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arXiv:1605.07710 · Journal of Physics A Mathematical and Theoretical

Efficient quantum circuits for Toeplitz and Hankel matrices

Anuradha Mahasinghe, J B Wang

Published May 24, 2016• 11 citations
arXivHTMLPDF

Abstract

Abstract Toeplitz and Hankel matrices have been a subject of intense interest in a wide range of science and engineering related applications. In this paper, we show that quantum circuits can efficiently implement sparse or Fourier-sparse Toeplitz and Hankel matrices. This provides an essential ingredient for solving many physical problems with Toeplitz or Hankel symmetry in the quantum setting with deterministic queries.

DOI
10.1088/1751-8113/49/27/275301

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Abstract Toeplitz and Hankel matrices have been a subject of intense interest in a wide range of science and engineering related applications. In this paper, we show that quantum circuits can efficiently implement sparse or Fourier-sparse Toeplitz and Hankel matrices. This provides an essential ingredient for solving many physical problems with Toeplitz or Hankel symmetry in the quantum setting with deterministic queries.
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Circulant matrices are an important family of operators, which have a wide range of applications in science and engineering-related fields. They are, in general, non-sparse and non-unitary. In this paper, we present efficient quantum circuits to implement circulant operators using fewer resources and with lower complexity than existing methods. Moreover, our quantum circuits can be readily extended to the implementation of Toeplitz, Hankel and block circulant matrices. Efficient quantum algorithms to implement the inverses and products of circulant operators are also provided, and an example application in solving the equation of motion for cyclic systems is discussed.
Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [Formula: see text] in a larger unitary transformation [Formula: see text] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [Formula: see text], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.
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.
Quantum computing leverages unitary matrices to perform reversible computations while preserving probability norms. However, many real-world applications involve non-unitary sparse matrices, posing a challenge for quantum implementation. This paper introduces a novel method for transforming a class of non-unitary sparse binary matrices into higher-dimensional permutation matrices, ensuring unitarity. Our approach is efficient in both space and time, ensuring practical applicability to large-scale problems. We demonstrate the utility of this transformation in constructing quantum gates and apply the method to model quantum finite state machines (QFSMs) derived from classical deterministic finite automata (DFAs). This work offers a practical pathway for integrating non-unitary transformations into quantum systems, with implications for the many applications that are based on sparse, non-unitary matrices. The significance of this work for automata theory and quantum computation is outlined.