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Solve a nonlinear ODE dy/dt = F(y)

Given access to a nonlinear vector field FF — in practice quadratic or polynomial — and a preparation unitary for the initial state, produce a quantum state proportional to y(T)y(T) or an estimate of an observable of it. Quantum time evolution is linear, so no quantum primitive acts on this contract directly.

Takes

Access oracles for the components of FF (for example a linear part F1F_1, a quadratic part F2F_2, a forcing term F0F_0), a preparation unitary for yiny_{\mathrm{in}}, the evolution time TT, and an error tolerance ε\varepsilon.

Returns

A normalized state ε\varepsilon-close to y(T)/∥y(T)∥y(T)/\lVert y(T)\rVert, a history state over [0,T][0,T], or an estimate of an observable of the solution.

This one, drawn

Solve a nonlinear ODE dy/dt = F(y)Embed a nonlinear system into a linear one · 6 ways throughEmbed a nonlinear system into a linear one · 6 ways through — click the line to open it hereSolve a linear ODE du/dt = A(t)u + b(t) · 9 ways throughSolve a linear ODE du/dt = A(t)u + b(t) · 9 ways through — click the line to open it hereChoose a time discretization or propagator approximation → Quantum linear solve · 2 parts inside, openChoose a time discretization or propagator approximation · 6 ways throughChoose a time discretization or propagator approximation · 6 ways through — click the line to open it hereQuantum linear solve · 5 ways throughQuantum linear solve · 5 ways through — click the line to open it hereSimulate Hamiltonian evolution → Estimate an observable · 2 parts inside, openSimulate Hamiltonian evolution · 3 ways throughSimulate Hamiltonian evolution · 3 ways through — click the line to open it hereEstimate an observable · 4 ways throughEstimate an observable · 4 ways through — click the line to open it hereEmbed a nonlinear system into a linear one · 6 ways through — click the name to read about itEmbed nonlinear as linearSolve a linear ODE du/dt = A(t)u + b(t) · 9 ways through — click the name to read about itSolve a linear ODE du/dt = A(t)u + b(t)Choose a time discretization or propagator approximation · 6 ways through — click the name to read about itDiscretize time or the propagatorQuantum linear solve · 5 ways through — click the name to read about itQuantum linear solveSimulate Hamiltonian evolution · 3 ways through — click the name to read about itSimulate Hamiltonian evolutionEstimate an observable · 4 ways through — click the name to read about itEstimate an observableNonlinear initial-value problem — you start hereLinear ODE system — 1 way arrive here, 3 lead onAnswer about the solution — you finish hereLinear system Ax = b — what one part hands to the nextEvolution circuit, input in hand — what one part hands to the next

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From Nonlinear initial-value problem to Answer about the solution

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Why this is a layer

Every route must first commit to a representation of the nonlinearity, and the available representations — tensor powers, phase-space densities, level sets, homotopy series — differ in which nonlinearities they admit, in whether the truncation provably converges, and in how the answer is read back. This is also the layer at which the hard lower bounds bite, so it is where an advantage claim lives or dies.

Ways to do this

4 methods recorded

  • Quantum Carleman linearization algorithm

    Carleman-linearize the quadratic ODE, discretize with forward Euler, assemble the whole history into one large sparse linear system, and solve that system with a quantum linear system algorithm. This is the route that made dissipative nonlinear ODEs tractable in evolution time.

  • Quantum simulation of the KvN representation

    Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

  • Level-set method for observables of nonlinear PDEs

    Use the exact level-set mapping to a linear PDE, solve the linear problem quantumly, and compute physical observables from it. For MM sets of initial data the cost does not grow with MM.

  • Homotopy-perturbation series, embedded as a linear ODE

    Embed the homotopy-perturbation series into a finite-dimensional linear ODE system and solve that with a quantum linear-ODE algorithm, obtaining a state ε\varepsilon-close to the normalized exact solution with Ω(1)Ω(1) success probability.

Routes that skip this layer

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In the Atlas

  • Quantum algorithms for nonlinear differential equations

    Given a system of nonlinear ordinary differential equations, in the primary algorithm's case a dissipative quadratic n-dimensional system, produce the solution at a chosen evolution time T to error ε, encoded in the amplitudes of a quantum state.