## Math 690: Quantum Scientific Computing (Spring 2024)

Instructor: Di Fang         Syllabus

Please find the Course webpage on Canvas! This page is generated from Canvas and intended solely for maintaining a record of the teaching schedule and may not be consistently updated.

Please find the references (textbooks) for the course in the syllabus.

The chapters are:

• Ch1 Basic QC Glossary (states, gates, measurements; relationship between quantum computing v.s. classical computing; no cloning theorem; EQA)
• Ch2 Hamiltonian Simulation & Trotterization
• Ch3 QFT & Quantum Phase Estimation (including applications of Trotter + QPE/QEEP, such as Hamiltonian learning, and HHL)
• Ch4 Grover Search & Amplitude Amplification
• Ch5 Block-encoding and its applications (including LCU, Qubitization, QSP/QSVT, optimal Hamiltonian simulation, QSLA by QSVT, non-unitary dynamics)

Tentative Course Plan:

Slides for the first few lectures (basic QC Glossary): basicQC.pdf
Some handwritten notes may be posted (not guaranteed to be typo-free); or References will be provided.

 Week Wednesday Friday 1 Course Policy; Philosophy/Motivation; History; Start of basic QC Glossary (braket notations)   [Lecture Notes] 2 QC Glossary continued: Quantum States (qubits); Unitary operators (quantum gates); No-cloning theorem (simple version) [Lecture Notes] common quantum gates; Universal gate sets; Solovay-Kitaev Theorem;  Quantum Circuits [Lecture Notes]  (References on the proof of Solovay-Kitaev Theorem: [Childs Ch2.3], [Nielsen-Chuang Appendix 3], and a mathematically rigor proof with all gaps filled.) 3 Measurements (measurement in different basis, partial measurement, observable); SWAP test; Relationship of Quantum Computing v.s. Classical Computing (reversible classical computing) [Lecture Notes] Garbage and Uncompute; Exponential Quantum Advantage; No-cloning theorem (full-version) [Lecture Notes] 4 Hamiltonian Simulation and Trotterization; different error metrics (special case v.s. worst case); no-fast-forwarding theorem [Lecture Notes] (Reference on no-fast-forwarding theorem: [Kothari's MS thesis] Analysis of Trotterization; Commutator Scaling; Examples [Lecture Notes] 5 Watch a talk of your choice and/or think about what topic you'd like to work on for the course project. Helpful resources (to find inspiration for project topics!) Phase kickback; Fourier transform on the Boolean cubic; Deutsch-Jozsa Algorithm; Bernstein-Vazirani Algorithm [Lecture Notes] 6 Hadamard test; Quantum Fourier transform; Quantum Phase Estimation [Lecture Notes] Analysis of the textbook version QPE, and a brief review of literature on the state-of-the-art QPE and QEEP results. [Lecture Notes] 7 Remarks of QPE (A brief review of literature on the state-of-the-arts QPE & QEEP results; How to get $$\log(\delta^{-1})$$ cost dependence);A taste of Hamiltonian learning and Heisenberg limit (one-qubit and two-qubit case)[Lecture Notes] Further reading/watching of QPE & QEEP. Many-body Hamiltonian learning;QLSP (Quantum Linear System Problem) and HHL (the Harrowâ€“Hassidimâ€“Lloyd algorithm)[Lecture Notes] 8 More on Controlled Rotation; Classical arithmetic circuit;Grover search; upper bound (Perspective  1) [Lecture Notes] Grover search --Analysis of upper bound (Perspective 2&3); Amplitude Amplification. [Lecture Notes] 9 Lower bound of Grover; Amplitude Estimation [Lecture Notes] Block-encoding, definition and understandings; Properties; linear combination of unitaries (LCU) - simple case [Lecture Notes] 10 Happy Spring Break! 11 linear combination of unitaries (LCU) - general case; near-optimal Hamiltonian simulation by truncated Taylor series; Motivation for OAA [Lecture Notes] Oblivious Amplitude Amplification (OAA); QSP (Quantum Signal Processing) and QSVT (quantum singular value tranformation); Optimal Hamiltonian Simulation by QSVT [Lecture Notes] 12 Other QSLAs -- talk recording by Robin Kothari  (using LCU + Chebyshev polynomial or Fourier approach for QLSA; Optional) Improved QLSA (Quantum Linear System Algorithm by QSVT; A dive into the idea of qubitization. 13 Prep for course project Prep for course project 14 Quantum (linear) Differential Equation Solvers -- QLSA-based; time-marching and LCHS based (post-QLSA algorithms) Project presentation(subject to change) 15 Project presentation Project presentation