We (myself, my son, my colleagues in my group and some other collaborators from certain universities) were concerning with construction of a fictituos probabilistic dynamics behind the ordinary differential equations. This was based on the fact that some dynamical systems like quantum dynamics or Liouville dynamics are governed by partial differential equations. However, it is also possible to construct an infinite degree of freedom system over the expectations or expected values of some operators such that these expectations are connected through some ordinary differential equations in infinite number since the operators appearing in the expectations are related to each other recursively through commutators without stopping. At the beginning even these expectation ODEs could have nonlinearities. After intense studies in our group we could be able to show that these ODEs can be structured in linear format by using certain appropriate basis operators. What we have observed was the fact that the lower number of ODEs were resulting in mostly nonlinear structures unless very specific characters are existing in the system under consideration. However infinite number of equations were capable of bringing linearity. This was a one way research. From probabilistic equations to ODEs. Later we focused on the reverse action. To find an appropriate probabilistic structure like partial differential equation became the target of our efforts. Four years ago, my son Emre Demiralp, joined with a group in University Michigan at Ann Arbor to work on neuroscientific problems. By his pushes on me we started to deal with the mathematical structure behind the brain activities. Since these problems were having too many parameters and necessitating huge number of data, the models might need causal and probabilistic aspects. However all systems were at most considered as dynamical systems which are governed by ODEs. If there would be a probabilistic structure it would be behind these mathematical objects. Our efforts intensified and focused on this issue. The last year our project group took some important steps and possibility of birth of a quantum mechanical background system seemed to be appearing on the horizon. This encouraged us to continue and eventually we could be able to develop a probabilistic structure not like quantum or Liouville mechanics but something new. There appeared an evolution operator instead of the Hamiltonian. A first order partial differential equation could have been constructed. Its solution was possessing wave function properties and so on. When this is done what we have seen was an infinite linear ordinary differential equation set with constant coefficient matrix. This was valid for all explicit ODEs, linear or nonlinear. Then we focused on to reveal this feature without using probabilistic toolsand considerations. Today we are at point that all these are accomplished. There is no nonlinear explicit ODEs but the nonlinear representation of infinitely linear structures. In other words, nonlinearity is the folded form of linearity. Many of well known ODEs are now under the attack of our group. Our major studies are directed to this area.
We (myself, my son, my colleagues in my group and some other collaborators from certain universities) were concerning with construction of a fictituos probabilistic dynamics behind the ordinary differential equations. This was based on the fact that some dynamical systems like quantum dynamics or Liouville dynamics are governed by partial differential equations. However, it is also possible to construct an infinite degree of freedom system over the expectations or expected values of some operators such that these expectations are connected through some ordinary differential equations in infinite number since the operators appearing in the expectations are related to each other recursively through commutators without stopping. At the beginning even these expectation ODEs could have nonlinearities. After intense studies in our group we could be able to show that these ODEs can be structured in linear format by using certain appropriate basis operators. What we have observed was the fact that the lower number of ODEs were resulting in mostly nonlinear structures unless very specific characters are existing in the system under consideration. However infinite number of equations were capable of bringing linearity. This was a one way research. From probabilistic equations to ODEs. Later we focused on the reverse action. To find an appropriate probabilistic structure like partial differential equation became the target of our efforts. Four years ago, my son Emre Demiralp, joined with a group in University Michigan at Ann Arbor to work on neuroscientific problems. By his pushes on me we started to deal with the mathematical structure behind the brain activities. Since these problems were having too many parameters and necessitating huge number of data, the models might need causal and probabilistic aspects. However all systems were at most considered as dynamical systems which are governed by ODEs. If there would be a probabilistic structure it would be behind these mathematical objects. Our efforts intensified and focused on this issue. The last year our project group took some important steps and possibility of birth of a quantum mechanical background system seemed to be appearing on the horizon. This encouraged us to continue and eventually we could be able to develop a probabilistic structure not like quantum or Liouville mechanics but something new. There appeared an evolution operator instead of the Hamiltonian. A first order partial differential equation could have been constructed. Its solution was possessing wave function properties and so on. When this is done what we have seen was an infinite linear ordinary differential equation set with constant coefficient matrix. This was valid for all explicit ODEs, linear or nonlinear. Then we focused on to reveal this feature without using probabilistic toolsand considerations. Today we are at point that all these are accomplished. There is no nonlinear explicit ODEs but the nonlinear representation of infinitely linear structures. In other words, nonlinearity is the folded form of linearity. Many of well known ODEs are now under the attack of our group. Our major studies are directed to this area.
