Quantum Singular Value Transformation for Solving the Time-Dependent Maxwell's Equations

This work presents a quantum algorithm for solving linear systems of equations of the form $\mathbf{A}{\frac{\mathbf{\partial f}}{\mathbf{\partial x}}} = \mathbf{B}\mathbf{f}$, based on the Quantum Singular Value Transformation (QSVT). The algorithm uses block-encoding of $A$ and applies an 21st-degree polynomial approximation to the inverse function $f(x) = 1/x$, enabling relatively shallow quantum circuits implemented on 9 qubits, including two ancilla qubits, corresponding to a grid size of 128 points. Phase angles for the QSVT circuit were optimized classically using the Adagrad gradient-based method over 100 iterations to minimize the solution cost. This approach was simulated in PennyLane and applied to solve a 1D benchmark case of Maxwell's equations in free space, with a Gaussian pulse as the initial condition, where the quantum-computed solution showed high fidelity of more than 99.9% when compared to the normalized classical solution. Results demonstrate the potential of QSVT-based linear solvers on simulators with full quantum state access. However, practical hardware implementations face challenges because accessing the complete quantum state is infeasible. This limitation restricts applicability to cases where only $O({poly}(n))$ observables are needed. These findings highlight both the promise and current limitations of using quantum algorithms, such as QSVT, to solve linear systems of equations, and they point to the need for the development of measurement-efficient algorithms for near-term quantum devices.