicosecond time scale. The Jahn–Teller effect on the dissymmetry vector is considered in detail. It is shown that the dissymmetry vector can be controlled using electric fields in the same way as three-dimensional spin if both static and alternating electric fields are in the plane of the molecule. The conducted studies indicate that the dipole moment of excited octupolar molecules is a promising candidate for the physical implementation of a qubit. [ABSTRACT FROM AUTHOR]

The decomposition of a square matrix into a sum of Pauli strings is a classical pre-processing step required to realize many quantum algorithms. Such a decomposition requires significant computational resources for large matrices. We present an exact and explicit formula for the Pauli string coefficients which inspires an efficient algorithm to compute them. More specifically, we show that up to a permutation of the matrix elements, the decomposition coefficients are related to the original matrix by a multiplication of a generalised Hadamard matrix. This allows one to use the Fast Walsh-Hadamard transform and calculate all Pauli decomposition coefficients in O (N 2 log ⁡ N) time and using O (1) additional memory, for an N × N matrix. A numerical implementation of our equation outperforms currently available solutions. [ABSTRACT FROM AUTHOR]

This study examines the spinor representations of TN (tangent and normal), NB (normal and binormal), TB (tangent and binormal) and TNB (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space E 3 . Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the C 3 complex vector space form a two-dimensional surface in the C 2 complex space. Additionally, each isotropic vector in C 3 space corresponds to two vectors in C 2 space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (TN , NB , TB and TNB)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. [ABSTRACT FROM AUTHOR]

In this paper, we develop an extended space-time geometry that includes an internal space built with division octonions with realization via Pauli matrices and Zorn (vectorial) matrices. Here, we extend a former field theory that used split octonion algebra to a division octonion algebra on a non-Riemannian manifold. The octonionic algebra is the last possible algebra allowed by the Hurwitz theorem. The interpretation of the "octonionic field" in this division octonionic algebra is not straightforward, however it behaves in a similar way to the quaternionic Yang–Mills fields. Former work suggests that this Yang–Mills-like octonionic field is associated with the field governing quarks within nucleons. In this work, we discover that the inclusion of octonionic fields in the geometry of an internal space necessarily excludes the quaternionic (Yang–Mills) fields in an extended non-Riemannian geometry. This is not what is expected from the standpoint of a "unified" field theory, which leads us to propose a different approach to Einstein's unified field theory. [ABSTRACT FROM AUTHOR]

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph C a y (P n , S P n ) of the generalized Pauli group P n on n -qubits; in fact, P n may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set S P n allows us to recognize the structure of central product of graphs in C a y (P n , S P n ). Using this approach, we are able to recursively construct the adjacency matrix of C a y (P n , S P n ) for each n ≥ 1 , and to explicitly describe its spectrum and the associated eigenvectors. It turns out that C a y (P n , S P n ) is a (3 n + 2) -regular bipartite graph on 4 n + 1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P 1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral. [ABSTRACT FROM AUTHOR]

We present a new geminal product wave function Ansatz where the geminals are not constrained to be strongly orthogonal or to be of seniority-zero. Instead, we introduce weaker orthogonality constraints between geminals that significantly lower the computational effort without sacrificing the indistinguishability of the electrons. That is to say, the electron pairs corresponding to the geminals are not fully distinguishable, and their product has yet to be antisymmetrized according to the Pauli principle to form a bona fide electronic wave function. Our geometrical constraints translate into simple equations involving the traces of products of our geminal matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is 2 × 2 and consists of either a Pauli matrix or a normalized diagonal matrix multiplied by a complex parameter to be optimized. With this simplified Ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced. A proof of principle is reported and confirms that the Ansatz is more accurate than strongly orthogonal geminal products while remaining computationally affordable. [ABSTRACT FROM AUTHOR]

