Journal of Physics A: Mathematical and Theoretical

Inspired by work by de Diego et al (Simoes et al 2021 Anal. Math. Phys. 11 1–28; Simoes et al 2022 J. Phys. A: Math. Theor. 55 1–53), we introduce constrained connection systems which are a generalization of kinetic nonholonomic mechanical systems. For a given point q∈Q, we prove that there exists a family of Riemannian metrics such that the geodesics for the constrained connection system starting from q are minimizers for the Riemannian metrics. We define Jacobi fields for constrained connection systems and derive Jacobi equations. Then two examples are investigated: a constrained mechanical system and a kinetic nonholonomic mechanical system. For the constrained mechanical system, we provide a characterization of the global form of the geodesic vector field; we also provide remarkable examples of Jacobi fields.

The electric response of an electrolytic cell submitted to an external electric field is investigated. It is assumed that the cell is limited by Ohmic electrodes, in which the conduction current at the electrodes is proportional to the surface electric field. In this framework, employing a model based on the equations of continuity for the ions and the equation of Poisson for the actual electric field inside the sample, we determine the relaxation time to reach the steady state. Our investigation generalises previous works devoted to the relaxation phenomena in electrolytic cell performed assuming blocking electrodes, in the presence of surface adsorption or in the presence of reversible trapping reaction. According to our analysis, the presence of a non-vanishing surface electric conductivity is responsible for reducing the relaxation time, with respect to that corresponding to blocking electrodes. The evolution of the bulk density of ions and of the electric potential profiles are deduced. Special attention is devoted to the electric current in the equilibrium state and during the transient time to reach the equilibrium state. An equivalent electric circuit of the cell is proposed, formed by a series of a bulk resistance with a parallel of surface capacitance and resistance due to the surface layers of thickness comparable with the Debye length. The influence of the rise time of the external power supply on the relaxation phenomenon is considered, supposing a simple exponential behaviour of the applied difference of potential.

We consider the classification problem of a high-dimensional mixture of two Gaussians with general covariance matrices. Using the replica method from statistical physics, we investigate the asymptotic behavior of a broad class of regularized convex classifiers in the limit where both the sample size n and the dimension p approach infinity while their ratio $\alpha=n/p$ remains fixed. This approach contrasts with traditional large-sample theory in statistics, which examines asymptotic behavior as $n\rightarrow\infty$ with p fixed. A key advantage of this asymptotic regime is that it provides precise quantitative guidelines for designing machine learning systems when both p and n are large but finite. Our focus is on the generalization error and variable selection properties of the estimators. Specifically, based on the distributional limit of the classifier, we construct a de-biased estimator to perform variable selection through an appropriate hypothesis testing procedure. Using $L_1$-regularized logistic regression as an example, we conduct extensive computational experiments to verify that our analytical findings align with numerical simulations in finite-sized systems. Additionally, we explore the influence of the covariance structure on the performance of the de-biased estimator.

We propose a set of generalized incompressible fluid dynamical equations, which interpolates between the Burgers and Navier-Stokes equations in two dimensions and study their properties theoretically and numerically. It is well-known that under the assumption of potential flows the multi-dimensional Burgers equations are integrable in the sense they can be reduced to the heat equation, via the so-called Cole-Hopf linearization. On the other hand, it is believed that the Navier-Stokes equations do not possess such a nice property. Take, for example, the 2D Navier-Stokes equations and rotate the velocity gradient by 90 degrees, we then obtain a system which is equivalent to the Burgers equations. Based on this observation, we introduce a system of generalized incompressible fluid dynamical equations by rotating velocity gradient through a continuous angle parameter. That way we can compare properties of an integrable system with those of non-integrable ones by relating them through a continuous parameter. Using direct numerical experiments we show how the flow properties change (actually, deteriorate) when we increase the angle parameter $\alpha$ from 0 to $\pi/2$. It should be noted that the case associated with least regularity, namely the Burgers equations ($\alpha=\pi/2$), is integrable via the heat kernel. We also formalize a perturbative treatment of the problem, which in principle yields the solution of the Navier-Stokes equations on the basis of that of the Burgers equations. The principal variations around the Burgers equations are computed numerically and are shown to agree well with the results of direct numerical simulations for some time.

