Quantum singular value transformation is a quantum algorithm primitive that unifies all existing quantum algorithms into a single framework thus simplifying quantum algorithm design and implementation.^[1] It applies polynomial functions to the singular values of matrices.^[2]

Algorithm

Input: A matrix ${\displaystyle A}$ whose Singular value decomposition is ${\displaystyle A=W\Sigma V^{\dagger ))$ where ${\displaystyle \Sigma }$ are the singular values of A

Input: A polynomial ${\displaystyle P}$

Output: A unitary where ${\displaystyle P}$ has been applied to the singular values of ${\displaystyle A}$: ${\displaystyle {\begin{bmatrix}WP(\Sigma )V^{\dagger }&.\\.&.\end{bmatrix))}$

1. Prepare a unitary ${\displaystyle U}$ that encodes ${\displaystyle A}$ on the top left side of ${\displaystyle U}$, that is ${\displaystyle U={\begin{bmatrix}A&.\\.&.\end{bmatrix))}$
2. Initialize an ${\displaystyle n}$ qubit state ${\displaystyle |0\rangle ^{\otimes n))$
3. If the polynomial is odd, first apply ${\displaystyle {\tilde {\Pi ))_{\phi _{1))U}$ and then ${\displaystyle \prod _{k=1}^{\frac {d-1}{2))\Pi _{\phi _{2k))U^{\dagger }{\tilde {\Pi ))_{\phi _{2k+1))U}$ to ${\displaystyle |0\rangle ^{\otimes n))$
4. If the polynomial is even apply ${\displaystyle \prod _{k=1}^{\frac {d}{2))\Pi _{\phi _{2k}-1}U^{\dagger }{\tilde {\Pi ))_{\phi _{2k))U}$ to ${\displaystyle |0\rangle ^{\otimes n))$

^[1]