Quantum Algorithm For Solving Nonlinear Algebraic Equations

TL;DR

This work develops a quantum algorithm to solve nonlinear algebraic equations where each $f_i$ is a homogeneous polynomial of even degree by embedding Newton's method within a quantum block-encoding framework. It constructs block encodings of the Jacobian and its inverse, prepares the right-hand side state, and executes iterative Newton updates to obtain an $ ilde{O}((ps\log(ps/\varepsilon))^{T+1})$-type runtime that is polylogarithmic in the number of variables $n$ for a fixed number of iterations $T$, while requiring only logarithmic qubits in $n$. The method generalizes to arbitrary polynomial types, including inhomogeneous terms, via a generalized gradient-encoding approach and a quantum gradient-descent extension. Motivating applications span physics (Gross–Pitaevskii, Lotka–Volterra) and algebraic geometry (variety intersections), suggesting a path toward quantum advantage in nonlinear science.

Abstract

Nonlinear equations are challenging to solve due to their inherently nonlinear nature. As analytical solutions typically do not exist, numerical methods have been developed to tackle their solutions. In this article, we give a quantum algorithm for solving a system of nonlinear algebraic equations, in which each equation is a multivariate polynomial of known coefficients. Building upon the classical Newton method and some recent works on quantum algorithm plus block encoding from the quantum singular value transformation, we show how to invert the Jacobian matrix to execute Newton's iterative method for solving nonlinear equations, where each contributing equation is a homogeneous polynomial of an even degree. A detailed analysis are then carried out to reveal that our method achieves polylogarithmic time in relative to the number of variables. Furthermore, the number of required qubits is logarithmic in the number of variables. In particular, we also show that our method can be modified with little effort to deal with polynomial of various types, thus implying the generality of our approach. Some examples coming from physics and algebraic geometry, such as Gross-Pitaevski equation, Lotka-Volterra equations, and intersection of algebraic varieties, involving nonlinear partial differential equations are provided to motivate the potential application, with a description on how to extend our algorithm with even less effort in such a scenario. Our work thus marks a further important step towards quantum advantage in nonlinear science, enabled by the framework of quantum singular value transformation.

Figures (2)

• Figure 1: Illustration of Newton's method in 1 variable case, aiming to find the root of the nonlinear equation $f(x) = 0.5x^2 - 3x +4$. First we initialize the guessed solution $x_0$. The tangent line to $f(x)$ at $x_0$ intersects the $x$-axis at the point $x_1$, which is the updated solution. The procedure is then iterated multiple times until the real root is found, or at least a desired approximation is reached. We note that since the equation might have more than 1 solution, the initialization is critical as the method would drive toward the closest root. On the same figure, we can see that if initially we begin at $x_m$ (on the left), then we end up with different root.
• Figure 2: Some example of algebraic varieties. Figure \ref{['fig:sub2']} features a one-dimensional algebraic variety. Figure \ref{['fig:sub3']} features two-dimensional algebraic variety.

Theorems & Definitions (13)

• Definition 1: Block Encoding Unitary
• Lemma 1: gilyen2019quantum
• Lemma 2: Block Encoding of Product of Two Matrices
• Lemma 3: camps2020approximate
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9: nghiem2023improved