Hamilton-Jacobi equations and scalar conservation laws develop shocks and caustics in finite time, beyond which physically relevant solutions are selected by viscosity and entropy conditions. Since quantum computation natively implements linear unitary evolution, post-shock quantum algorithms require linear representations that respect these selection principles. This paper presents a self-contained account of two such selection-correct linear lifts. We first examine the quadratic Hamilton-Jacobi and viscous Burgers equations, where the Cole-Hopf transform maps the viscous regularization to the heat equation, yielding the correct vanishing-viscosity limit. We then address general convex Hamiltonians using entropy penalization and soft-min Hopf-Lax formulas to convert dynamic-programming minimizations into linear positive integral operators. We derive the discrete-time linear dynamics, prove soft-min error bounds, and demonstrate that discretization yields positive matrices with controllable operator norms. These linear maps are then explicitly embedded into unitaries via Halmos dilation, providing a direct route to quantum simulation compatible with Schrödingerization-based pipelines.
Hamilton-Jacobi equations and scalar conservation laws develop shocks and caustics in finite time, beyond which physically relevant solutions are selected by viscosity and entropy conditions. Since quantum computation natively implements linear unitary evolution, post-shock quantum algorithms require linear representations that respect these selection principles. This paper presents a self-contained account of two such selection-correct linear lifts. We first examine the quadratic Hamilton-Jacobi and viscous Burgers equations, where the Cole-Hopf transform maps the viscous regularization to the heat equation, yielding the correct vanishing-viscosity limit. We then address general convex Hamiltonians using entropy penalization and soft-min Hopf-Lax formulas to convert dynamic-programming minimizations into linear positive integral operators. We derive the discrete-time linear dynamics, prove soft-min error bounds, and demonstrate that discretization yields positive matrices with controllable operator norms. These linear maps are then explicitly embedded into unitaries via Halmos dilation, providing a direct route to quantum simulation compatible with Schrödingerization-based pipelines.
We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians. These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation. Our method is based on an entropy penalization method which generalizes the Cole-Hopf transform from quadratic to general convex Hamiltonians, allowing an approximation of viscous Hamilton-Jacobi dynamics by a discrete-time linear dynamics which approximates a linear heat-like parabolic equation, and thus can also extend to continuous-time dynamics. This makes the method suitable for quantum simulation. The validity of these results hold for arbitrary nonlinearity that correspond to convex Hamiltonians, and for arbitrarily long times, thus obviating a chief obstacle in most quantum algorithms for nonlinear partial differential equations. We provide quantum algorithms-both analog and digital-for extracting pointwise values, gradients, minima, and function evaluations at the minimizer of the viscosity solution, without requiring nonlinear updates or full state reconstruction.
We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians. These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation. Our method is based on an entropy penalisation method proposed by Gomes and Valdinoci, which generalises the Cole-Hopf transform from quadratic to general convex Hamiltonians, allowing a reformulation of viscous Hamilton-Jacobi dynamics by a discrete-time linear dynamics which approximates a linear heat-like parabolic equation, and can also extend to continuous-time dynamics. This makes the method suitable for quantum simulation. The validity of these results hold for arbitrary nonlinearity that correspond to convex Hamiltonians, and for arbitrarily long times, thus obviating a chief obstacle in most quantum algorithms for nonlinear partial differential equations. We provide quantum algorithms, both analog and digital, for extracting pointwise values, gradients, minima, and function evaluations at the minimiser of the viscosity solution, without requiring nonlinear updates or full state reconstruction.