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
Abstract The focus of this paper is to address the structured development of numerical algorithms for water wave simulations which might execute on future quantum computers. To carry out this goal we address the feasibility of numerically simulating nonlinear wave equations in the quantum domain. Many researchers express the opinion that quantum computers may be available in the next 5 to 10 years. It therefore seems appropriate to begin preparation of the key ideas and mathematical structure that would support the development of quantum algorithms and the writing of an actual code on a hypothetical quantum computer, although the latter goal will be addressed in later papers. A number of efforts in the direction of quantum computation have already been made and some standardization of the notation has occurred. In particular Google, IBM and others have made some progress in this regard to hardware and a number of investigators have made progress with algorithms. The achievement of “quantum supremacy” has recently been advertised by Google. However, it goes without saying that considerable further progress needs to be made, particularly with regard to “quantum error correction.” We do not pursue the details of quantum computers here. Instead we study the construction of the possible quantum physics for nonlinear wave equations. The goal is to extend our understanding of the quantization of nonlinear wave equations. The problem of deep-water waves begins with the nonlinear Schroedinger (NLS) equation and its physical hierarchy: This includes the NLS, the Dysthe, the Trulsen-Dysthe, the Yan Li and the Zakharov equations. A full discussion of the quantization of some of these equations and their nearby integrable counterparts is beyond the scope of the present paper and will be presented at a later date. Here we study the simplest nonlinear, integrable wave equation, that found by Korteweg and deVries [1895]. The approach discussed herein extends the method of Heisenberg from nonlinear Hamiltonian ordinary differential equations (odes) to Hamiltonian nonlinear partial differential equations (pdes) by the development of the “matrix mechanics” appropriate for a particular nonlinear wave equation. I focus at first on the integrable Hamiltonian system for KdV and then extend this to a perturbed Hamiltonian system. The beauty of the approach is that a Hamiltonian nonlinear pde can be solved with quasiperiodic Fourier series [Osborne, 2023], an idea already used by Heisenberg for nonlinear odes in his book of 1930 [Heisenberg, 1930]. Indeed, it was his use of these Fourier series that led to matrix mechanics. This paper emphasizes the mathematical and physical structure of the quantum mechanics of the “classical integrable Hamiltonian equations” of water wave motion. We address a particular water wave equation, the Korteweg-deVries (KdV) equation, which we “quantize” using the method of Heisenberg [1930]. This leads to the “matrix mechanics” of the KdV equation: We thus find the fundamental equations of quantum mechanics which form the starting point for the development of a quantum algorithm for solving the KdV equation.
As quantum computing moves toward scientific computing applications, nonlinear differential equations remain a central challenge since quantum evolution is intrinsically linear. In this work, we introduce a reduced basis algorithm (RBA) for polynomial nonlinear ordinary differential equations (ODEs) and spatially discretized partial differential equations (PDEs). After time discretization, the method composes the resulting polynomial update map over m timesteps, identifies the reduced monomial basis appearing in this composed map, and constructs a linear RBA operator whose action recovers the exact m-timestep nonlinear dynamics. Thus, at the level of the chosen discrete update rule, the method introduces no additional approximation error beyond the time discretization error. The qubit number requirement is governed by the size of the reduced monomial basis. For an n-dimensional polynomial ODE system of degree p>1, the lifted register requires at most qmODE=O(nmlogp) qubits in the full basis scenario. For PDEs discretized on ND grid points, a locality-based construction requires at most qmPDE=O(DlogN+nmD+1logp) qubits. Hence, the dependence on the grid size remains logarithmic, while the nonlinear overhead is controlled by local reduced basis size. The main computational burden is moved from the quantum computer to a classical preprocessing step, where the reduced monomial basis and RBA operator are constructed for the chosen timestep window. Through numerical tests on the Lorenz system and the one-dimensional Burgers equation, we verify that the RBA reproduces the corresponding discrete time nonlinear dynamics exactly, while exposing the trade-off between timestep composition, reduced basis growth, and locality.
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE) dXt=(A(t)Xt+F(t)Xt⊗s)dt+B(t)dWt for an RN-valued random process Xt, aiming at quantum speed-up with respect to the dimension N. We introduce auxiliary processes Ytj that approximate Xt⊗j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of Xt by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order J in CL, and we can estimate the target expectation with a polylog(N) number of queries to unitaries encoding coefficient matrices A, F, and B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as J increases.
We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse n-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with k-th degree polynomial vector fields for arbitrary (finite) values of k. The steps involve: 1) mapping the k-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the k-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on k and demonstrate the framework on an example.