BEGIN:VCALENDAR VERSION:2.0 PRODID:-//linuxsoftware.nz//NONSGML Joyous v1.4//EN BEGIN:VEVENT SUMMARY:Solving Nonlinear Equations and Correcting Errors on Quantum Compu ters DTSTART:20260715T123000Z DTEND:20260715T140000Z DTSTAMP:20260710T224631Z UID:021eabd6-c77b-4968-b845-624980c81697 SEQUENCE:2 CREATED:20260709T083322Z DESCRIPTION:Quantum computing takes advantage of inherent properties of qu antum mechanics to perform computations that would be intractable on class ical 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 intrinsic ally connected research directions. We develop a quantum algorithm for sol ving certain nonlinear differential equations\, and propose quantum error correction methods suitable for both near-term noisy intermediate-scale qu antum (NISQ) devices and future fault-tolerant quantum computers. Our firs t contribution is a quantum algorithm for the forward scattering transform . This spectral analysis technique computes scattering data from the initi al conditions of nonlinear integrable systems\, and is the first and most computationally demanding step of the inverse scattering transform\, a cla ssical method that decomposes certain nonlinear partial differential equat ions into sequences of linear steps. Our algorithm achieves a quadratic sp eedup in the spatial discretization compared to classical methods. Additio nally\, it applies broadly to entire hierarchies of integrable nonlinear e quations sharing the same spatial Lax operator. As a concrete demonstratio n\, we compute the reflection coefficients for the Korteweg-de Vries equat ion\, a prototypical nonlinear wave equation with applications in fluid me chanics\, plasma physics\, and other fields. Our second contribution is qu antum GRAND (Guessing Random Additive Noise Decoding)\, to our knowledge t he first practical maximum-likelihood decoder for quantum random linear co des. These codes achieve optimal error correction performance asymptotical ly\, but have historically been impractical because their lack of algebrai c structure prevented the development of efficient decoders. Rather than r elying on code structure\, QGRAND exploits noise statistics to identify th e most likely error patterns. We extend QGRAND to the fault-tolerant setti ng by modeling the full error process during syndrome extraction\, and ach ieve a threshold error rate of approximately 5.5 × 10−3 . The technique s we develop also enable partial error correction matched to available har dware capabilities. These results confirm theoretically optimal quantum co des as a viable option for fault-tolerant quantum computing. LAST-MODIFIED:20260709T083338Z LOCATION:Online URL:http://df.vps.tecnico.ulisboa.pt/pt/eventos/solving-nonlinear-equation s-and-correcting-errors-on-quantum-computers/ X-ALT-DESC;FMTTYPE=text/html:

Quantum computing t akes advantage of inherent properties of quantum mechanics to perform comp utations that would be intractable on classical computers. However\, curre nt quantum devices are susceptible to noise\, and error correction techniq ues are needed to make reliable computation possible. With this thesis\, w e present contributions along two intrinsically connected research directi ons. We develop a quantum algorithm for solving certain nonlinear differen tial 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 q uantum algorithm for the forward scattering transform. This spectral analy sis technique computes scattering data from the initial conditions of nonl inear integrable systems\, and is the first and most computationally deman ding step of the inverse scattering transform\, a classical method that de composes certain nonlinear partial differential equations into sequences o f linear steps. Our algorithm achieves a quadratic speedup in the spatial discretization compared to classical methods. Additionally\, it applies br oadly to entire hierarchies of integrable nonlinear equations sharing the same spatial Lax operator. As a concrete demonstration\, we compute the re flection coefficients for the Korteweg-de Vries equation\, a prototypical nonlinear wave equation with applications in fluid mechanics\, plasma phys ics\, and other fields.

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

Rather than re lying on code structure\, QGRAND exploits noise statistics to identify the most likely error patterns. We extend QGRAND to the fault-tolerant settin g by modeling the full error process during syndrome extraction\, and achi eve a threshold error rate of approximately 5.5 × 10−3 . The techniques we develop also enable partial error correction matched to available hard ware capabilities. These results confirm theoretically optimal quantum cod es as a viable option for fault-tolerant quantum computing.

END:VEVENT END:VCALENDAR