Quantization and a Green’s Function for Systems of Linear Ordinary Differential Equations

P. B. Guest

Yang Xiao, Liming Yang, Chang Shu +2 more

In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via spatial discretization and then integrate it, thereby converting the task of solving the PDE into computing the integral for the driving function f(x). These solvers rely on the quantum amplitude estimation algorithm, which requires the driving function f(x) to be within the range of [0, 1] and necessitates the construction of a quantum circuit for the oracle R that encodes f(x). This construction can be highly complex, even for simple functions like f(x) = x. An important exception arises for the specific case of f(x) = sin2(mx + c), which can be encoded more efficiently using a set of Ry rotation gates. To address these challenges, we expand the driving function f(x) as a Fourier series and propose the quantum Fourier ODE Solver. This approach not only simplifies the construction of the oracle R but also removes the restriction that f(x) must lie within [0,1]. The proposed method was evaluated by solving several representative linear and nonlinear PDEs, including the Navier–Stokes (N–S) equations. The results show that the quantum Fourier ODE solver produces results that closely match both analytical and reference solutions.

Metin Demіralp

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.

J.-P. Amiet

Vinod P. Gehlot, Mark J. Balas, Marco B. Quadrelli +3 more

Abstract Quantum computing and quantum information science is a burgeoning engineering field at the cusp of solving challenging robotic applications. This paper introduces a hybrid (gate-based) quantum computing and classical computing architecture to solve the motion propagation problem for a robotic system. This paper presents the quantum-classical architecture for linear differential equations defined by two types of linear operators: unitary and non-unitary system matrices, thereby solving any linear ordinary differential equations. The ability to encode information using bits or qubits—is essential in any computation problem. The results in this paper also introduce two novel approaches to encoding any arbitrary state vector or any arbitrary linear operator using qubits. Unlike other algorithms that solve ordinary differential equations (ODEs) using purely quantum or classical architectures, the ODE solver presented in this paper leverages the best of quantum and classical computing paradigms.