Quantum algorithms speeding up classical counterparts are proposed for the problems:
1. Recognition of eigenvalues with fixed precision. Given a quantum circuit generating unitary mapping $U$ and a complex number the problem is to determine is it an eigenvalue of $U$ or not.
2. Given a molecular structure find thermodynamic functions like partitioning function, entropy, etc. for a gas consisting of such molecules.
3. Recognition of molecular structures. Find a molecular structure given its spectrum.
4. Recognition of electronic devices. Given an electronic device that can be used only as a black box how to recognize its internal construction?
We consider mainly structures generating sparse spectrums. These algorithms require the time from about square root to logarithm of the time of classical analogs and for the first three problems give exponential memory saving. Say, the time required for distinguishing two devices with the same given spectrum is about seventh root of the time of direct classical method, for the recognition of eigenvalue - about sixth root. Thus microscopic quantum devices can recognize molecular structures and physical properties of environment faster than big classical computers.
Quantum dynamics controlled by a time-dependent coupling constant are studied. It is proven that an energy eigenstate expectation value of work done by the system in a quench process cannot exceed the work in the corresponding quasi-static process, if and only if the energy eigenvalue is a concave function of the coupling constant. We propose this concavity of energy eigenvalues as a new universal criterion for quantum dynamical systems to satisfy the second law of thermodynamics. We argue simple universal conditions on quantum systems for the concavity, and show that every energy eigenvalue i…
With the help of quantum mechanics one can formulate a model of associative memory with optimal storage capacity. I generalize this model by introducing a parameter playing the role of an effective temperature. The corresponding thermodynamics provides criteria to tune the efficiency of quantum pattern recognition. I show that the associative memory undergoes a phase transition from a disordered high-temperature phase with no correlation between input and output to an ordered, low-temperature phase with minimal input-output Hamming distance.
We apply the asymptotic iteration method (AIM) to obtain the solutions of Schrodinger equation in the presence of Poschl-Teller (PT) potential. We also obtain the solutions of Dirac equation for the same potential under the condition of spin and pseudospin (p-spin) symmetries. We show that in the nonrelativistic limits, the solution of Dirac system converges to that of Schrodinger system. Rotational-Vibrational energy eigenvalues of some diatomic molecules are calculated. Some special cases of interest are studied such as s-wave case, reflectionless-type potential and symmetric hyperbolic PT p…
Non-Hermitian Hamiltonians possessing a discrete real spectrum motivated a remarkable research activity in quantum physics and new insights have emerged. In this paper we formulate concepts of statistical thermodynamics for systems described by non-Hermitian Hamiltonians with real eigenvalues. We mainly focus on the case where the energy and another observable are the conserved quantities. The notion of entropy and entropy inequalities are central in our approach, which treats equilibrium thermodynamics.
Analytical solutions of the Schrodinger equation for the generalized trigonometric Poschl Teller potential by using an appropriate approximation to the centrifugal term within the framework of the Functional Analysis Approach have been considered. Using the energy equation obtained, the vibrational partition function was calculated and other relevant thermodynamic properties. More so, we use the concept of the superstatistics to also evaluate the thermodynamics properties of the system. It is noted that the well-known normal statistics results are recovered in the absence of the deformation pa…
A version of the second law of thermodynamics states that one cannot lower the energy of an isolated system by a cyclic operation. We prove this law without introducing statistical ensembles and by resorting only to quantum mechanics. We choose the initial state as a pure quantum state whose energy is almost E_0 but not too sharply concentrated at energy eigenvalues. Then after an arbitrary unitary time evolution which follows a typical "waiting time", the probability of observing the energy lower than E_0 is proved to be negligibly small.
