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arXiv:2509.16777

Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation

Park, Seonggeun

Published Sep 20, 2025• 0 citations
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Abstract

Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order $O(M^{-3/2})$, up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where $M$ denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple $(F_h, |r_h\rangle, \langle l_h|)$ can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from $O(\log M)$ to $O((\log M)^2)$, depending on the stage of the pipeline.

DOI
10.48550/arxiv.2509.16777

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We introduce a universal dilation framework to simulate arbitrary non-Hermitian dynamics x ˙ = L ( t ) x on unitary quantum processors. By imposing algebraic moment-matching conditions on an ancilla, our framework generates diverse families of dilation schemes that unify existing methods and broaden the design space for hardware-aware implementations. A tight-binding dilation that maps the ancilla to a 1D lattice with nearest-neighbor hopping is derived, overcoming the connectivity bottlenecks of dilation schemes. This construction handles general dynamics, including gain, without rescaling and achieves near-optimal complexity, as demonstrated by benchmarks on dissipative wave propagations.
Nonautonomous linear ordinary differential equations of the form v˙(t)=A(t) v(t)\dot{v}(t) = A(t)\, v(t)v˙(t)=A(t)v(t), where A(t)A(t)A(t) is non-skew-symmetric, are often used to describe nonunitary dynamics in a variety of fields that range from open quantum system dynamics to economic modeling. Because quantum computing hardware is designed to natively implement unitary transformations, existing algorithms for solving such equations on quantum hardware are based on the assumption that the nonunitary propagator is known, and use dilation techniques to embed the nonunitary dynamics within the unitary dynamics of a larger system. However, with the exception of cases where the nonunitary propagator is known in closed form, it needs to be calculated and manipulated on a classical computer at each time step. In this paper, we propose a quantum algorithm that does not require a priori knowledge of the explicit nonunitary propagator and effectively performs the dilation on the quantum hardware. Our algorithm combines a dilation scheme that uses singular value decomposition (SVD) to write the nonunitary propagator as a sum of unitaries with simulating the dynamics of the SVD factors on the quantum hardware. The population-only time-convolutionless quantum master equation describing photoinduced charge transfer in a solvated molecular triad is used as a demonstrative example of the applicability of the algorithm and its sensitivity to noise.
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We present a universal framework for simulating NNN-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs dXt=A(t)Xt dt+∑j=1JBj(t)Xt dWtj dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j dXt​=A(t)Xt​dt+∑j=1J​Bj​(t)Xt​dWtj​ to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.
The Schrödingerisation method combined with the autonomozation technique in \cite{cjL23} converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into two higher dimension. Despite the success of Schrödingerisation techniques, they typically require the black box of the sparse Hamiltonian simulation, suitable for continuous-variable based analog quantum simulation. For qubit-based general quantum computing one needs to design the quantum circuits for practical implementation. This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor (PEC) boundary conditions and time-dependent source terms, based on Schrödingerization and autonomozation, with corresponding computational complexity analysis. Through initial value smoothing and high-order approximation to the delta function, the increase in qubits from the extra dimensions only requires minor rise in computational complexity, almost log⁡log⁡1/ε\log\log {1/\varepsilon}loglog1/ε where ε\varepsilonε is the desired precision. Our analysis demonstrates that quantum algorithms constructed using Schrödingerisation exhibit polynomial acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format.
ABSTRACT Nagy's unitary dilation theorem in the operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block‐encoding techniques, based on finite‐dimensional scenarios. In this study, we delve into the recently devised Schrödingerization approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of , which arises in wide‐ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerization approach is adaptable to both finite‐ and infinite‐dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite‐dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous‐variable version of the Schrödingerization procedure which makes it suitable for analog quantum computing. Furthermore, by discretizing continuous variables, the Schrödingerization method can also be effectively employed in finite‐dimensional scenarios suitable for qubit‐based quantum computing.