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Are there efficient quantum algorithms that given a d-sparse hessian $H \in \mathbb{C}^{N \times N}$ decompose it into a sum of unitaries (e.g. Pauli matrices)?

$$H = \sum_i^q a_i U_i$$

If an algorithm exists, how could I contruct a quantum circuit for it?

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  • $\begingroup$ are Hessian matrices arbitrary symmetric matrices, or do they have some further structure? $\endgroup$
    – glS
    Jul 5, 2022 at 11:22
  • $\begingroup$ They are arbitary in general. Though for my case real symmetric sparse matrices also suffice. They are constructed from Finite Element problems, I don't think that helps though. $\endgroup$
    – consthatza
    Jul 5, 2022 at 11:41
  • $\begingroup$ In the original VQLS paper, arxiv.org/pdf/1909.05820.pdf, there is a section (Appendix A) that showed how you can express a sparse matrices as LCU efficiently. $\endgroup$
    – KAJ226
    Jul 6, 2022 at 18:14
  • $\begingroup$ @KAJ226 I have that in mind. There are some oracles there which are assumed to be known. Do I have to figure them out based on linear algebra? $\endgroup$
    – consthatza
    Jul 7, 2022 at 10:12

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