Takes

A block-encoding of $A(t)$ with a normalization $\alpha_A \ge \max_t \lVert A(t)\rVert$, preparation unitaries for $u_0$ and $b$, the evolution time $T$, and an error tolerance $\varepsilon$.

Returns

A state proportional to $u(T)$, or a history state, together with separately stated matrix-query and initial-state-query complexity.

This is the pivot of the cluster. Methods fulfilling it split into two structurally different families: those that assemble one large linear system and call a quantum linear solver, and those that never form a linear system at all, reducing instead to Hamiltonian simulation or to repeated singular value amplification. The two families differ in how many queries they make to the initial-state preparation oracle, and that difference is structural rather than a matter of constants — when the initial state is expensive to prepare, it dominates the end-to-end cost.

• Linear multistep method, all-at-once encoding

  The first quantum algorithm for general linear ODEs. Discretize with a high-order linear multistep method, lay every time step out at once against a clock register so that one state $\sum_j |t_j\rangle|x_j\rangle$ holds the whole history, and solve the resulting sparse linear system with a quantum linear system algorithm. Going high-order is what buys the $\Delta t$ scaling: plain Euler costs at least $O(\Delta t^4)$ here.

• Taylor propagator, all-at-once encoding

  Encode a truncated Taylor series of the propagator into a single sparse linear system approximating the whole evolution, then solve it with a quantum linear system algorithm. This is what brought the precision dependence down to polynomial in $\log(1/\varepsilon)$.

• Chebyshev spectral method, global collocation

  The route that brought $\mathrm{poly}(\log(1/\varepsilon))$ precision to linear ODEs with **time-dependent** coefficients, which is what its abstract says was missing: "no such algorithm was previously known for differential equations with time-dependent coefficients". It does it by not stepping. The solution is approximated globally by a truncated Chebyshev series, the coefficients are fixed by collocating the differential equation at Chebyshev nodes, and the resulting sparse system goes to a high-precision quantum linear system algorithm. The exponential precision is bought by smoothness rather than by the solver: it is the $C^{\infty}$ convergence of the Chebyshev series that makes the series length logarithmic in $1/\varepsilon$.

• Krovi's reanalysis of the all-at-once encoding

  Reanalyses the all-at-once propagator encoding and shows that the norm of the matrix exponential, rather than the eigenvector condition number, characterizes the run time. It still forms a global linear system and still calls a quantum linear solver.

• Dyson propagator, all-at-once encoding

  Encode the Dyson series in a system of linear equations and solve it via the optimal quantum linear equation solver, extending the all-at-once approach to genuinely time-dependent generators.

• Time-marching with uniform singular value amplification

  Propagate the solution one step at a time and defeat the exponentially vanishing success probability by repeatedly invoking uniform singular value amplification, improved further by a compression gadget lemma. Fang, Lin and Tong present it explicitly as a design path alternative to solvers based on quantum linear systems algorithms.

• LCHS — linear combination of Hamiltonian simulation

  Express a general non-unitary evolution operator as a linear combination of unitary evolution operators, each of which solves a Hamiltonian simulation problem, rather than converting the problem into a dilated linear system. An, Liu and Lin state that the method can achieve optimal cost in terms of state preparation.

• LCHS with the improved kernel

  A family of identities expressing non-unitary evolution as a linear combination of unitary evolutions, built on the kernel $f(z) = 1/(C_β e^{(1+iz)^β})$ with $β ∈ (0,1)$ and $C_β = 2π e^{-2^β}$. The kernel decays at a near-exponential rate $e^{-c|k|^β}$, replacing the original Cauchy kernel's quadratic decay and exponentially enhancing accuracy.

• Schrödingerisation (linear PDEs as Schrödinger equations)

  A simple change of variable — the warped phase transformation, which introduces one extra variable — recasts any linear PDE or ODE system into a system of Schrödinger equations in real time, which ordinary Hamiltonian simulation then runs. The original solution is recovered from the auxiliary dimension.

• Quantum algorithms for linear differential equations

  Given a first-order linear differential equation d/dt x = A(t)x + b(t) with N-dimensional vectors x and b and an N×N matrix A, and given an initial condition x(0), produce the solution x(t) at a later time t to precision ε, in the sense that the normalized vector x(t)/‖x(t)‖ returned is at distance at most ε from the exact solution.

• Quantum algorithm for simulating the wave equation

  Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.