Multivariate Quantum Signal Processing

@misc{low2026multivariate,
  title = {{Multivariate Quantum Signal Processing}},
  author = {Guang Hao Low},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01125},
  eprint = {2610.01125},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01125},
}

TY  - UNPB
TI  - Multivariate Quantum Signal Processing
AU  - Guang Hao Low
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01125
UR  - https://arxiv.org/abs/2610.01125
AB  - We develop Multivariate Quantum Signal Processing for designing matrix-valued transformations of multiple noncommuting nonnormal block-encoded matrices. Our framework equips quantum transducers with the analytic structure of multiport Infinite-Impulse-Response (IIR) filters and establishes a transfer-function methodology for designing quantum query algorithms. Efficiently computable analytic bounds that hold uniformly across the oracle promise yield efficient quantum circuits with logarithmic error dependence and near-unity success probability, paralleling the approximation-theoretic guarantees of univariate QSP/QSVT. We provide a complete characterization of achievable transfer functions and a principled library of analytic IIR components. Our modular designs greatly improve or resolve the per-oracle query complexity of open multi-oracle problems: (1) Hamiltonian simulation under H=∑_j H_j with weighted quasinorm cost ≈ t⟨λ⃗,C⃗⟩_1/2+𝒪(|C⃗|_1log(1/ε)) for block-encoding normalizations λ⃗ and costs C⃗; (2) sparse simulation with optimal time--norm Θ(√d|H|_1→2t) and additive error dependence; (3) solving linear systems and (4) differential equations under minimal matrix assumptions with optimal weighted state-preparation and multi-term cost; (5) ground-state preparation and energy estimation; (6) generalized eigenvalue problems. We apply these to exactly simulate downfolded Hamiltonians with leading cost t⟨λ⃗_eff,C⃗_eff⟩_1/2 further reduced by fractional dynamic correlation energy contributions from coarse-grained Hamiltonians, and show similar classical-data-input-reduction results for multiscale graph classification via Kron reduction. We also show the online generalization with anisotropic Kalman filtering of streaming quantum data.
ER  -