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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 10 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 Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation: A Weyl Calculus Approach
Develops Weyl-calculus-based LCHS formulas for analytic f(A) yielding O(log 1/ε) quantum eigenvalue transformation and 2.1× cheaper time-dependent ODE simulation.
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Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS
Quantum circuit framework for advection-diffusion PDEs with Robin and periodic boundary conditions via LCHS, including LCU error analysis and gate complexity showing potential quantum advantage in high dimensions.
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Quantum Eigenvalue Transformations for Arbitrary Matrices
n-regular block encodings let QSP apply degree-n polynomials directly to the eigenvalues of any square matrix, with an efficient conversion from standard block encodings.
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Quantum Simulation of Non-Unitary Dynamics via Amplitude-Phase Separation
Introduces Amplitude-Phase Separation (APS) decomposition for quantum simulation of non-unitary dynamics, with complementary error scaling advantages in time-independent cases and unification of prior methods like LCH...
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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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Quantum Simulation of Non-Hermitian Special Functions and Dynamics via Contour-based Matrix Decomposition
CBMD decomposes non-Hermitian evolution operators into Hermitian LCU terms via a matrix residue theorem, matching known optimal query bounds and offering a route to polynomial matrix functions.
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From Meta Idea to Advanced Mathematical Discovery -- Human-AI Co-Discovery of Sign-Embedding Quantum Algorithms
Human-AI collaboration expanded a meta-idea on rational approximation into sign-embedding quantum algorithms for matrix problems, with humans retaining final judgment on routes and refinements.
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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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