Analytic continuation from (3, 1) signature Minkowski to (2, 2) signature Klein space has emerged as a useful tool for the understanding of scattering amplitudes and flat space holography. Under this continuation, past and future null infinity merge into a single boundary ( J) which is the product of a null line with a (1, 1) signature torus. The Minkowskian S-matrix continues to a Kleinian S-vector which in turn may be represented by a Poincaré-invariant vacuum state |C⟩ in the Hilbert space built on J. |C⟩ contains all information about S in a novel, repackaged form. We give an explicit construction of |C⟩ in a Lorentz/conformal basis for a free massless scalar. J separates into two halves J± which are the asymptotic null boundaries of the regions timelike and spacelike separated from the origin. |C⟩ is shown to be a maximally entangled state in the product of the J± Hilbert spaces.

In this work, we define a family of probability densities involving the generalized trigonometric functions defined by Drábek and Manásevich (1999 Differ. Integral Equ. 12 773–88), which we name generalized trigonometric densities (GTDs). We show their relationship with the generalized stretched Gaussians and other types of laws such as logistic, hyperbolic secant, and raised cosine probability densities. We prove that, for a fixed generalized Fisher information, this family of densities is of minimal Rényi entropy. Moreover, we introduce generalized moments via the mean of the power of a deformed cumulative distribution. The latter is defined as a cumulative of the power of the probability density function, this second parameter tuning the tail weight of the deformed cumulative distribution. These generalized moments coincide with the usual moments of a deformed probability distribution with a regularized tail. We show that, for any bounded probability density, there exists a critical value for this second parameter below which the whole subfamily of generalized moments is finite for any positive value of the first parameter (power of the moment). In addition, we show that such generalized moments satisfy remarkable properties like order relation w.r.t. the first parameter, or adequate scaling behavior. Then we highlight that, if we constrain such a generalized moment, both the Rényi entropy and generalized Fisher information achieve respectively their maximum and minimum for the GTDs. Finally, we emphasis that GTDs and cumulative moments can be used to formally characterize heavy-tailed distributions, including the whole family of stretched Gaussian densities.

A lemma by Chen et al (2017 J. Phys. A: Math. Theor. 50 475304) provides a necessary condition on the structure of any complex Hadamard matrix in a set of four mutually unbiased bases in C6. The proof of the lemma is shown to contain a mistake, ultimately invalidating three theorems derived in later publications.

McNulty and Weigert (2024 J. Phys. A: Math. Theor. submitted) voiced suspicions to the lemma 11(v) Part 6 in Chen and Yu (2017 J. Phys. A: Math. Theor. 50 475304) and three theorems derived in later publications (Liang et al 2019 Quantum Inf. Process. 18 352; Liang et al 2019 Linear Multilinear Algebra 69 2908–25; Chen et al 2021 Quantum Inf. Process. 20 353). For these suspicions, we reprove that any 6×6 complex Hadamard matrix whose number of real elements more than 22 does not belong to a set of four mutually unbiased bases. We show that the number of 2×2 complex Hadamard submatrices of any H2-reducible matrix is not 10,…,16,18. We also put forward some possible directions for further development.

The Nelson model describes non-relativistic particles coupled to a relativistic Bose scalar field. In this article, we study the renormalized version of the Nelson model with massless bosons in Davies’ weak coupling limit. Our main result states that the two-body Coulomb potential emerges as an effective pair interaction between the particles, which arises from the exchange of virtual excitations of the quantum field.

