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Quantum block encoding for semiseparable matrices

Giacomo Antonioli, Paola Boito, Gianna M. Del Corso, Margherita Porcelli

Abstract

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. This process takes polylogarithmic time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.

Quantum block encoding for semiseparable matrices

Abstract

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs ancillary qubits. This process takes polylogarithmic time and has an error of , where is the matrix size.
Paper Structure (16 sections, 9 theorems, 52 equations, 12 figures)

This paper contains 16 sections, 9 theorems, 52 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Block encoding
  3. One-pair semiseparable matrices
  4. Overview on QBE for structured and unstructured matrices
  5. The unstructured case: FABLE
  6. Block encoding of diagonal matrices and their inverses
  7. Block encoding of a linear combination of matrices
  8. Block encoding of a matrix product
  9. QBE for semiseparable matrices
  10. QBE of matrix L
  11. QBE of matrix Du
  12. QBE of matrix Dz
  13. The final block encoding
  14. Some considerations about the encoding
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Theorem 3

Let $A$ be an $N\times N$ symmetric tridiagonal matrix of the form with $b_i\neq 0$ for $i=0,\ldots,N-2$. Then the inverse of $A$ is a one-pair matrix. Conversely, the inverse of an invertible one-pair matrix is an irreducible symmetric tridiagonal matrix.

Figures (12)

  • Figure 1: Controlled rotations of the powers of two
  • Figure 2: Circuit for the block encoding of a diagonal matrix
  • Figure 3: Circuit for the fast inversion of a Diagonal Matrix
  • Figure 4: Quantum circuits illustrating how to combine two block encodings for matrices $A$ and $B$ to create a block encoding for their sum and difference, respectively.
  • Figure 5: Circuit realizing $U_L$, the $(N, \log(N)+1, 0)-$ block encoding of $L$.
  • ...and 7 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Proposition 4
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • ...and 7 more