A new highly symmetrical model of the compact Lie algebra \mathfrak {g}^c_2 is provided as a twisted ring group for the group \mathbb {Z}_2^3 and the ring \mathbb {R}\oplus \mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in \mathfrak {su}(2) and of the Gell-Mann matrices in \mathfrak {su}(3). As a bonus, the split Lie algebra \mathfrak {g}_2 is also seen as a twisted ring group. [ABSTRACT FROM AUTHOR]

In this paper, we first study the maximally commutative set in prime dimensional systems, which is a set of generalized Pauli matrices, and it can be used to detect the local discrimination of generalized Bell states. We give a simple characterization of prime dimensional maximally commutative sets, that is, a subset of a set of generalized Bell states, whose second subscript is a multiple of the first subscript. Furthermore, some sets of generalized Bell states which can be locally distinguishable by one-way local operation and classical communication (LOCC) are constructed by using the structural characteristics of prime dimensional maximally commutative sets. Based on these distinguishable generalized Bell states, we propose a (t, n)-threshold quantum secret sharing scheme. Compared with the existing quantum secret sharing scheme, it can be found that there are enough distinguishable states to encode classical information in our scheme, the dealer only needs to send entangled particles once to make the participants get their secret share, which makes the secret sharing process more efficient than the existing schemes. Finally, we prove that this protocol is secure under dishonest participant attack, interception-and-resend attack and entangle-and-measure attack. [ABSTRACT FROM AUTHOR]

Using spin-1/2 state (qubit state) description by the conditional probability distributions of dichotomic random variables, we construct the irreducible Lie su(2) algebra representations by means of the probability distribution functions. We obtain explicit 2×2-matrices of the Lie algebra elements and express them in terms of Pauli matrices. Then we discuss the possibility to obtain the probability representation of arbitrary Lie algebra, using the symmetry Lie group of quantum system states. Finally, we write Pauli matrices in the probability representation of dichotomic random variables. [ABSTRACT FROM AUTHOR]

Motivation Imaging Mueller polarimetry has already proved its potential for biomedicine, remote sensing, and metrology. The real-time applications of this modality require both video rate image acquisition and fast data post-processing algorithms. First, one must check the physical realizability of the experimental Mueller matrices in order to filter out non-physical data, i.e. to test the positive semi-definiteness of the 4 × 4 Hermitian coherency matrix calculated from the elements of corresponding Mueller matrix pixel-wise. For this purpose, we compared the execution time for the calculations of (i) eigenvalues, (ii) Cholesky decomposition, (iii) Sylvester's criterion, and (iv) coefficients of the characteristic polynomial (two different approaches) of the Hermitian coherency matrix, all calculated for the experimental Mueller matrix images (600 pixels × 700 pixels) of mouse uterine cervix. The calculations were performed using C++ and Julia programming languages. Results Our results showed the superiority of the algorithm (iv) based on the simplification via Pauli matrices over other algorithms for our dataset. The sequential implementation of latter algorithm on a single core already satisfies the requirements of real-time polarimetric imaging. This can be further amplified by the proposed parallelization (e.g. we achieve a 5-fold speed up on six cores). Availability and implementation The source codes of the algorithms and experimental data are available at https://github.com/pogudingleb/mueller_matrices. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in [4,5] to construct a complex wave function ψ ∈ H s ( R d) , s = 1 , 2 , associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density. [ABSTRACT FROM AUTHOR]

We derive a new eight dimensional matrix representation of the Maxwell equations for a linear homogeneous medium and extend it to the case of a linear inhomogneous medium. This derivation starts ab initio with the Maxwell equations and uses arguments based on the algebra of the Pauli matrices. This process leads automatically to the matrix representation based on the Riemann–Silberstein–Weber (RSW) vector. The new representation for the homogeneous medium is a direct sum of four Pauli matrix blocks. This aspect of the new representation should make it suitable for studying the propagation of electromagnetic waves in a linear inhomogeneous medium adopting the techniques of quantum mechanics treating the inhomogeneity as a perturbation. The new representation is used to rederive the Mukunda–Simon–Sudarshan matrix substitution rule for transition from the Helmholtz scalar wave optics to the Maxwell vector wave optics.

Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system's noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. This work proposes a modified representation for a single-qubit depolarization channel. Our modified channel uses two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per channel execution. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model's accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era. [ABSTRACT FROM AUTHOR]

We introduce the notion of (renormalized) quantum uncertainty and reveal its basic features. In terms of this quantity, we completely characterize the minimum and maximum quantum uncertainty states for qubit systems involving Pauli matrices. It turns out that the minimum quantum uncertainty states consist of both certain pure states and certain mixed states, in sharp contrast to the case of conventional Heisenberg uncertainty relation. The maximum quantum uncertainty states are H -type magic states arising from the stabilizer formalism of quantum computation, and can be obtained from minimum quantum uncertainty states via the T -gate. [ABSTRACT FROM AUTHOR]

In this study, the mean-field approximation (MFA) is applied to the study of the XZ model for spin-1/2 in terms of Pauli spin matrices and their exponentials to obtain the magnetization components for the ferromagnetic (FM) and antiferromagnetic (AFM) cases. The details of obtaining the series expansion for the exponential Pauli spin matrices by using their fundamental properties and commutativity relations are presented. The partition function, free energy, and magnetization are obtained for a given coordination number in a simple form as hyperbolic tangent functions. The thermal variations of the magnetization components in the x- and z-directions are studied to obtain the phase diagrams in possible planes, which only present the second-order phase transitions. The reentrant behavior is also observed in the AFM case. [ABSTRACT FROM AUTHOR]

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum Stot=3ℏ/2, (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the "spinor" or the "vector" form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space. [ABSTRACT FROM AUTHOR]

The mean-field approximation (MFA) of spin-1/2 XYZ model was studied by using the Pauli spin matrices and their exponentials which led to very nice hyperbolic tangent functions in nonlinear form. The magnetization components Mx, My and Mz were obtained for the ferromagnetic (FM) case and then it was modified for the antiferromagnetic (AFM) case by introducing the sublattices. The bilinear exchange interaction parameters Jx, Jy and Jz, and the external magnetic fields Hx, Hy and Hz were considered along the three-dimensions for various coordination numbers q = 3, 4 and 6. The thermal variations of the magnetizations and thus the phase diagrams were obtained to illustrate the behaviours of phase transition lines in the AFM case. [ABSTRACT FROM AUTHOR]

Spinors are components of a complex vector space that can be related to Euclidean space in both geometry and physics. In essence, the forms of usage include quaternions that are equivalent to Pauli spin matrices, which may be produced by thinking of a quaternion matrix as the compound. This study's objective is the spinor structure that forms based on the quaternion algebra. In this work, first, spinors have been mathematically presented. Then, Padovan and Perrin spinors have been defined using the Padovan and Perrin quaternions. Later, we defined the algebraic structure for these spinors. Finally, we have established certain identities such as the Binet formulas and generating functions for Padovan and Perrin spinors. [ABSTRACT FROM AUTHOR]

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians. [ABSTRACT FROM AUTHOR]

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the proposed spin matrices. The fermionic quantum Heisenberg model is constructed using the proposed spin matrices, and comparative studies against simulation results using the Pauli spin matrices are conducted. Further analysis of the key findings as well as discussions on extending the proposed spin matrix framework to describe hypothetical bosonic systems (spin-1 particles) are provided. [ABSTRACT FROM AUTHOR]

