Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials $e^{-tA}$ on quantum computers. In this paper, we develop LCHS formulas for computing general matrix functions $f(A)$ when $f$ is analytic on the numerical range of $A$, with $A$ possibly non-normal. The essential technical tool is Weyl calculus, which reduces the construction of LCHS formulas for noncommuting operators to scalar Fourier approximation problems. Our construction yields a quantum eigenvalue transformation algorithm with optimal $\mathcal{O}(\log\frac{1}ε)$ query complexity scaling. Furthermore, our Weyl-calculus-based theory gives rise to an ansatz-free convex optimization framework that directly produces discrete LCHS formulas. This circumvents the inefficiencies of traditional quadrature rules and yields formulas highly optimized for coherent implementation on quantum computers. In addition, both our theory and optimization framework apply to the simulation of time-dependent dissipative ODE $\frac{\mathrm{d}}{\mathrm{d} t} ψ(t) = -A(t)ψ(t)$, for which we achieve a $2.1\times$ cost reduction over prior art.
Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials e−tA on quantum computers. In this paper, we develop LCHS formulas for computing general matrix functions f(A) when f is analytic on the numerical range of A, with A possibly non-normal. The essential technical tool is Weyl calculus, which reduces the construction of LCHS formulas for noncommuting operators to scalar Fourier approximation problems. Our construction yields a quantum eigenvalue transformation algorithm with optimal O(logε1) query complexity scaling. Furthermore, our Weyl-calculus-based theory gives rise to an ansatz-free convex optimization framework that directly produces discrete LCHS formulas. This circumvents the inefficiencies of traditional quadrature rules and yields formulas highly optimized for coherent implementation on quantum computers. In addition, both our theory and optimization framework apply to the simulation of time-dependent dissipative ODE dtdψ(t)=−A(t)ψ(t), for which we achieve a 2.1× cost reduction over prior art.
We discover two complementary linear-combination-of-Hermitian-matrices (LCHM) formulations to achieve a general non-normal matrix eigenvalue transformation g(A). Firstly, for A=L+iH with Hermitian L and H, the vanilla LCHM formula represents g(A) as a kernel integral of g(i(H+kL)), and it contains linear-combination-of-Hamiltonian-simulation (LCHS) [An, Liu, Lin, Phys. Rev. Lett. 2023] as the special case for matrix exponentials. Secondly, for the angular Hermitian Xθ=cosθL+sinθH, the Weyl LCHM formula expresses g(A) via integrating g(eiθ(Xθ±i(I−Xθ2)1/2)). For the matrix power g(A)=Am, the Fourier projection of Weyl LCHM gives Am=π2∫0πeimθTm(Xθ)dθ=N2∑j=0N−1eimθjTm(Xθj),θj=Nπj,for every N>m with Chebyshev polynomial of Hermitian Tm(Xθ) and N samples. The discrete formula is exact, introduces no truncation and angular quadrature error, and offers O(1) post-selection weights.
LCHM formulas lead to new quantum eigenvalue transformation (QET) algorithms. For a degree-d polynomial pd(A) on ∣ψ⟩, our QET algorithm can achieve optimal Θ(d) circuit depth and optimal O(∣∣pd∣∣∞/∣∣pd(A)∣ψ⟩∣∣) post-selection repetitions. LCHM-based QETs unify various quantum linear algebraic problems with near-optimal O(dlog(d/ε)) Clifford+T gates, including driven ODEs (reduced to standard LCHS), iterative methods, resolvents, log(I+A), (λI+A)ν, Sign and ReLU transforms, and Faber approximation on noncircular domains.
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for non-normal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation (LCHS) method [An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, A−k, and the exponential of the matrix inverse, e−A−1. The latter can be interpreted as the solution of a mass-matrix differential equation of the form Au′(t)=−u(t). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting A, reducing the computational complexity.
Linear combination of Hamiltonian simulation (LCHS) provides an asymptotically optimal framework for the quantum simulation of linear non-unitary dynamics. The LCHS approach approximately implements the Peano--Baker propagator through an integral representation determined by a kernel function. We establish an LCHS approximation theorem for a broadened class of admissible kernel functions, defined by natural complex-analytic conditions. In the standard time-independent block-encoding access model, we explicitly construct, for each target error ε, the unique kernel function in this class minimizing the query-cost functional and show that the minimum of this functional has the asymptotic expansion π2e(logε1−loglogε1+o(1)) as ε↓0. The provably optimal leading coefficient 2e/π improves by a factor of approximately 2.1 on the coefficient obtained empirically by Low and Somma through numerical minimization over their family of optimally scaling kernel functions. We construct a spectrally convergent quadrature with explicit bounds on operator error and linear combination of unitaries (LCU) subnormalization, proving that this asymptotic improvement persists after discretization into a finite linear combination of unitaries suitable for implementation. Our results are based on Hardy space theory and convex analysis.
Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity O(κAlogε1) without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.
Abstract. Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, [Formula: see text], and the exponential of the matrix inverse, [Formula: see text]. The latter can be interpreted as the solution of a mass-matrix differential equation of the form [Formula: see text]. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting [Formula: see text], thereby reducing the computational complexity.
The Linear Combination of Hamiltonian Simulation (LCHS) framework simulates dissipative linear dynamics by representing time evolution as an integral over unitary operators, which is discretized by quadrature and implemented via Hamiltonian simulation. While existing analyses achieve near-optimal scaling in time and precision using norm-based quantities of the dissipative generator, we show that implementing the Hamiltonian simulation steps with Multi-Product Formulas (MPFs) yields commutator-sensitive error and complexity bounds. We demonstrate that the quadrature rule affects not only discretization error but also commutator structure and query complexity. This dependence is quantified through post-quadrature analysis for abstract MPF error profiles and for general time-independent and local Hamiltonians using known commutator-sensitive MPF error estimates. We compare uniform trapezoidal and free-scale sinh--sinh quadrature, showing improved quadrature-cardinality scaling for the latter, and illustrate the framework with applications to fractional diffusion, advection--diffusion, and open quantum systems.