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I am currently writing my Bachelor’s thesis on Quantum Algorithms for Solving Differential Equations. I have a background in both differential equations (including partial differential equations) and quantum algorithms from my coursework.

One approach I have come across is discretizing the differential equation and then solving the resulting linear system using a quantum algorithm such as HHL (Harrow-Hassidim-Lloyd). However, I have not studied HHL in my courses and would like to build a solid understanding of it and other quantum linear solvers.

I am looking for resources that cover:

1. Quantum Linear Solvers (e.g., HHL)

  • Intuitive explanations of how these algorithms work and why they provide exponential speedup.
  • Rigorous treatments with formal proofs of their complexity, error analysis, and conditions for speedup.

2. State-of-the-art Quantum Algorithms for Differential Equations

  • What are the latest quantum algorithms that use the discretization + quantum linear solver approach?
  • Any reviews or papers discussing the feasibility and limitations of quantum differential equation solvers.

3. General Resources or Books

  • While I suspect there may not be books specifically on "quantum algorithms for differential equations," any books or survey papers that provide an intuitive understanding of this topic would be helpful.

I would appreciate any recommendations, whether they are lecture notes, papers, surveys, or even book chapters that are relevant. Thanks in advance!

Omeglac
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1 Answers1

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As indicated, generally one of the best resources for finding up-to-date or nearly up-to-date state-of-the-art algorithms is S. Jordan's curated Quantum Algorithm Zoo.

  • The Zoo's section on "Solving Linear Differential Equations" lists about 20 or so references - including Berry et al.'s Quantum algorithm for linear differential equations with exponentially improved dependence on precision, which may be the state-of-the-art.

  • Berry et al. relies on the Quantum Linear Systems Algorithm - e.g., the HHL algorithm, for which, again, the Zoo provides a number of resources in a section titled "Linear Systems". This particular forum also has a lot of other links to various expositions on the HHL algorithm - some have liked Morell et al.'s Step-by-Step HHL Algorithm Walkthrough to Enhance the Understanding of Critical Quantum Computing Concepts. For HHL in particular, there are also a number of good videos - I myself learned much of the algorithm from Lloyd's discussion here, while Sev Gharibian's video here is a more formal derivation.

  • Much of the improvements in the algorithms come from improvements in the Hamiltonian simulation of the underlying system. Again, the Zoo has a lot of resources in a section titled "Approximation and Simulation Algorithms." Any video on the topic by Childs is very good - I liked this one, for example. His lecture notes are good as well.

Otherwise the question right now is pretty broad and may be hard to answer in any more focused manner.

Mark Spinelli
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