To address the risk of spoofing the information exchanged by users and reduce the trustworthiness of third party, we propose a quantum information fair exchange protocol based on three-particle GHZ states. This protocol achieves quantum multi-party fair exchange of secret information through the utilization of three-particle GHZ states, Pauli matrix, a semi-trusted third party, and polarization state of a single photon. By leveraging the unique physical properties of the GHZ state, this protocol achieves fair exchange of secret information with experimental results that align with theoretical derivation, thus demonstrating its feasibility. Additionally, it provides superdense coding which enhances transmission efficiency while also being resistant to false signal attacks, interception retransmission attacks, and entanglement attacks, ultimately improving security. Furthermore, compared with the classic fair exchange protocol, which requires a trusted third party, it only requires a semi-trusted third party, which raises the security from "computational security" to "unconditional security". The implementation of this protocol resolves the risk of users who hand over information first, presenting a novel solution for digital currency trading, information swapping, and multi-party secure computing. [ABSTRACT FROM AUTHOR]

This paper investigates quantum error correction schemes for fully-correlated noise channels on an n-qubit system, where error operators take the form W ⊗ n , with W being an arbitrary 2 × 2 unitary operator. In previous literature, a recursive quantum error correction scheme can be used to protect k qubits using (k + 1) -qubit ancilla. We implement this scheme on 3-qubit and 5-qubit channels using the IBM quantum computers, where we uncover an error in the previous paper related to the decomposition of the encoding/decoding operator into elementary quantum gates. Here, we present a modified encoding/decoding operator that can be efficiently decomposed into (a) standard gates available in the qiskit library and (b) basic gates comprised of single-qubit gates and CNOT gates. Since IBM quantum computers perform relatively better with fewer basic gates, a more efficient decomposition gives more accurate results. Our experiments highlight the importance of an efficient decomposition for the encoding/decoding operators and demonstrate the effectiveness of our proposed schemes in correcting quantum errors. Furthermore, we explore a special type of channel with error operators of the form σ x ⊗ n , σ y ⊗ n and σ z ⊗ n , where σ x , σ y , σ z are the Pauli matrices. For these channels, we implement a hybrid quantum error correction scheme that protects both quantum and classical information using IBM's quantum computers. We conduct experiments for n = 3 , 4 , 5 and show significant improvements compared to recent work. [ABSTRACT FROM AUTHOR]

The author expounds or proves some results connected with Segal's chronometric theory. He gives short proofs of results about linear representation of the group of nondegenerate complex -matrices on the Minkowski space-time and on the universal covering of the Lie group of unitary -matrices, i.e., on the Einstein Universe, as well as about the Cayley transform of Lie algebras of Lie groups of unitary matrices into these groups. In comparison with the structure of conformal infinity for the Minkowski space, the structure of the set of unitary -matrices, which do not admit the Cayley transform, is found. Some problems are suggested. [ABSTRACT FROM AUTHOR]

In the present paper, we develop a theory of thinking based on an attempt to formalize the construction of mental representations as described in psychoanalytic theory. In previous work, we described Freud's and Matte Blanco's structural Unconscious in a formal model in which the properties of unconscious representations are captured by particular sets-infinite singletons-that can be derived in first-order logic language. Here, we afford the issue of the finitization of unconscious representations by assuming that the mind can form an all-purpose modality, originating from abstraction from infinite singletons; in this way, a symmetric prelogical setting for mental representations is formally created, and this is interpreted in a quantum spin model by a modal (necessity) projector. Then, by introducing time, one can describe the links that mental representations can establish with reality, and hence finitize the representations. The modality is so split into finite components, here termed positive, negative and irreal; the splitting of the modality is traced back to the decomposition of the spin observables by means of the Pauli matrices, which can offer a quantum semantics to the method applied. Here, we suggest that the development of the modal approach and its quantum logic implementation can be considered as a proper formalization of some aspect of the psychoanalytic theory of thinking proposed by Bion; namely, we will show that the process of abstraction leading from raw data to preconceptions, and therefore to the definition of the content-container relationship, is adequately captured by our model, and further correspondences can be detected with Bion's theory about links and transformations, implying different ways in which the mind can get in touch with both internal and external reality. [ABSTRACT FROM AUTHOR]

