Exact and Efficient Circuit Construction for Block Encoding Matrix Polynomials

Abstract:We propose a direct and stable circuit compiling algorithm that explicitly and exactly constructs block encodings of matrix polynomials for Hermitian matrices. In a unified framework, our algorithm is directly applicable to both standard input models: block encoding and Hamiltonian simulation. For a target polynomial of degree $d$, our classical algorithm achieves a near-optimal time complexity of $\mathcal{O}(d\log d)$. Numerical results confirm this asymptotic scaling for polynomial degrees up to $10^7$ in about a minute on a standard CPU. We achieve this by developing a general-purpose diagonal block encoding of function values, which bridges standard quantum-state preparation techniques with an interpolation-based QSP framework.

Quantum Physics (quant-ph); Numerical Analysis (math.NA)

arXiv:2608.15161 [quant-ph]

(or arXiv:2608.15161v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.15161 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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