Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision

Abstract

We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time. The complexity of the algorithm is polynomial in the logarithm of the inverse error, an exponential improvement over previous quantum algorithms for this problem. Our result builds upon recent advances in quantum linear systems algorithms by encoding the simulation into a sparse, well-conditioned linear system that approximates evolution according to the propagator using a Taylor series. Unlike with finite difference methods, our approach does not require additional hypotheses to ensure numerical stability.

Time-dependent Hamiltonian simulation withL1-norm scaling

“…Quantum computers can simulate various physical systems, including condensed matter physics [3], quantum field theory [29], and quantum chemistry [2,14,47]. The study of quantum simulation has also led to the discovery of new quantum algorithms, such as algorithms for linear systems [28], differential equations [9], semidefinite optimization [11], formula evaluation [22], quantum walk [15], and ground-state and thermal-state preparation [20,42].…”

Quantum Spectral Methods for Differential Equations

“…Since it uses a finite difference approximation, the complexity of Berry's algorithm as a function of the solution error ǫ is poly(1/ǫ) [5]. Reference [10] improved this to poly(log(1/ǫ)) by using a high-precision QLSA based on linear combinations of unitaries [15] to solve a linear system that encodes a truncated Taylor series. However, this approach assumes that A(t) and f (t) are timeindependent so that the solution of the ODE can be written as an explicit series, and it is unclear how to generalize the algorithm to time-dependent ODEs.…”

“…For ODEs with special structure, some prior results already show how to avoid a local approximation and thereby achieve complexity poly(log(1/ǫ)). When A(t) is anti-Hermitian and f (t) = 0, we can directly apply Hamiltonian simulation [9]; if A and f are time-independent, then [10] uses a Taylor series to achieve complexity poly(log(1/ǫ)). However, the case of general time-dependent linear ODEs had remained elusive.…”

“…Furthermore, we assume that A(t), f (t), and γ are provided by black-box subroutines (which serve as abstractions of efficient computations). In particular, following essentially the same model as in [10] (see also Section 1.1 of [15]), suppose we have an oracle O A (t) that, for any t ∈ [0, T ] and any given row or column specified as input, computes the locations and values of the nonzero entries of A(t) in that row or column. We also assume oracles O x and O f (t) that, for any t ∈ [0, T ], prepare normalized states |γ and |f (t) proportional to γ and f (t), and that also compute γ and f (t) , respectively.…”

Quantum algorithm for the Vlasov equation

“…Provided κ is small, QLSAs obtain a dramatic speedup, however they output the result vector encoded as the amplitudes of a quantum state, which does not allow for efficient extraction of all the information, unlike in the classical case. Despite these caveats, QLSAs suggest the potential for a quantum speedup of linear computations, including solving systems of linear ordinary differential equations [17,18].…”