We show that Pauli quantum logic gates can be achieved with composite solitons of a generalized Manakov system. We first prove that the complex superposition principle applies to the generalized Manakov system where the linear combination of two composite solutions, with complex coefficients, generates a third composite solution. The superposition is then written in terms of a complex rotation matrix that generates the three Pauli matrices for certain values of relative phases. This maps the rotation matrix to rotations of qubits on the Bloch sphere. Possible physical realizations in Bose–Einstein condensates and optical fibers are discussed. • Show that superposition principle applies to Manakov system with complex coefficients. • Generating the three Pauli matrices from the complex rotation matrix. • Proposing two experimental realizations of the Pauli gates with composite solitons in Bose–Einstein condensation and birefringent optical fibers. [ABSTRACT FROM AUTHOR]

We consider the characteristic time operator T introduced in Galapon (2002) [26] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian H with some growth condition, T satisfies the canonical relation [ T , H ] | ψ 〉 = i ħ | ψ 〉 for | ψ 〉 in a dense subspace of the Hilbert space. While T is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set T. In the neighborhood of each time t in T , T is still canonically conjugate to H and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of T , and is proportional to the Pauli matrix σ y. Thus, one can construct a quantum clock that tells the time in the neighborhood of T by measuring a compatible observable. • Semibounded discrete Hamiltonians provide a self-adjoint clock time operator. • Time can only be accessed in the neighborhood of a particular set of times. • The two-dimensional projection of this operator saturates the uncertainty relation. [ABSTRACT FROM AUTHOR]

We give a new way to obtain the standard isomorphisms of complex Clifford algebras, known as the tensor product of Pauli matrices, by representing the complex Clifford algebras on the space of complex valued functions defined over a finite subset of the Cantor set. [ABSTRACT FROM AUTHOR]

The transformation law of the Pauli spinors under rotations, linked to 1/2-spin particles, leads to a mathematical problem that can be rephrased and solved in purely linear algebraic terms. We present it as a possible approach to be incorporated in a first course of quantum mechanics when introducing the concept of the spin. This concept was very elusive at early times and had an intricate development commonly overlooked in modern texts. We also take the opportunity to make some historical comments. [ABSTRACT FROM AUTHOR]

Pauli established the standard view that the spin of the electron was a completely abstract non-classical angular momentum that could not be thought of as the rotation of anything. Here, we give a pedagogical presentation of old work by Belifante (1939), recently updated by Ohanian (1986), which shows that, contrary to Pauli's edict, the spin of the electron can be viewed as the rotational angular momentum in the wave field of the electron. [ABSTRACT FROM AUTHOR]

In this study, Pauli-Blocking has been observed in a 0D/2D MoS2 quantum dots/graphene (MoS2-QDs/graphene) hybrid system. For the observation of room temperature Pauli-Blocking in the 0D/2D system, a photodetector device based on n-type MoS2-QDs and CVD grown graphene has been fabricated using a facile and lithography free technique. The current-voltage characteristics of the device have been performed at room temperature. The fabricated device shows a negative response under visible light (λ ∼ 400 to 700 nm) illumination. The dark to photo current ratio of the device shows variation up to two orders of magnitude. This negative response, which results decrease in current under visible light illumination, may be attributed to the Pauli-Blocking due to high absorbance of photon energy in visible light range. Furthermore, it is believed that the present study may provide an insight into understanding the Pauli-Blocking in 0D/2D hybrid system at room temperature. [ABSTRACT FROM AUTHOR]

In this paper, we outline the research conducted by the first named author and his associates on the axiom-preserving, thus isotopic completion of quantum mechanics into hadronic mechanics according to the historical legacy by A. Einstein, B. Podolsky and N. Rosen that quantum mechanics is not a complete theory and review the ensuing exact representation of the magnetic moment and spin of the Deuteron in its ground state thanks to the isotopic completion of Pauli's matrices with an explicit and concrete content of D. Bohm's hidden variable λ . We then outline the independent studies conducted by the second named author on the representation of the conventional Pauli's matrices via geometric Clifford algebras. We finally show that the combination of the two studies allows a mathematically rigorous, numerically exact and time invariant geometric representation of the magnetic moment, spin and hidden variable of the Deuteron in its ground state. [ABSTRACT FROM AUTHOR]

