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A Block-Encoding Method for Constant and Variable Delay Linear Delay Differential Equations

Snayhin Sharma

Published Nov 26, 2025• 0 citations
Publisher

Abstract

Abstract Delay Differential Equations (DDEs) are used to model many systems in medicine, chemistry, and environmental science, and are characterized by the derivative of the function being dependent on the solution in both the present and the past. They are utilized in realistic systems where the transfer between states is not instantaneous, or there is delayed activation of some component. While much literature has been focused on creating quantum algorithms for ODEs, there is no such research considering DDEs. We extend the method of solving lin ear delay functions as a matrix equation to first-order DDEs by creating custom quantum block-encoding oracles for the coefficient matrix resulting from both constant and variable-delays. The findings now allow this new class of equations to be solved faster than linear time with quantum linear system solvers.

DOI
10.21203/rs.3.rs-8169166/v1

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Abstract Delay Differential Equations (DDEs) are used to model many systems in medicine, chemistry, and environmental science, and are characterized by the derivative of the function being dependent on the solution in both the present and the past. They are utilized in realistic systems where the transfer between states is not instantaneous, or there is delayed activation of some component. While much literature has been focused on creating quantum algorithms for ODEs, there is no such research considering DDEs. We extend the method of solving lin ear delay functions as a matrix equation to first-order DDEs by creating custom quantum block-encoding oracles for the coefficient matrix resulting from both constant and variable-delays. The findings now allow this new class of equations to be solved faster than linear time with quantum linear system solvers.
Non-Markovian dynamics is ubiquitous in both quantum and classical systems, but the numerical computation of the time-delay dynamics is demanding. In this work, we propose an efficient quantum algorithm for solving linear distributed delay differential equations and identify the condition under which it applies. Using the linear chain trick, the distributed delay differential equations can be embedded into ordinary differential equations augmented with auxiliary variables, when the kernel function is characterized by a phase-type distribution. Employing the Schrödingerization method, the resulting equations can be embedded into the Schrödinger equation and efficiently solved by Hamiltonian simulation. Although this embedding requires the augmented differential equation to be semi-stable, we show that it is satisfied if and only if the original distributed-delay differential equations are semi-stable. The query complexity to obtain the normalized solution state of the NNN-dimensional delay system ∣x(t)⟩≡x(t)/∣∣x(t)∣∣|\mathbf{x}(t)\rangle\equiv\mathbf{x(t)}/\vert\vert\mathbf{x}(t)\vert\vert∣x(t)⟩≡x(t)/∣∣x(t)∣∣ is O((st∣∣H∣∣max⁡+log⁡ε−1/log⁡log⁡ε−1)∣∣x(0)∣∣/∣∣x(t)∣∣)\mathcal{O}((st\vert\vert H\vert\vert_{\max}+\logε^{-1}/\log\logε^{-1})\vert\vert\mathbf{x}(0)\vert\vert/\vert\vert\mathbf{x}(t)\vert\vert)O((st∣∣H∣∣max​+logε−1/loglogε−1)∣∣x(0)∣∣/∣∣x(t)∣∣) with εεε, ggg, HHH, and sss being the allowable error, the dimension of the auxiliary variables associated with each kernel function, the Hamiltonian operator, and its sparsity, respectively. The gate complexity is given by this quantity multiplied by O(m+log⁡(N(1+gs)))\mathcal{O}(m+\log(N(1+gs)))O(m+log(N(1+gs))), where mmm is the number of precision bits. To demonstrate the efficacy of our method, we present its applications to the generalized master equation and to the Redfield equation of the dephasing model.
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Padé approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix exponentials. The proposed algorithm approximates the matrix exponential by the diagonal Padé approximation, which is then encoded into a large, block-sparse linear system and solved via quantum linear system algorithms (QLSA). The detailed quantum circuit is given based on quantum oracle access to the matrix, the inhomogeneous term, and the initial state. The complexity of the proposed algorithm is analyzed. Compared to the method based on Taylor approximation, which approximates the matrix exponential using a k -th order Taylor series, the proposed algorithm improves the approximation order k from two perspectives: 1) the explicit complexity dependency on k is improved, and 2) a smaller k suffices for the same precision. Numerical experiments demonstrate the advantages of the proposed algorithm comparing to other related algorithms.
Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.
Solving linear ordinary differential equations (ODEs) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed nonunitary dynamics into intrinsically unitary quantum circuits. In this Letter, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called nondiagonal density matrix encoding, which leverages the inherent nonunitary dynamics of Lindbladians to encode general linear ODEs into the nondiagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.
In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries (LCU\mathrm{LCU}LCU)-based block-encodings for the first-order derivative and Laplacian operators in the non-local N=2nN=2^nN=2n-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense LCU\mathrm{LCU}LCU amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm (QLSA\mathrm{QLSA}QLSA). For a ddd-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity O(dn3α(k)log⁡(1/ε)){O}(dn^3α^{(k)}\log(1/\varepsilon))O(dn3α(k)log(1/ε)), where α(k)α^{(k)}α(k) is the subnormalisation factor of the order-kkk SLAC block-encoding and ε\varepsilonε denotes the algorithmic approximation error.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.