Differential algebraic description and analysis of spin dynamics

Huynh T. T. Tran, Hieu Trung Nguyen, Thanh Long Vu +1 more

Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

Xiao-Chuan Wu, Ran Cheng, Di Xiao

Arkopal Dutt, Anirban Chowdhury, Kristan Temme +1 more

We introduce a quantum algorithm for simulating the dynamics of electrical circuits consisting of resistors, inductors and capacitors (aka RLC circuits) along with power sources. Given oracle access to the connectivity of the circuit and values of the electrical elements, our algorithm prepares a quantum state that encodes voltages and current values either at a specified time or the history of their evolution over a time-interval. For an RLC circuit with $N$ components, our algorithm runs in time $\textsf{polylog}(N)$ under mild assumptions on the connectivity of the circuit and values of its components. This provides an exponential speed-up over classical algorithms that take $\textsf{poly}(N)$ time in the worst-case. Our algorithm can be used to estimate energy across a set of components or dissipated power in $\textsf{polylog}(N)$ time, a problem that we prove is BQP-hard and therefore unlikely to be efficiently solved by classical algorithms. The main challenge in simulating the dynamics of RLC circuits is that they are governed by differential-algebraic equations (DAEs), a coupled system of differential equations with hidden algebraic constraints. Consequentially, existing quantum algorithms for ordinary differential equations cannot be directly utilized. We therefore develop a quantum DAE solver for simulating the time-evolution of linear DAEs. For RLC circuits, we employ modified nodal analysis to create a system of DAEs compatible with our quantum algorithm. We establish BQP-hardness by demonstrating that any network of classical harmonic oscillators, for which an energy-estimation problem is known to be BQP-hard, is a special case of an LC circuit. Our work gives theoretical evidence of quantum advantage in simulating RLC circuits and we expect that our quantum DAE solver will find broader use in the simulation of dynamical systems.

J.-P. Amiet, M. B. Cibils

An overview of isomorphic correspondences between spin algebra and algebras of functions on the sphere is given. In addition, the authors show that in one case the induced product between the spin function and another function is expressible by the action of a first-order differential operator correcting the ordinary product. This deformation leads to the geometric quantization of classical spin. The classical limit is studied for a class of correspondences.

V Sinivee