Optimal Scaling Quantum Linear Systems Solver via Discrete Adiabatic Theorem

Explore an advanced quantum computing lecture on solving linear systems using a discrete adiabatic theorem approach. Delve into the development of an asymptotically optimal quantum algorithm with linear complexity in the condition number, matching known lower bounds. Examine the rigorous proof of the discrete adiabatic theorem, its application to quantum linear systems, and the algorithm's simplified implementation. Investigate the constant factors, gate count complexities, and potential applications. Compare this method to existing suboptimal approaches and understand its advantages in terms of precision and efficiency.

Intro
Why do we care?
Quantum linear systems problem
Complexity scaling
Continuous adiabatic algorithm
Adiabatic approach to QLSP
Non-symmetric case
Adiabatic walk
Norm of differences
Multistep gap
Discrete adiabatic theorem
Summation by parts formula
Contour integrals for bounds
Multiple eigenvalues problem
Numerical testing for constant factor
Filtering solution
LCU with two qubits
Putting it all together
Lower bound
Conclusions