In this paper we show how almost cosymplectic structures are a natural framework to study thermodynamical systems. Indeed, we are able to obtain the same evolution equations obtained previously by Gay-Balmaz and Yoshimura [8] using variational arguments. The proposed geometric description allows us to apply geometrical tools to discuss reduction by symmetries, the Hamilton-Jacobi equation or discretization of these systems.

A model for a lattice of coupled cat maps has been recently introduced. This new and specific choice of the coupling makes the description especially easy and nontrivial quantities as Lyapunov exponents determined exactly. We studied the ergodic property of the dynamics along such a chain for a local perturbation. While the perturbation spreads across a front growing ballistically, the position and momentum profiles show large fluctuations due to chaos leading to diffusive transport in the phase space. It provides an example where the diffusion can be directly inferred from the microscopic chaos.

This work outlines a consistent method of identifying subsystems in finite-dimensional Hilbert spaces, independent of the underlying inner-product structure. Such Hilbert spaces arise in PT-symmetric quantum mechanics, where a non-Hermitian Hamiltonian is made self-adjoint by changing the inner product using the so-called ‘metric operator’. This is the framework of pseudo-Hermitian quantum mechanics. For composite quantum systems in this framework, defining subsystems is generally considered feasible only when the metric operator is chosen to have a tensor product form so that a partial trace operation can be well defined. In this work, we use arguments from algebraic quantum mechanics to show that the subsystems can be well-defined in every metric space—irrespective of whether or not the metric is of tensor product form. This is done by identifying subsystems with a decomposition of the underlying C∗-algebra into commuting subalgebras. Although the choice of the metric is known to have no effect on the system’s statistics, we show that different choices of the metric can lead to inequivalent subsystem decompositions. Each of the subsystems can be tomographically constructed and these subsystems satisfy the no-signalling principle. With these results, we put all the choices of the metric operator on an equal footing for composite systems.

Quantum algorithms based on the variational principle have found applications in diverse areas with a huge flexibility. But as the circuit size increases the variational landscapes become flattened, causing the so-called Barren plateau phenomena. This will lead to an increased difficulty in the optimization phase, due to the reduction of the cost function parameters gradient. One of the possible solutions is to employ shallower circuits or adaptive ansätze. We introduce a systematic procedure for ansatz building based on approximate Quantum Mutual Information (QMI) with improvement on each layer obtained by repeated applications of the previously defined Quantum Information Driven Ansatz (QIDA) approach. The objective is to recover the correlation that is discarded by the standalone QIDA method, including it in additional layers. Our approach generates a layered-structured ansatz, where each layer entangler map is defined on the qubit couples selected based on their QMI values. Following the rationale that shallower circuits may mitigate the phenomenon of barren plateaus, the Multi-QIDA ansatz partitions the optimization landscape into smaller steps. The method employs an iterative construction and optimization strategy, progressively extending the circuit depth to guide the optimization towards the appropriate potential well. We benchmarked our approach on various configurations of the Heisenberg model Hamiltonian, ranging between 9 and 12 qubits, demonstrating significant improvements in the accuracy of the ground state energy calculations, in terms of deviation from the best results and convergence, compared to traditional heuristic ansatz methods. Our results show that the Multi-QIDA method reduces the computational complexity while maintaining high precision, making it a promising tool for quantum simulations in lattice spin models.

We analyse a generalised Fokker–Planck equation in which both the nonlinear terms and the diffusivity are non-trivially dependent on the density and its derivatives. The key feature of the equation is its integrability, for it is linearisable through a Cole–Hopf transformation. We determine solutions of travelling wave and multi-kink type by resorting to a geometric construction in the regime of small viscosity. The resulting asymptotic solutions are time-dependent Heaviside step functions representing classical (viscous) shock waves. As a result, line segments in the space of independent variables arise as resonance conditions of exponentials and represent shock trajectories. We then discuss fusion and fission dynamics exhibited by the multi-kinks by drawing parallels in terms of shock collisions and scattering processes between particles, which preserve total mass and momentum. Finally, we propose Bäcklund transformations and examine their action on the solutions to the equation under study.

