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arXiv:1903.05608

A quantum algorithm for solving systems of nonlinear algebraic equations

Qian, Peng, Huang, Wei-Cong, Gui‐Lu Long

Published Mar 13, 2019• 2 citations
arXivHTMLPDF

Abstract

We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of the corresponding classical ones when the set of equations has many variables and not so high orders. Different with most popular way of representing the results by state vector, we get the results in computation basis which is readable directly.

DOI
10.48550/arxiv.1903.05608

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We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables nnn, which means our algorithm's complexity is less than O(n3)O(n^{3})O(n3) of the corresponding classical ones when the set of equations has many variables and not so high orders. Different with most popular way of representing the results by state vector, we get the results in computation basis which is readable directly.
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.
We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse nnn-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with kkk-th degree polynomial vector fields for arbitrary (finite) values of kkk. The steps involve: 1) mapping the kkk-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the kkk-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on kkk and demonstrate the framework on an example.
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix AAA and a vector $\stackrel{\ensuremath{\rightarrow}}{b}$, find a vector $\stackrel{\ensuremath{\rightarrow}}{x}$ such that $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$. We consider the case where one does not need to know the solution $\stackrel{\ensuremath{\rightarrow}}{x}$ itself, but rather an approximation of the expectation value of some operator associated with $\stackrel{\ensuremath{\rightarrow}}{x}$, e.g., ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ for some matrix MMM. In this case, when AAA is sparse, $N\ifmmode\times\else\texttimes\fi{}N$ and has condition number $\ensuremath{\kappa}$, the fastest known classical algorithms can find $\stackrel{\ensuremath{\rightarrow}}{x}$ and estimate ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ in time scaling roughly as $N\sqrt{\ensuremath{\kappa}}$. Here, we exhibit a quantum algorithm for estimating ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ whose runtime is a polynomial of log(N)\mathrm{log}(N)log(N) and $\ensuremath{\kappa}$. Indeed, for small values of $\ensuremath{\kappa}$ [i.e., polylog(N)\mathrm{poly}\mathrm{log}(N)polylog(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as du/dt=G0+G1eiud{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}du/dt=G0​+G1​eiu, where u{\bf u}u denotes a vector of nnn complex variables evolving with ttt, G0G_0G0​ is an nnn-dimensional complex vector, G1G_1G1​ is an n×nn \times nn×n complex matrix and eiue^{i{\bf u}}eiu denotes the vector with entries {eiuj}\{e^{iu_j}\}{eiuj​}. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of u{\bf u}u, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.
Nonlinear equations are generally challenging to solve. As analytical solutions typically do not exist, numerical methods have been developed to find their solutions. In this article, building upon the classical Newton method and the recent quantum singular value transformation framework, we give a quantum algorithm for solving a system of nonlinear algebraic equations, in which each equation is a homogeneous polynomial of an even degree. A detailed analysis is then carried out to reveal that our method achieves polylogarithmic time 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 generalized to deal with polynomials of various types. Some potential applications stem from various contexts, such as the Gross-Pitaevski equation, the Lotka-Volterra equations, and the intersection of algebraic varieties. Our work thus marks a significant step forward toward quantum advantage in nonlinear science, highlighting a promising avenue for future exploration.
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