In this work, we consider the one‐dimensional Dirac operator DQ+Qsu(x)=1iσ2ddx+Q(x)+Qs(x)u(x),$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s}u(x)=\left(\frac{1}{\mathrm{i}}{\sigma}_2\frac{d}{dx}+\mathrm{Q}(x)+{\mathrm{Q}}_s(x)\right)u(x), $$where σ2$$ {\sigma}_2 $$ is Pauli's matrix, Q$$ \mathrm{Q} $$ is a 2×2$$ 2\times 2 $$‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, Qs$$ {\mathrm{Q}}_s $$ is a singular potential consisting of N$$ N $$ delta distributions Qs(x)=∑j=1NAjδx−xj,$$ {\mathrm{Q}}_s(x)=\sum \limits_{j=1}^N{\mathrm{A}}_j\delta \left(x-{x}_j\right), $$Aj$$ {\mathrm{A}}_j $$ (j=1,⋯,N$$ j=1,\cdots, N $$) are 2×2$$ 2\times 2 $$‐matrices representing the strengths of Dirac deltas, and u=u1u2⊤$$ u={\left({u}^1\kern0.5em {u}^2\right)}^{\top } $$ is a two‐spinor. We associate to the operator DQ+Qs$$ {\mathfrak{D}}_{\mathrm{Q}+{\mathrm{Q}}_s} $$ an unbounded in L2ℝ,ℂ2$$ {L}^2\left(\mathbb{R},{\mathbb{C}}^2\right) $$ symmetric operator denoted by DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$, where Ω=xjj=1N$$ \Omega ={\left\{{x}_j\right\}}_{j=1}^N $$ is the support of singular potential Qs$$ {\mathrm{Q}}_s $$. The operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ includes only the regular potential Q$$ \mathrm{Q} $$ together with certain interaction conditions at each point xj∈Ω$$ {x}_j\in \Omega $$. The paper presents a method for determining the discrete spectrum of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ for arbitrary potential Q$$ \mathrm{Q} $$ whose entries are given by L∞(ℝ)$$ {L}^{\infty}\left(\mathbb{R}\right) $$‐functions. The eigenvalues λn$$ {\lambda}_n $$ of the operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ are the zeros of a dispersion equation ηΩ(λ)=0$$ {\eta}_{\Omega}\left(\lambda \right)=0 $$, where the characteristic function ηΩ$$ {\eta}_{\Omega} $$ is determined explicitly in terms of power series involving the spectral parameter λ$$ \lambda $$. The construction of the characteristic function ηΩ$$ {\eta}_{\Omega} $$ from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator DQ,Ω$$ {\mathcal{D}}_{\mathrm{Q},\Omega} $$ from the zeros of certain approximate function η˜Ω$$ {\tilde{\eta}}_{\Omega} $$ which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method. [ABSTRACT FROM AUTHOR]

In modern computing environments, speed of execution of any operations are gaining more attentions. The faster processing of the unit operations, opens the room for the applications such as cryptography, data communication etc. Hence, in this work a new software-oriented approach is proposed to use in the environments that requires faster executions. This proposed method speeds up the computations of the logical operations – bitwise XOR. This paper first elaborates on the procedure of how to construct the bitwise XOR vector space efficiently using recursive approach based on Pauli's matrix. The corresponding time complexity for the construction of the bitwise XOR vector space is computed by master theorem and is O (log 2 (n / 2 n)). Later, the paper presents on the implementation of this bitwise XOR vector space constructed to facilitate the XOR – Free operations. For the evidential proof, the proposed bitwise XOR-free operations are implemented on standard key stream generation scheme, SNOW 3G. The average speed up gained by the bitwise XOR-Free operations on the SNOW 3G is 1.36 x and the corresponding throughput attained is 21.85 Mbps. [ABSTRACT FROM AUTHOR]