We study numerically, the distribution of the zeros of the grand partition function of k-mers on a k × L strip in the complex fugacity (z) plane. Using transfer matrix methods, we find that our results match the analytical predictions of Heilmann and Leib for k = 2. However, for k = 3, the zeros are confined within a bounded region, suggesting a fundamental difference in critical behavior. This indicates that trimers belong to a distinct universality class in some finite geometries. We observe that the density of zeros along multiple line segments in the complex plane reveals a richer structure than in the dimer case. Our findings emphasize the role of geometric constraints in shaping the statistical mechanics of k-mer models and set the stage for further studies in higher-dimensional lattices.

Recently multiple families of spin chain models were found, which have a free fermionic spectrum, even though they are not solvable by a Jordan-Wigner transformation. Instead, the free fermions emerge as a result of a rather intricate construction. In this work we consider the quantum circuit formulation of the problem. We construct circuits using local unitary gates built from the terms in the local Hamiltonians of the respective models, and ask the question: which circuit geometries (sequence of gates) lead to a free fermionic spectrum? Our main example is the 4-fermion model of Fendley, where we construct free fermionic circuits with various geometries. In certain cases we prove the free fermionic nature, while for other geometries we confirm it numerically. Surprisingly, we find that many standard brickwork circuits are not free fermionic, but we identify certain symmetric constructions which are. We also consider a recent generalization of the 4-fermion model and obtain the factorization of its transfer matrix, and subsequently derive a free-fermionic circuit for this case as well.

In this work we study the validity of the rotating wave approximation of an ideal system composed of two harmonic oscillators evolving with a quadratic Hamiltonian and arbitrarily strong interaction. We prove its validity for arbitrary states by bounding the error introduced. We then restrict ourselves to the dynamics of Gaussian states and are able to fully quantify the deviation of arbitrary pure Gaussian states that evolve through different dynamics from a common quantum state. We show that this distance is fully determined by the first and second moments of the statistical distribution of the number of excitations created from the vacuum during an appropriate effective time-evolution. We use these results to completely control the dynamics for this class of states, therefore providing a toolbox to be used in quantum optics and quantum information. Applications and potential physical implementations are also discussed.

Though Cliffords and matchgates are both examples of classically simulable circuits, they are considered simulable for different reasons. The celebrated Gottesman-Knill explains the simulability Cliffords, and the efficient simulability of matchgates is understood via Pfaffians of antisymmetric matrices. We take the perspective that by studying Clifford-matchgate hybrid circuits, we expand the set of known simulable circuits and reach a better understanding of what unifies these two circuit families. While the simulability of Clifford conjugated matchgate circuits for single qubit outputs has been briefly considered, the simulability of Clifford and matchgate hybrid circuits has not been generalized up to this point. In this paper extend that work, studying simulability of marginals as well as Pauli expectation values of Clifford and matchgate hybrid circuits. We describe a hierarchy of Clifford circuits, and find that as we consider more general Cliffords, we lose some amount of simulability of bitstring outputs. We then show that the known simulability of Pauli expectation values of Clifford circuits acting on product states can be generalized to Clifford circuits acting after any matchgate circuit. We conclude with general discussion about the relationship between Cliffords and matchgates, and show that both circuit families can be understood as being Gaussian.

Fractional evolution equations lack generally accessible and well-converged codes excepting anomalous diffusion. A particular equation of strong interest to the growing intersection of applied mathematics and quantum information science and technology is the fractional Schrödinger equation, which describes sub-and super-dispersive behavior of quantum wavefunctions induced by multiscale media. We derive a computationally efficient sixth-order split-step numerical method to converge the eigenfunctions of the FSE to arbitrary numerical precision for arbitrary fractional order derivative. We demonstrate applications of this code to machine precision for classic quantum problems such as the finite well and harmonic oscillator, which take surprising twists due to the non-local nature of the fractional derivative. For example, the evanescent wave tails in the finite well take a Mittag-Leffer-like form which decay much slower than the well-known exponential from integer-order derivative wave theories, enhancing penetration into the barrier and therefore quantum tunneling rates. We call this effect fractionally enhanced quantum tunneling. This work includes an open source code for communities from quantum experimentalists to applied mathematicians to easily and efficiently explore the solutions of the fractional Schrödinger equation in a wide variety of practical potentials for potential realization in quantum tunneling enhancement and other quantum applications.

