How to choose values of phi for Hamiltonian simulation with Quantum Singular Value Transform?

Question

I am reading the review, Grand Unification of Quantum Algorithms, which covers the area known as "Quantum Singular Value Transform (QSVT)."

I am really trying to understand it behind the context of Hamiltonian Simulation, and have read many papers and watched a few videos outlining the methods.

I understand that we can block encode an $N \times M$ matrix $H$ into a unitary $U$.

We can then construct $V_{\phi}(\theta) = \prod_{n=0}^{d - 1} R_\phi(\theta)$ for $\vec{\phi} \in \mathbb{R}^d$ where $R_\phi(\theta)$ is a rotation about axis $\phi$ by angle $\theta$.

This forms the matrix

$ \begin{bmatrix} P(a) & iQ(a)\sqrt{1-a^2} \\ iQ^*(a)\sqrt{1-a^2} & P^*(a) \end{bmatrix} $

where $deg(P), deg(Q) \leq d$ with a few more constraints that aren't important for now.

Then note that $e^{-iHt} = cos(Ht) - isin(Ht)$. This then allows us to break the problem into simulation of $cos(Ht)$ and $-isin(Ht)$.

We then use the Jacobi-Anger function expansion to approximate $cos(Ht)$ and $sin(Ht)$ up to some degree $d$.

These approximations are then used as the function $P(a)$ by picking some $\vec{\phi}$.

But how exactly would you pick the $\vec{\phi}$ for this case? I found some sources mentioning Remez-Type exchanges, but I can't find any implementations.

Thanks for any help!

Answer 1

• Appendix C in "Grand Unification of Quantum Algorithms" mentioned the pyqsp open source package.

• QSPPACK package contains an implementation of the Remez exchange algorithm. This package is associated with the paper Efficient phase-factor evaluation in quantum signal processing

• Angle Sequence package which implements the method described in Finding Angles for Quantum Signal Processing with Machine Precision