Solving Nonlinear Equations and Correcting Errors on Quantum Computers

Quantum computing takes advantage of inherent properties of quantum mechanics to perform computations that would be intractable on classical computers. However, current quantum devices are susceptible to noise, and error correction techniques are needed to make reliable computation possible. With this thesis, we present contributions along two intrinsically connected research directions. We develop a quantum algorithm for solving certain nonlinear differential equations, and propose quantum error correction methods suitable for both near-term noisy intermediate-scale quantum (NISQ) devices and future fault-tolerant quantum computers.

Our first contribution is a quantum algorithm for the forward scattering transform. This spectral analysis technique computes scattering data from the initial conditions of nonlinear integrable systems, and is the first and most computationally demanding step of the inverse scattering transform, a classical method that decomposes certain nonlinear partial differential equations into sequences of linear steps. Our algorithm achieves a quadratic speedup in the spatial discretization compared to classical methods. Additionally, it applies broadly to entire hierarchies of integrable nonlinear equations sharing the same spatial Lax operator. As a concrete demonstration, we compute the reflection coefficients for the Korteweg-de Vries equation, a prototypical nonlinear wave equation with applications in fluid mechanics, plasma physics, and other fields.

Our second contribution is quantum GRAND (Guessing Random Additive Noise Decoding), to our knowledge the first practical maximum-likelihood decoder for quantum random linear codes. These codes achieve optimal error correction performance asymptotically, but have historically been impractical because their lack of algebraic structure prevented the development of efficient decoders.

Rather than relying on code structure, QGRAND exploits noise statistics to identify the most likely error patterns. We extend QGRAND to the fault-tolerant setting by modeling the full error process during syndrome extraction, and achieve a threshold error rate of approximately 5.5 × 10−3 . The techniques we develop also enable partial error correction matched to available hardware capabilities. These results confirm theoretically optimal quantum codes as a viable option for fault-tolerant quantum computing.