The process of speciation, where an ancestral species divides in two or more new species, involves several geographic, environmental and genetic components that interact in a complex way. Understanding all these elements at once is challenging and simple models can help unveiling the role of each factor separately. The Derrida-Higgs model describes the evolution of a sexually reproducing population subjected to mutations in a well mixed population. Individuals are characterized by a string with entries ±1 representing a haploid genome with biallelic genes. If mating is restricted by genetic similarity, so that only individuals that are sufficiently similar can mate, sympatric speciation, i.e. the emergence of species without geographic isolation, can occur. Only four parameters rule the dynamics: population size N , mutation rate µ , minimum similarity for mating qmin and genome size B . In the limit B → ∞ , speciation occurs if the simple condition q min > (1 + 4µN) ⁻¹ is satisfied. However, this condition fails for finite genomes, and speciation does not occur if the genome size is too small. This indicates the existence of a critical genome size for speciation. In this work, we develop an analytical theory of the distribution of similarities between individuals, a quantity that defines how tight or spread out is the genetic content of the population. This theory is carried out in the absence of mating restrictions, where evolution equations for the mean and variance of the similarity distribution can be derived. We then propose a heuristic description of the speciation transition which allows us to numerically calculate the critical genome size for speciation as a function of the other model parameters. The result is in good agreement with the simulations of the model and may guide further investigations on theoretical conditions for species formation.

We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system's symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system's free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.

We discuss minimal covariant quantum space-time M^{13}_0, which is defined through the minimal doubleton representation of so(4,2). An elementary definition in terms of generators and relations is given. This space is shown to admit a semi-classical interpretation as quantized twistor space CP^{1,2}, viewed as a quantized S^2-bundle over a 3+1-dimensional k=-1 FLRW space-time. In particular we find an over-complete set of (quasi-) coherent states, with a large hierarchy between the uncertainty scale and the geometric curvature scale. This provides an interesting background for the IKKT model, leading to a \hs-extended gravitational gauge theory, which is free of ghosts due to the constraints arising from the doubleton representation.

Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case, it corresponds to having a time-dependent Hamiltonian. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e. quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrödingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.

We provide a mathematically and conceptually robust notion of quantum superpositions of graphs. The formalism may be useful for the purpose of modelling a ‘fully quantum internet’ and states of geometry in quantum gravity. We argue that quantum superpositions of graphs do require node names for their correct alignment, which we demonstrate through a no-signalling argument. The need for node names was often overlooked or disagreed upon in previous proposals. Still, node names should remain a fiducial construct, serving a similar purpose to the labelling of points through a choice of coordinates in continuous space. Graph renamings, aka isomorphisms, are understood as a change of coordinates on the graph. We explain how to impose renaming invariance at the level of operators over quantum superpositions of graphs, in a way that still allows us to talk about local operators.

We have developed a general algebraic method for constructing a nonredundant Wigner function for N-ququart systems using the structure of the underlying algebraic ring and the associated non-euclidian geometry of the 4^N x 4^N pseudo phase-space. Using the stabilizer formalism we show that the Wigner function satis es the tomographic condition and possesses a generalized covariance under some speci c Clifford symplectic transformations allowing to represent a wide class of ququart stabilizer states in form of delta-functions. We establish a relation bateen the Wigner function of N-ququart states represented in 4^N x 4^N grid with its 2N- qubit counterpart in a F_2^{2N} x F_2^{2N} discrete phase space and discuss the corresponding drawbacks.