Riverlane’s Dr. Hari Krovi’s new research, published in Quantum Journal in February 2023 in collaboration with MIT and the US Department of Energy, represents significant progress in developing quantum algorithms that can be applied to a significantly larger class of linear and nonlinear differential equations with greater speed and accuracy. The researcher proposed a quantum approach that significantly generalized and improved over previous work for inhomogeneous linear and nonlinear ordinary differential equations (ODE).
Classical physics modeling has assisted us in understanding the origins of the cosmos and observing profound space occurrences that occurred light years ago. However, it has still posed a challenge for effectively modeling highly unpredictable physical processes on Earth. From modeling intricate weather patterns to enhanced aerodynamics during turbulence, depicting these models has still been challenging.
Theoretically, the more nonlinear the system, the more complicated the differential equations. The solution is to solve these equations with more powerful quantum algorithms. However, such algorithms have hitherto been limited in speed, accuracy, and functionality.
Quantum Algorithms for differential equations
Several articles have proposed quantum methods for differential equations. A study provided a quantum approach for solving linear inhomogeneous equations using high-order methods. Another study employed truncated Taylor series, resulting in an exponential improvement in the dependence of the solution error. Another spectrum approach was used to solve time-dependent linear differential equations using a quantum algorithm.
A recent study on a quantum approach for solving dissipative nonlinear differential equations utilizing the Carleman linearization has been published. This approach is efficient when the ratio of nonlinearity to dissipation is less than one. When the time scale of the simulation is short, a quantum algorithm for nonlinear differential equations is described in another experiment. Unfortunately, this was derived heuristically rather than rigorously.
The practical application cases apply to both non-linear and linear systems. They include precise simulations of fluid dynamics in viscosity and turbulence, weather modeling, and plasma physics simulations applied to inertial confinement fusion, which aids in creating clean nuclear energy.
The complexity of Diagonalization
According to observations, using the closest diagonalizable matrix results in an exponentially worse inaccuracy. As a result, the authors chose to extend over a wide range of non-diagonalizable matrices while maintaining the logarithmic dependence on error. The researchers defined the matrices for which the technique is efficient by limiting the gate and query complexity. Diagonalization is required to derive a bound on the condition number and a bound on the solution error. To avoid the need for diagonalization, the study presented distinct and better analyses for both components.
Use of Non-Diagonalizable Approach
The researcher initially provided a quantum technique for solving time-independent linear inhomogeneous equations, expanding their technique to incorporate numerous non-diagonalizable and even singular matrices that their algorithm does not cover, making it possible to do so.
The researcher also used the truncated Taylor series to build a quantum linear system whose solution yields a quantum state proportionate to the linear differential equation solution. They also use the “ramp” to increase the likelihood of success with the approach. According to the researcher, the linear system was created using block encoding techniques since it is easier to evaluate using the techniques established here. Furthermore, they used their linear ODE technique to solve nonlinear differential equations via Carleman linearization, which improved the dependence on the error caused by using the Euler method. Carleman linearization does not require that the linear ODE be diagonalizable. This means that the linear ODE algorithm used to solve Carleman linearized ODEs may deal with non-normal and non-diagonalizable matrices.
The study concluded that they had exponentially enhanced the dependency on the error of prior studies in nonlinear differential equations. Their approach is also efficient for nonlinear differential equations when F1 is non-normal (and non-diagonalizable) due to a negative log norm. The study also emphasized that expanding the nonlinear differential equations algorithm to more general applications would be interesting; however, before that, one must broaden the definition of R and demonstrate Carleman linearization’s convergence to these more general matrices.
The study builds on a previous publication published in the PNAS journal by Krovi and colleagues in August 2021.
Read the full journal article here.