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arXiv:1608.02184 · Physical Review A

Asymptotic quantum algorithm for the Toeplitz systems

Lin‐Chun Wan, Chao‐Hua Yu, Shi‐Jie Pan, Fei Gao, Qiaoyan Wen, Su‐Juan Qin

Published Jun 15, 2018• 57 citations
arXivHTMLPDF

Abstract

Solving the Toeplitz systems, which involves finding the vector $x$ such that ${T}_{n}x=b$ given an $n\ifmmode\times\else\texttimes\fi{}n$ Toeplitz matrix ${T}_{n}$ and a vector $b$, has a variety of applications in mathematics and engineering. In this paper, we present a quantum algorithm for solving the linear equations of Toeplitz matrices, in which the Toeplitz matrices are generated by discretizing a continuous function. It is shown that our algorithm's complexity is nearly $O(\ensuremath{\kappa}\mathrm{poly}(logn))$, where $\ensuremath{\kappa}$ and $n$ are the condition number and the dimension of ${T}_{n}$, respectively. This implies our algorithm is exponentially faster than its classical counterpart if $\ensuremath{\kappa}=O(\mathrm{poly}(logn))$. Since no assumption on the sparseness of ${T}_{n}$ is demanded in our algorithm, it can serve as an example of quantum algorithms for solving nonsparse linear systems.

DOI
10.1103/physreva.97.062322

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Solving the Toeplitz systems, which involves finding the vector xxx such that Tnx=b{T}_{n}x=bTn​x=b given an n\ifmmode×\else\texttimes\finn\ifmmode\times\else\texttimes\fi{}nn\ifmmode×\else\texttimes\fin Toeplitz matrix Tn{T}_{n}Tn​ and a vector bbb, has a variety of applications in mathematics and engineering. In this paper, we present a quantum algorithm for solving the linear equations of Toeplitz matrices, in which the Toeplitz matrices are generated by discretizing a continuous function. It is shown that our algorithm's complexity is nearly O(\ensuremathκpoly(logn))O(\ensuremath{\kappa}\mathrm{poly}(logn))O(\ensuremathκpoly(logn)), where \ensuremathκ\ensuremath{\kappa}\ensuremathκ and nnn are the condition number and the dimension of Tn{T}_{n}Tn​, respectively. This implies our algorithm is exponentially faster than its classical counterpart if \ensuremathκ=O(poly(logn))\ensuremath{\kappa}=O(\mathrm{poly}(logn))\ensuremathκ=O(poly(logn)). Since no assumption on the sparseness of Tn{T}_{n}Tn​ is demanded in our algorithm, it can serve as an example of quantum algorithms for solving nonsparse linear systems.
Solving the Toeplitz systems, which is to find the vector xxx such that Tnx=bT_nx = bTn​x=b given a n×nn\times nn×n Toeplitz matrix TnT_nTn​ and a vector bbb, has a variety of applications in mathematics and engineering. In this paper, we present a quantum algorithm for solving the Toeplitz systems, in which a quantum state encoding the solution with error ϵ\epsilonϵ is generated. It is shown that our algorithm's complexity is nearly linear in the condition number, and polylog in the dimensions nnn and in the inverse error ϵ−1\epsilon^{-1}ϵ−1. This implies our algorithm is exponentially faster than the best classical algorithm for the same problem if the condition number of TnT_nTn​ is O(poly(log n))O(\textrm{poly}(\textrm{log}\,n))O(poly(logn)). Since no assumption on the sparseness of TnT_nTn​ is demanded in our algorithm, the algorithm can serve as an example of quantum algorithms for solving non-sparse linear systems.
Toeplitz matrix-vector multiplication is widely used in various fields, including optimal control, systolic finite field multipliers, multidimensional convolution, etc. In this paper, we first present a non-asymptotic quantum algorithm for Toeplitz matrix-vector multiplication with time complexity O ( κ polylog n ) , where κ and 2n are the condition number and the dimension of the circulant matrix extended from the Toeplitz matrix, respectively. For the case with an unknown generating function, we also give a corresponding non-asymptotic quantum version that eliminates the dependency on the L 1-norm ρ of the displacement of the structured matrices. Due to the good use of the special properties of Toeplitz matrices, the proposed quantum algorithms are sufficiently accurate and efficient compared to the existing quantum algorithms under certain circumstances.
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.
Abstract Solving linear systems of equations is of vital importance in various fields of science. With the scale of calculation soaring, classical computers are intractable, and the advantage of quantum computing is reflected. Quantum algorithm has exponential acceleration in solving the linear systems A x ⃗ = b ⃗ for sparse and well-conditioned coefficient matrix A with dimension N × N . In this paper, we present a quantum algorithm to solve matrix equations of the form AX = B , where X = ( x ⃗ 1 , x ⃗ 2 , … , x ⃗ n ) , B = ( b ⃗ 1 , b ⃗ 2 , … , b ⃗ n ) , x ⃗ k and b ⃗ k are unit vectors. The problem can be intuitively treated as solving n linear systems simultaneously, which will needs O ( n log N ) complexity by the famous HHL algorithm. We demonstrate that in our algorithm the scale of complexity is O ( log n N ) . In the error analysis part, phase estimation is discussed in details.
Quantum computing enables the efficient resolution of complex problems, often outperforming classical methods across various applications. In 2009, Harrow, Hassidim and Lloyd proposed an algorithm for solving linear systems of equations, demonstrating exponential speedup (under ideal conditions) with a complexity of poly(log⁡N)poly(\log N)poly(logN), in contrast to classical approaches, which in the general case exhibit a complexity of O(N3)O(N^3)O(N3), although they can achieve O(N)O(N)O(N) in specific cases involving sparse matrices. This algorithm holds promise for advancements in machine learning, the solution of differential equations, linear regression, and cryptographic analysis. However, its structure is intricate, and there is a notable lack of detailed instructional materials in the literature. In this context, this paper presents a tutorial addressing the physical and mathematical foundations of the HHL algorithm, aimed at undergraduate students, explaining its theoretical construction and its implementation for solving linear equation systems. After discussing the underlying mathematical and physical concepts, we present numerical examples that illustrate the evolution of the quantum circuit. Finally, the algorithm's complexity, limitations, and future prospects are analyzed. The examples are compared with their classical simulations, allowing for an operational assessment of the algorithm's performance.