The dynamical feature for a p‐wave superconductor model in different parameter regions in terms of the appearance of topologically gapless edge modes in reel geometry is investigated. First, the parameter region with gapless edge modes versus a parameter and quasi‐momentum is shown. The parameter diagram can be reflected by the expectations of Pauli matrices in global manners. In another view, the dynamical feature of the excitation behave differently in the parameter regions with topological gapless edge modes and not. And the cusps of dynamical return rate vanish as the parameter pass the topological phase boundary slowly enough. It is found that the dynamics in the parameter region with gapless edge modes behaves differently to that without edge modes and related mostly to the eigenenergy gap between the pre‐and post‐quench eigenstates. The cusps of the dynamical return rate behave robustly against the noise occurs during evolution in the lattice until localization behavior dominates. This work benefits detecting topological edge modes by dynamical manners. [ABSTRACT FROM AUTHOR]

This paper is concerned with open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices for finite-level systems. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics according to a quasilinear Hudson–Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This setting includes the zero-Hamiltonian isolated system dynamics as a particular case, where the system variables are constant in time, which makes them potentially applicable as a quantum memory. In a more realistic case of nonvanishing system-field coupling, we define a memory decoherence time when a mean-square deviation of the system variables from their initial values becomes relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the decoherence time maximization over the energy parameters of the system and obtain a condition under which the zero Hamiltonian provides a suboptimal solution. This optimization problem is also discussed for a direct energy coupling interconnection of such systems. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Euler's and De Moivre's formulas for the 4 × 4 matrix representation of Pauli quaternions. Moreover, we provide De Moivre's formula for the light-like Pauli quaternions. Additionally, we give the nth roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas. [ABSTRACT FROM AUTHOR]

The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group. [ABSTRACT FROM AUTHOR]

Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical Z 4 -linear Kerdock codes maps to a unitary 2-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock 2-design with symplectic transvections, and show that this process produces an ϵ -approximate unitary 3-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact 3-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock 2-design mixes edges of the same type, and the transvections mix the types. More precisely, on m qubits, the process samples O (log (N 5 / ϵ)) random transvections, where N = 2 m , followed by a random Kerdock 2-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, 2-qubit transvections exactly map to the Mølmer–Sørensen gates that form the native 2-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate 3-design that only involves such 2-qubit transvections. [ABSTRACT FROM AUTHOR]

In practical engineering, the use of Pauli algebra can provide a computational advantage, transforming conventional vector algebra to straightforward matrix manipulations. In this work, the Pauli matrices in cylindrical and spherical coordinates are reported for the first time and their use for representing a three-dimensional vector is discussed. This method leads to a unified representation for 3D multivectors with Pauli algebra. A significant advantage is that this approach provides a representation independent of the coordinate system, which does not exist in the conventional vector perspective. Additionally, the Pauli matrix representations of the nabla operator in the different coordinate systems are derived and discussed. Finally, an example on the radiation from a dipole is given to illustrate the advantages of the methodology. [ABSTRACT FROM AUTHOR]

We give an intriguing construction for establishing a sequence of unital, commutative, associative number systems by introducing the Z 2 -graded algebra S L n (R). We show how the algebra S L n (R) arises naturally from the Pauli matrices. Moreover, we investigate eigenvalues of elements in S L n (R) and the matrix algebra M m (S L n (R)). [ABSTRACT FROM AUTHOR]

We have developed a software library that simulates noisy quantum logic circuits. We represent quantum states by their density matrices in the Pauli basis, and incorporate possible errors in initialization, logic gates, memory and measurement using simple models. Our quantum simulator is implemented as a new backend on IBM's open-source Qiskit platform. In this paper, we provide its description, and illustrate it with some simple examples. [ABSTRACT FROM AUTHOR]