Discriminative Canonical Correlation Analysis (DCCA) is a powerful supervised feature extraction technique for two sets of multivariate data, which has wide applications in pattern recognition. DCCA consists of two parts: (i) mean-centering that subtracts the sample mean from the sample; (ii) solving the generalized eigenvalue problem. The cost of DCCA is expensive when dealing with a large number of high-dimensional samples. To solve this problem, here we propose a quantum DCCA algorithm. Specifically, we devise an efficient method to compute the mean of all samples, then use block-Hamiltonia…
We investigate the behavior of two coupled non-linear photonic cavities, in presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly non-classical properties and its dyna…
The entropy produced when a system undergoes an infinitesimal quench is directly linked to the work parameter susceptibility, making it sensitive to the existence of a quantum critical point. Its singular behavior at $T=0$, however, disappears as the temperature is raised, hindering its use as a tool for spotting quantum phase transitions. Notwithstanding the entropy production can be split into classical and quantum components, related with changes in populations and coherences. In this paper we show that these individual contributions continue to exhibit signatures of the quantum phase trans…
Fluctuations of thermodynamic quantities become non-negligible and play an important role when the system size is small. We develop finite-time thermodynamics of fluctuations in microscopic heat engines whose environmental temperature and mechanical parameter are driven periodically in time. Within the slow-driving regime, this formalism universally holds in a coarse-grained time scale whose resolution is much longer than the correlation time of the fluctuations, and is shown to be consistent with the relation analogous to the fluctuation-dissipation relation. Employing a geometric argument, a…
The quantum master equation obtained from two different thermodynamic arguments is seriously nonlinear. We argue that, for quantum systems, nonlinearity occurs naturally in the step from reversible to irreversible equations and we analyze the nature and consequences of the nonlinear contribution. The thermodynamic nonlinearity naturally leads to canonical equilibrium solutions and extends the range of validity to lower temperatures. We discuss the Markovian character of the thermodynamic quantum master equation and introduce a solution strategy based on coupled evolution equations for the eige…
Accurate solution of the Schrödinger equation with Deng-Fan potential is presented by means of Nikiforov-Uvarov method. A modified Pekeris-type approximation is proposed for the centrifugal term, from a linear combination of the $r \to 0$ and $r \to r_e$ limits. It can potentially offer a series of approximations (depending on an adjustable parameter $λ$). The existing approximations in the literature can then be recovered in certain special cases. Its efficiency and feasibility is demonstrated by a critical comparison of eigenvalues produced at various $λ$'s for four molecules, \emph{viz.}, H…
In this study, the Schrodinger equation (SE) with screened Kratzer potential (SKP) in the presence of external magnetic and AB-flux fields is investigated using the factorization method. The eigenvalue and eigenfunction for the system are obtained in closed form. It is found that the present of the magnetic field partially removes the degeneracy when the screening parameter of the potential was small but the addition of the AB field removed the degeneracy faster and better. The magnetization and magnetic susceptibility of the system are evaluated at zero and finite temperatures and other therm…
A shortcut to adiabaticity (STA) is concerned with the fast and robust manipulation of the dynamics of a quantum system that reproduces the effect of an adiabatic process. A recently proposed method enables the generation of shortcuts from a prescribed slow dynamics by simply rescaling the time variable of the quantum evolution operator [B. L. Bernardo, Phys. Rev. Research 2, 013133 (2020)]. This time-rescaling method does not demand knowledge about the eigenvalues and eigenstates of the Hamiltonian, and in many cases no additional coupling fields. Here, we use this approach to study the probl…
Despite its enormous empirical success, the formalism of quantum theory still raises fundamental questions: why is nature described in terms of complex Hilbert spaces, and what modifications of it could we reasonably expect to find in some regimes of physics? Here we address these questions by studying how compatibility with thermodynamics constrains the structure of quantum theory. We employ two postulates that any probabilistic theory with reasonable thermodynamic behavior should arguably satisfy. In the framework of generalized probabilistic theories, we show that these postulates already i…
The problem of time is a notable obstacle towards the recognition of quantum theory as the ultimate fundamental description of nature. Quantum theory may not be complete if founded upon classical notions. Louis de Broglie, seeming to be more or less convinced about the ontology of his proposed matter waves, tried to develop a theory of sub-quantum degrees of freedom relying on statistical thermodynamics. He realized a quantum particle as a fluctuating dense corpuscle formed via non-linear effects from a sub-quantum medium. A wave on the medium guides the vibrating corpuscle. He argued that an …
Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $ϕ_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, via the expression $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρ{L(v)}^\dagger \, \dm(v)$. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where $L$ was the identity. Under some mild assumptions on the quantum channel $ϕ_L$ we analyze the eigenvalue property for $ϕ_L$ and we define entropy for such…
Quantum tunneling events occurring through biochemical bonds are capable to generate quantum correlations between bonded systems, which in turn makes the conventional second law of thermodynamics approach insufficient to investigate these systems. This means that the utilization of these correlations in their biological functions could give an evolutionary advantage to biomolecules to an extent beyond the predictions of molecular biology that are generally based on the second law in its standard form. To explore this possibility, we first compare the tunneling assisted quantum entanglement sha…
