Computational modeling of cellular systems, where reactants are governed by biochemical equations and physical representations, requires extensive classical computing resources. These limitations significantly constrain the system size and spatiotemporal scales of simulations. A key challenge lies in the "curse of dimensionality", where the number of possible reaction terms grows exponentially with the number of species, and the computation of reaction rates involving many-body interactions becomes intractable in polynomial time on classical computers. In this work, we introduce a quantum algorithmic framework designed to overcome these challenges, leveraging the architecture of quantum computing to simultaneously compute reaction rates and track the spatiotemporal dynamical evolutions of subcellular systems. We generalize the reaction-diffusion equation (RDE) for multiscale systems with arbitrary species count, encompassing higher-order interactions. Our approach achieves two principal quantum advantages: (i) an exponential quantum speedup in reaction-rate computation, contingent on the efficient preparation of polynomially accurate ground states on a quantum computer, and (ii)) a quadratic scaling in spatial grid points and polynomial scaling in the number of species for solving nonlinear RDEs, contrasting sharply with classical methods that scale exponentially with the system's degrees of freedom. To our knowledge, this represents the first efficient quantum algorithm for solving multiscale reaction-diffusion systems. This framework opens the door to simulations of biologically relevant subcellular processes across previously inaccessible spatial and temporal scales, with profound implications for computational biology, soft matter physics, and biophysical modeling.
Computational modeling of cellular systems, where reactants are governed by biochemical equations and physical representations, requires extensive classical computing resources. These limitations significantly constrain the system size and spatiotemporal scales of simulations. A key challenge lies in the "curse of dimensionality", where the number of possible reaction terms grows exponentially with the number of species, and the computation of reaction rates involving many-body interactions becomes intractable in polynomial time on classical computers. In this work, we introduce a quantum algorithmic framework designed to overcome these challenges, leveraging the architecture of quantum computing to simultaneously compute reaction rates and track the spatiotemporal dynamical evolutions of subcellular systems. We generalize the reaction-diffusion equation (RDE) for multiscale systems with arbitrary species count, encompassing higher-order interactions. Our approach achieves two principal quantum advantages: (i) an exponential quantum speedup in reaction-rate computation, contingent on the efficient preparation of polynomially accurate ground states on a quantum computer, and (ii)) a quadratic scaling in spatial grid points and polynomial scaling in the number of species for solving nonlinear RDEs, contrasting sharply with classical methods that scale exponentially with the system's degrees of freedom. To our knowledge, this represents the first efficient quantum algorithm for solving multiscale reaction-diffusion systems. This framework opens the door to simulations of biologically relevant subcellular processes across previously inaccessible spatial and temporal scales, with profound implications for computational biology, soft matter physics, and biophysical modeling.
High-dimensional fractional reaction-diffusion equations have numerous applications in the fields of biology, chemistry, and physics, and exhibit a range of rich phenomena. While classical algorithms have an exponential complexity in the spatial dimension, a quantum computer can produce a quantum state that encodes the solution with only polynomial complexity, provided that suitable input access is available. In this work, we investigate efficient quantum algorithms for linear and nonlinear fractional reaction-diffusion equations with periodic boundary conditions. For linear equations, we analyze and compare the complexity of various methods, including the second-order Trotter formula, time-marching method, and truncated Dyson series method. We also present a novel algorithm that combines the linear combination of Hamiltonian simulation technique with the interaction picture formalism, resulting in optimal scaling in the spatial dimension. For nonlinear equations, we employ the Carleman linearization method and propose a block-encoding version that is appropriate for the dense matrices that arise from the spatial discretization of fractional reaction-diffusion equations.
Whole-cell simulation, modeling all of a cell's functional systems over its life cycle, is an outstanding challenge in computational biology. Even the simplest living cell contains thousands of interacting proteins and metabolites (on the order of trillions of atoms) whose full functional dynamics spans roughly five orders of magnitude in space (nm to μm) and nearly nineteen in time (fs to hours). Further, many of the governing physical and chemical properties remain incompletely characterized. Simulating such complex systems at fully atomistic resolution over a full cell cycle is computationally intractable on classical architectures, raising a central question: Can quantum computing offer a viable path to whole-cell simulations that integrate molecular- and systems-level complexity? This Perspective examines the potential of quantum computing across three hierarchical scales: atomistic-molecular modeling, metabolic and regulatory networks, and whole-cell spatial modeling. We present a complexity analysis comparing classical and quantum algorithms for representative biological problems, identifying regimes of substantial theoretical speedup under specified algorithmic assumptions. We highlight algorithmic developments designed to leverage both near-term exploratory and fault-tolerant quantum architectures, and discuss practical bottlenecks: data encoding overhead, system conditioning, measurement constraints, and hybrid quantum-HPC integration. Together, these results outline a roadmap for quantum-accelerated whole-cell modeling and the biological insights such multiscale frameworks may eventually enable.
The Fokker-Planck equation models rare events across sciences, but blue{direct solution of the PDE is intractable for classical computers due to } its high-dimensional nature. Classical stochastic methods circumvent this curse-of-dimensionality, and serve as the de facto standard for practicing computational scientists. Quantum algorithms for such non-unitary dynamics often suffer from exponential decay in success probability. We introduce a quantum algorithm that overcomes this bottleneck for estimating reaction rates {and dynamical correlation functions more generally}. Using a sum-of-squares representation, we develop a Gaussian linear combination of Hamiltonian simulations (Gaussian-LCHS) to represent the non-unitary propagator with O(t∥H∥log(1/ε)) queries to its block encoding. Crucially, we pair this with {a} novel technique to directly estimate matrix elements without exponential decay. For η pairwise interacting particles discretized with N plane waves per degree of freedom, we estimate reactive flux to error ε using O((η5/2tβαV+η3/2t/βN)/ε) quantum gates, where αV=maxr∣V′(r)/r∣. We further prove that under comparable worst-case analytical guarantees, the sharpest classical bounds for estimating reaction rates via simulation of the associated overdamped Langevin dynamics scale as O(tη2eΩ(η)/ε4), yielding an exponential improvement in η, a quartic speedup in ε, and quadratic speedup in the time horizon t. While classical algorithms may outperform these bounds in practice, this work demonstrates a rigorous route toward quantum advantage for high-dimensional dissipative dynamics.
The description of quantum many-body dynamics is extremely challenging on classical computers, as it can involve many degrees of freedom. On the other hand, the time evolution of quantum states is a natural application for quantum computers that are designed to efficiently perform unitary transformations. In this paper, we study quantum algorithms for response functions, relevant for describing different reactions governed by linear response. We focus on nuclear-physics applications and consider a qubit-efficient mapping on the lattice, which can efficiently represent the large volumes required for realistic scattering simulations. For the case of a contact interaction, we develop an algorithm for time evolution based on the Trotter approximation that scales logarithmically with the lattice size, and is combined with quantum phase estimation. We eventually focus on the nuclear two-body system and a typical response function relevant for electron scattering as an example. We also investigate ground-state preparation and examine the total circuit depth required for a realistic calculation and the hardware noise level required to interpret the signal.
Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.