Abstract We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.
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.
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.
Linear combination of Hamiltonian simulation (LCHS) is an efficient method for simulating dissipative dynamics on quantum computers. The choice of the LCHS kernel controls the cost of the algorithm. We prove that the kernel family proposed by Ni and Ying~\cite{NY26} minimizes the LCHS query cost in the minimax sense. For target accuracy
ε, the minimum cost is asymptotically
π2e(lnε1−lnlnε1+o(1)), with optimal leading constant
π2e. Our lower-bound proof combines weak duality with a realization lemma that approximates extremal dual functions by matrix elements of dissipative evolutions.
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.