Optimization Based Approach for Quantum Signal Processing and Its Energy Landscape
summary
An optimization-based algorithm reliably finds QSP phase factors at standard double precision for degrees beyond 10,000, outperforming algebraic root-finding that needs very high precision. The energy landscape has finite global minimizers (via Laurent-factorization) and a distinguished maximal solution — the Chebyshev “base factor” — that lies close to the simple initial guess and yields degree-independent local Hessian bounds and fast local convergence. A sweeping optimizer exploiting matrix product state structure shows rapid practical convergence and a conjectured decay of phase coefficients matching Chebyshev/Fourier decay.
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chapters (22)
- 0:00Intro
- 0:20Joint work with
- 0:50Outline
- 1:20Grand unification of quantum algorithms
- 2:23Quantum signal processing
- 4:45Goal of OSP (real case)
- 7:25Algorithms for finding phase factors
- 10:12Optimization based formulation
- 11:57Symmetric QSP
- 13:33Example: solve linear systems
- 14:48Applications
- 15:50Streamlining the process of finding phase factors
- 17:11Optimization landscape
- 19:20Uniqueness of symmetric phase factor
- 20:55Global minimizer and (PQ) pair
- 24:01Key: Laurent polynomials
- 27:30Condition number and the magnitude
- 33:54Matrix product state structure of QSP
- 35:07Gradient calculation
- 35:34Fast convergence (very few sweeps)
- 36:04Convergence from
- 41:07Open question: decay behavior of the phase sequen
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