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arXiv:2609.19676

Linear combination of Schrödingerization for quantum linear systems with optimal matrix-query complexity

Yin Yang, Yue Yu, Long Zhang

quant-phSubmitted Sep 17, 2026• Announced Sep 18, 2026• 0 citations
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Abstract

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 $\mathcal{O}(κ_A\log\frac1\varepsilon)$ without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.

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Schrödingerization for quantum linear systems problems

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We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded $L^2$ kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses $\mathcal O(κ_A\log(1/\varepsilon))$ queries to each original input oracle, with constant success probability and $\ell^2$ state error at most $\varepsilon$. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.

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Simultaneously Query-Optimal Quantum Linear-System Algorithm

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Quantum linear-system algorithms access the coefficient matrix through a block-encoding oracle $U_A$ and the right-hand side through a state-preparation oracle $U_b$. The optimal query complexities of these two oracles are known separately: $O(κ\log(1/\varepsilon))$ queries to $U_A$ and $O(κ/s)$ queries to $U_b$, where $s=α\|A^{-1}b\|$. Whether a single algorithm can achieve both bounds simultaneously has remained open. We resolve this question affirmatively. Given a constant-factor estimate of $s$, we give a quantum linear-system algorithm that prepares the normalized solution to error $\varepsilon$ using simultaneously $ O\left(κ\log\frac1\varepsilon\right)$ queries to $U_A$ and $O\left(κ/{s}\right)$ queries to $U_b$. The algorithm separates state preparation from precision refinement, so only the matrix-query count depends on the target accuracy. We also prove a matching same-instance worst-case lower bound: for every quantum linear-system algorithm and every solution-norm scale, there exists a single Hermitian instance on which that algorithm requires both $Ω(κ\log(1/\varepsilon))$ matrix queries and $Ω(κ/s)$ state-preparation queries. Thus the two oracle complexities can be optimized simultaneously, and the resulting bounds are jointly optimal.

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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.

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Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation: A Weyl Calculus Approach

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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.

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Quantum linear system algorithm with optimal queries to initial state preparation

Guang Hao Low, Yuan Su

Quantum algorithms for linear systems produce the solution state A − 1 | b ⟩ by querying two oracles: O A that block encodes the coefficient matrix and O b that prepares the initial state. We present a quantum linear system algorithm making Θ ( 1 / p ) queries to O b , which is optimal in the success probability, and O ( κ log ⁡ ( 1 / p ) ( log ⁡ log ⁡ ( 1 / p ) + log ⁡ ( 1 / ϵ ) ) ) queries to O A , nearly optimal in all parameters including the condition number and accuracy. Notably, our complexity scaling of initial state preparation holds even when p is not known a priori . This contrasts with recent results achieving O ( κ log ⁡ ( 1 / ϵ ) ) complexity to both oracles, which, while optimal in O A , is highly suboptimal in O b as κ can be arbitrarily larger than 1 / p . In various applications such as solving differential equations, preparing ground states of operators with real spectra, and estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on p using a block preconditioning scheme to nearly match or outperform best p

arXiv:2410.18178•Quantum•Mar 23, 2026•1 citation
Quantum linear systems algorithms (QLSAs) aim to solve linear systems A\bbx=\bbbA\bb{x}=\bb{b}A\bbx=\bbb 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 \bbx\bb{x}\bbx 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(κAlog⁡1ε)\mathcal{O}(κ_A\log\frac1\varepsilon)O(κA​logε1​) without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.
We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded L2L^2L2 kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses O(κAlog⁡(1/ε))\mathcal O(κ_A\log(1/\varepsilon))O(κA​log(1/ε)) queries to each original input oracle, with constant success probability and ℓ2\ell^2ℓ2 state error at most ε\varepsilonε. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.
Quantum linear-system algorithms access the coefficient matrix through a block-encoding oracle UAU_AUA​ and the right-hand side through a state-preparation oracle UbU_bUb​. The optimal query complexities of these two oracles are known separately: O(κlog⁡(1/ε))O(κ\log(1/\varepsilon))O(κlog(1/ε)) queries to UAU_AUA​ and O(κ/s)O(κ/s)O(κ/s) queries to UbU_bUb​, where s=α∥A−1b∥s=α\|A^{-1}b\|s=α∥A−1b∥. Whether a single algorithm can achieve both bounds simultaneously has remained open. We resolve this question affirmatively. Given a constant-factor estimate of sss, we give a quantum linear-system algorithm that prepares the normalized solution to error ε\varepsilonε using simultaneously $ O\left(κ\log\frac1\varepsilon\right)$ queries to UAU_AUA​ and O(κ/s)O\left(κ/{s}\right)O(κ/s) queries to UbU_bUb​. The algorithm separates state preparation from precision refinement, so only the matrix-query count depends on the target accuracy. We also prove a matching same-instance worst-case lower bound: for every quantum linear-system algorithm and every solution-norm scale, there exists a single Hermitian instance on which that algorithm requires both Ω(κlog⁡(1/ε))Ω(κ\log(1/\varepsilon))Ω(κlog(1/ε)) matrix queries and Ω(κ/s)Ω(κ/s)Ω(κ/s) state-preparation queries. Thus the two oracle complexities can be optimized simultaneously, and the resulting bounds are jointly optimal.
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.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O~(νLt/ε)\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)O(νLt/ε), where the constant ννν depends on the problem parameters, L\mathcal{L}L involves a time integral of upper bounds on the norms of evolution operators, and εεε is the error. In particular, νLν\mathcal{L}νL is linear in ttt for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε)Ω(ν\mathcal{L} t/ε)Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
Linear combination of Hamiltonian simulation (LCHS) provides an efficient method for implementing matrix exponentials e−tAe^{-tA}e−tA on quantum computers. In this paper, we develop LCHS formulas for computing general matrix functions f(A)f(A)f(A) when fff is analytic on the numerical range of AAA, with AAA 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(log⁡1ε)\mathcal{O}(\log\frac{1}ε)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 ddtψ(t)=−A(t)ψ(t)\frac{\mathrm{d}}{\mathrm{d} t} ψ(t) = -A(t)ψ(t)dtd​ψ(t)=−A(t)ψ(t), for which we achieve a 2.1×2.1\times2.1× cost reduction over prior art.
Quantum algorithms for linear systems produce the solution state A − 1 | b ⟩ by querying two oracles: O A that block encodes the coefficient matrix and O b that prepares the initial state. We present a quantum linear system algorithm making Θ ( 1 / p ) queries to O b , which is optimal in the success probability, and O ( κ log ⁡ ( 1 / p ) ( log ⁡ log ⁡ ( 1 / p ) + log ⁡ ( 1 / ϵ ) ) ) queries to O A , nearly optimal in all parameters including the condition number and accuracy. Notably, our complexity scaling of initial state preparation holds even when p is not known a priori . This contrasts with recent results achieving O ( κ log ⁡ ( 1 / ϵ ) ) complexity to both oracles, which, while optimal in O A , is highly suboptimal in O b as κ can be arbitrarily larger than 1 / p . In various applications such as solving differential equations, preparing ground states of operators with real spectra, and estimating and transforming eigenvalues of non-normal matrices, we can further improve the dependence on p using a block preconditioning scheme to nearly match or outperform best p