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Quantum Algorithm for the Toeplitz Systems

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

Published Aug 7, 2016• 1 citation
OpenAlex

Abstract

Solving the Toeplitz systems, which is to find the vector $x$ such that $T_nx = b$ given a $n\times n$ 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 Toeplitz systems, in which a quantum state encoding the solution with error $\epsilon$ is generated. It is shown that our algorithm's complexity is nearly linear in the condition number, and polylog in the dimensions $n$ and in the inverse error $\epsilon^{-1}$. This implies our algorithm is exponentially faster than the best classical algorithm for the same problem if the condition number of $T_n$ is $O(\textrm{poly}(\textrm{log}\,n))$. Since no assumption on the sparseness of $T_n$ is demanded in our algorithm, the algorithm can serve as an example of quantum algorithms for solving non-sparse linear systems.

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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.
Solving the Toeplitz systems, which involves finding the vector xxx such that Tnx=b{T}_{n}x=bTn​x=b given an $n\ifmmode\times\else\texttimes\fi{}n$ 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 $O(\ensuremath{\kappa}\mathrm{poly}(logn))$, where $\ensuremath{\kappa}$ 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 $\ensuremath{\kappa}=O(\mathrm{poly}(logn))$. 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.
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.
In the quantum linear systems problem (QLSP), we are given query access to a ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
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.
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.
We propose fast and practical quantum-inspired classical algorithms for solving linear systems. Specifically, given sampling and query access to a matrix A∈Rm×nA\in\mathbb{R}^{m\times n}A∈Rm×n and a vector b∈Rmb\in\mathbb{R}^mb∈Rm, we propose classical algorithms that produce a data structure for the solution x∈Rnx\in\mathbb{R}^{n}x∈Rn of the linear system Ax=bAx=bAx=b with the ability to sample and query its entries. The resulting xxx satisfies ∥x−A+b∥≤ε∥A+b∥\|x-A^{+}b\|\leqε\|A^{+}b\|∥x−A+b∥≤ε∥A+b∥, where ∥⋅∥\|\cdot\|∥⋅∥ is the spectral norm and A+A^+A+ is the Moore-Penrose inverse of AAA. Our algorithm has time complexity O~(κF4/κε2)\widetilde{O}(κ_F^4/κε^2)O(κF4​/κε2) in the general case, where κF=∥A∥F∥A+∥κ_{F} =\|A\|_F\|A^+\|κF​=∥A∥F​∥A+∥ and κ=∥A∥∥A+∥κ=\|A\|\|A^+\|κ=∥A∥∥A+∥ are condition numbers. Compared to the prior state-of-the-art result [Shao and Montanaro, arXiv:2103.10309v2], our algorithm achieves a polynomial speedup in condition numbers. When AAA is sss-sparse, our algorithm has complexity O~(sκlog⁡(1/ε))\widetilde{O}(s κ\log(1/ε))O(sκlog(1/ε)), matching the quantum lower bound for solving linear systems in κκκ and 1/ε1/ε1/ε up to poly-logarithmic factors [Harrow and Kothari]. When AAA is sss-sparse and symmetric positive-definite, our algorithm has complexity O~(sκlog⁡(1/ε))\widetilde{O}(s\sqrtκ\log(1/ε))O(sκ​log(1/ε)). Technically, our main contribution is the application of the heavy ball momentum method to quantum-inspired classical algorithms for solving linear systems, where we propose two new methods with speedups: quantum-inspired Kaczmarz method with momentum and quantum-inspired coordinate descent method with momentum. Their analysis exploits careful decomposition of the momentum transition matrix and the application of novel spectral norm concentration bounds for independent random matrices. Finally, we also conduct numerical experiments for our algorithms on both synthetic and real-world datasets, and the experimental results support our theoretical claims.