Improved quantum algorithms for linear and nonlinear differential equations

Abstract

We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here can also be exponentially faster for certain classes of diagonalizable matrices.Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

Time-marching based quantum solvers for time-dependent linear differential equations

“…First, our method does not require the diagonalizability of A(t), and the complexity is independent of the condition number κ V . This generalizes the result of [48] to equations with time-dependent matrix coefficients.…”

“…This QLSA-based strategy was first proposed by Berry [8], which successfully avoids the pitfall in Section 1.3. It has been adopted by various quantum linear differential equation solvers [14,26,48,50], and has been applied to solve nonlinear differential equations using linearization techniques [2,34,36,45,46,51,52,64].…”

“…) which diverges as δ → 0. However, the technique in [48] is only applicable when A is time-independent.…”

“…Compared to the state-of-the-art results based on the QLSA in [26] and [48], our algorithm has several advantages, which we summarize in Table 1. First, our method does not require the diagonalizability of A(t), and the complexity is independent of the condition number κ V .…”

“…This is because the construction of the adiabatic Hamiltonian corresponding to the linear system uses the initial state, and thus each query to the adiabatic Hamiltonian also queries the initial state. As a result, the quantum differential equation solvers in both [26,48] need to query the initial state for O(κ) = O(T ) times. The relation between κ and T is rooted in the no-fast-forwarding theorem and cannot be generally improved.…”