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Register a new userQuantum algorithm for linear differential equations with exponentially improved dependence on precision
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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.
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We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $Ntimes N$ matrix $mathcal{M}$, an $N$-dimensional vector $textbf{emph{b}}$, and an initial vector $textbf{emph{x}}(0)$, obtain a target vector $textbf{emph{x}}(t)$ as a function of time $t$ according to the constraint $dtextbf{emph{x}}(t)/dt=mathcal{M}textbf{emph{x}}(t)+textbf{emph{b}}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a $4times4$ linear differential equation using a 4-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.
Quantum computers are known to provide an exponential advantage over classical computers for the solution of linear differential equations in high-dimensional spaces. Here, we present a quantum algorithm for the solution of nonlinear differential equations. The quantum algorithm provides an exponential advantage over classical algorithms for solving nonlinear differential equations. Potential applications include the Navier-Stokes equation, plasma hydrodynamics, epidemiology, and more.
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $mathrm{poly}(1/epsilon)$, where $epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $mathrm{poly}(d, log(1/epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.
While there has been extensive previous work on efficient quantum algorithms for linear differential equations, analogous progress for nonlinear differential equations has been severely limited due to the linearity of quantum mechanics. Despite this obstacle, we develop a quantum algorithm for initial value problems described by dissipative quadratic $n$-dimensional ordinary differential equations. Assuming $R < 1$, where $R$ is a parameter characterizing the ratio of the nonlinearity to the linear dissipation, this algorithm has complexity $T^2mathrm{poly}(log T, log n, log 1/epsilon)/epsilon$, where $T$ is the evolution time and $epsilon$ is the allowed error in the output quantum state. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in $T$. We achieve this improvement using the method of Carleman linearization, for which we give a novel convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for $R ge sqrt{2}$. Finally, we discuss potential applications of this approach to problems arising in biology as well as in fluid and plasma dynamics.
Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an $Ntimes{N}$ matrix $A$ and a vector $vec b$, find the vector $vec x$ that satisfies $Avec x = vec b$. It has been shown that using the algorithm one could obtain the solution encoded in a quantum state $|x$ using $O(log{N})$ quantum operations, while classical algorithms require at least O(N) steps. If one is not interested in the solution $vec{x}$ itself but certain statistical feature of the solution ${x}|M|x$ ($M$ is some quantum mechanical operator), the quantum algorithm will be able to achieve exponential speedup over the best classical algorithm as $N$ grows. Here we report a proof-of-concept experimental demonstration of the quantum algorithm using a 4-qubit nuclear magnetic resonance (NMR) quantum information processor. For all the three sets of experiments with different choices of $vec b$, we obtain the solutions with over 96% fidelity. This experiment is a first implementation of the algorithm. Because solving linear systems is a common problem in nearly all fields of science and engineering, we will also discuss the implication of our results on the potential of using quantum computers for solving practical linear systems.
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