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Fourier transform-based linear combination of Hamiltonian simulation
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abstract
Linear combination of Hamiltonian simulation (LCHS) connects the general linear non-unitary dynamics with unitary operators and serves as the mathematical backbone of designing near-optimal quantum linear differential equation algorithms. However, the existing LCHS formalism needs to find a kernel function subject to complicated technical conditions on a half complex plane. In this work, we establish an alternative formalism of LCHS based on the Fourier transform. Our new formalism completely removes the technical requirements beyond the real axis, providing a simple and flexible way of constructing LCHS kernel functions. Specifically, we construct a different family of the LCHS kernel function, providing a $1.81$ times reduction in the quantum differential equation algorithms based on LCHS, and an $8.27$ times reduction in its quantum circuit depth at a truncation error of $\epsilon \le 10^{-8}$. Additionally, we extend the scope of the LCHS formula to the scenario of simulating linear unstable dynamics for a short or intermediate time period.
Forward citations
Cited by 4 Pith papers
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Optimal quantum simulation of linear non-unitary dynamics
A query-optimal quantum algorithm for non-unitary linear dynamics using generalized LCHS with approximate exponential-decay kernels and exponentially convergent uniform quadrature.
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Optimal Quantum Eigenvalue Transformation via Linear Combinations of Hermitian Matrices
The paper derives the exact angular projection A^m = (2/N)Σ_j e^{imθ_j}T_m(Re(e^{-iθ_j}A)) and a quantum algorithm realizing matrix polynomial transforms with Θ(d) depth and optimal post-selection overhead.
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Quantum Simulation of Non-Hermitian Special Functions and Dynamics via Contour-based Matrix Decomposition
CBMD decomposes non-Hermitian operators via contour residues to enable optimal-query quantum simulation of first-order dynamics and special functions such as Bessel and Airy evolutions without requiring diagonalizability.
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Circuit-Efficient Randomized Quantum Simulation of Non-Unitary Dynamics with Observable-Driven and Symmetry-Aware Designs
A randomized compilation of LCHS for non-unitary dynamics, with an observable-driven variant and a symmetry-aware sampler, claims reduced ancilla and circuit depth at the cost of more repetitions.
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