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:20260711T223404Z UID:021eabd6-c77b-4968-b845-624980c81697 SEQUENCE:1 CREATED:20260709T083310Z DESCRIPTION: Quantum computing takes advantage of inherent properties of q uantum mechanics to perform computations that would be intractable on clas sical computers. However\, current quantum devices are susceptible to nois e\, and error correction techniques are needed to make reliable computatio n possible. With this thesis\, we present contributions along two intrinsi cally connected research directions. We develop a quantum algorithm for so lving certain nonlinear differential equations\, and propose quantum error correction methods suitable for both near-term noisy intermediate-scale q uantum (NISQ) devices and future fault-tolerant quantum computers. Our fir st contribution is a quantum algorithm for the forward scattering transfor m. This spectral analysis technique computes scattering data from the init ial conditions of nonlinear integrable systems\, and is the first and most computationally demanding step of the inverse scattering transform\, a cl assical method that decomposes certain nonlinear partial differential equa tions into sequences of linear steps. Our algorithm achieves a quadratic s peedup in the spatial discretization compared to classical methods. Additi onally\, it applies broadly to entire hierarchies of integrable nonlinear equations sharing the same spatial Lax operator. As a concrete demonstrati on\, we compute the reflection coefficients for the Korteweg-de Vries equa tion\, a prototypical nonlinear wave equation with applications in fluid m echanics\, plasma physics\, and other fields. Our second contribution is q uantum GRAND (Guessing Random Additive Noise Decoding)\, to our knowledge the first practical maximum-likelihood decoder for quantum random linear c odes. These codes achieve optimal error correction performance asymptotica lly\, but have historically been impractical because their lack of algebra ic structure prevented the development of efficient decoders. Rather than relying on code structure\, QGRAND exploits noise statistics to identify t he most likely error patterns. We extend QGRAND to the fault-tolerant sett ing by modeling the full error process during syndrome extraction\, and ac hieve a threshold error rate of approximately 5.5 × 10−3 . The techniqu es we develop also enable partial error correction matched to available ha rdware capabilities. These results confirm theoretically optimal quantum c odes as a viable option for fault-tolerant quantum computing. LAST-MODIFIED:20260709T083310Z LOCATION:Online URL:http://df.vps.tecnico.ulisboa.pt/en/events/solving-nonlinear-equations -and-correcting-errors-on-quantum-computers/ X-ALT-DESC;FMTTYPE=text/html:

Quantum computing takes advantage of inherent properties of quantum mechanics to perform com putations that would be intractable on classical computers. However\, curr ent quantum devices are susceptible to noise\, and error correction techni ques are needed to make reliable computation possible. With this thesis\, we present contributions along two intrinsically connected research direct ions. We develop a quantum algorithm for solving certain nonlinear differe ntial equations\, and propose quantum error correction methods suitable fo r both near-term noisy intermediate-scale quantum (NISQ) devices and futur e fault-tolerant quantum computers.

Our first contribution is a quantum algorithm for the forward scattering transform. This spectral anal ysis technique computes scattering data from the initial conditions of non linear integrable systems\, and is the first and most computationally dema nding step of the inverse scattering transform\, a classical method that d ecomposes 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 b roadly to entire hierarchies of integrable nonlinear equations sharing the same spatial Lax operator. As a concrete demonstration\, we compute the r eflection coefficients for the Korteweg-de Vries equation\, a prototypical nonlinear wave equation with applications in fluid mechanics\, plasma phy sics\, and other fields.

Our second contribution is quantum GRAN D (Guessing Random Additive Noise Decoding)\, to our knowledge the first p ractical maximum-likelihood decoder for quantum random linear codes. These codes achieve optimal error correction performance asymptotically\, but h ave historically been impractical because their lack of algebraic structur e 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.

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