We develop a quantum algorithm for solving the dynamics of the nonlinear viscous Burgers equation. We apply the Carleman linearization procedure on the spatially discretized equation, followed by a padding scheme that allows implementation on qubit registers. Existing Carleman-based quantum algorithms commonly formulate the lifted linear differential equation in an oracle model. Here we decompose the padded generator into diagonal masks and reversible arithmetic permutations using the Permutation Matrix Representation (PMR), which we show to be naturally compatible with the Linear Combination of Hamiltonian Simulations (LCHS) algorithm. Under the assumptions required by LCHS - most importantly positive semidefiniteness of the Hermitian part of the linear generator, possibly after a stabilizing shift - the algorithm prepares a normalized quantum state proportional to the solution of the truncated lifted system; the stabilizing shift introduces an exponential postselection overhead, which we quantify and mitigate through a rescaling scheme. We show that our algorithm scales with the off-diagonal norm of the Carleman generator instead of the matrix norm, which can be advantageous for other generators that are diagonally dominant. We also extend the PMR scheme to general fluid equations that may contain higher-order derivatives or nonlinear terms, or may involve multiple fluid variables or spatial dimensions. The construction illustrates how PMR can serve as a convenient Hamiltonian-simulation primitive for a broader class of LCU-based algorithms.
From fluid flow and transport to collective dynamics, numerical simulation of nonlinear partial differential equations underpins modern scientific computing. Extending this capability to quantum computers remains a longstanding challenge because nonlinear and non-Hermitian evolution is fundamentally incompatible with conventional Hamiltonian-based quantum simulation. Here we experimentally realize the time evolution of nonlinear fluid dynamics on a quantum processor using a hybrid variational framework for the viscous and inviscid Burgers equations. Our approach directly encodes the nonlinear dynamics into a variational optimization procedure, avoiding the enlarged linear embeddings and truncation overhead associated with Carleman linearization-based quantum algorithms. We further demonstrate convection-dominated dynamics corresponding to Reynolds numbers of order 102. We encode the governing evolution into parametrized quantum circuits and iteratively reconstruct the time-dependent field through quantum-classical optimization. By introducing a zero-noise extrapolation method without additional circuit-folding overhead, we accurately execute deep error-circuits with entangling-gate counts beyond those typical of Hadamard test circuits. We accurately reconstruct the time evolution across multiple timesteps despite hardware noise and finite device coherence. Our results constitute, to our knowledge, the first experimental realization of nonlinear time propagation on a quantum processor, extending quantum simulation beyond predominantly linear settings and establishing a route toward quantum computation for nonlinear continuum dynamics.
Quantum algorithms for nonlinear partial differential equations remain challenging because nonlinear dynamics are not directly amenable to unitary quantum simulation. Building on the level-set formulation, we construct a quantum algorithm and provide an explicit gate-level implementation for solving scalar conservation laws. The nonlinear equation is first lifted to a linear Liouville equation, discretized by finite differences, and then embedded into a unitary evolution through Schrödingerisation. We further develop quantum procedures for estimating relevant observables from the evolved state. Error bounds and gate-complexity estimates are established for the complete algorithm. The resulting complexity comparison demonstrates a quantum advantage for observable estimation in sufficiently high spatial dimensions, under standard assumptions on state preparation and oracle access. Finally, numerical experiments validate the accuracy, multidimensional applicability, and predicted scaling of the proposed method.
We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form A=L†L, a structure that arises naturally in standard discretizations of elliptic operators. Our main tool is the Kannai transform, which represents the diffusion semigroup e−AT, where T is the final simulation time, as a Gaussian-weighted superposition of unitary wave propagators. For target accuracy ε, this representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity O(∥A∥Tlog(1/ε)), up to the standard dependence on state-preparation and output-norm factors, improving the scaling in ∥A∥, T, and ε compared with generic Hamiltonian-simulation-based methods. We instantiate the method for the heat equation and biharmonic diffusion under non-periodic physical boundary conditions, and further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for the viscous Hamilton--Jacobi equations. In the long-time regime, under a spectral-gap assumption, the same framework gives a structured quantum linear solver by exploiting convergence to the steady state. For normalized positive definite systems σ(A)⊂[1,κ], the solver outputs an ε-approximation to the state proportional to x=A−1b with query complexity O(∥x∥∥b∥κlog2ε∥x∥∥b∥).