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Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices

Weitao Lin, Guojing Tian, Xiaoming Sun

TL;DR

It is proved that the quantum algorithm inspired by the classical multi-row iteration method converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)].

Abstract

In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing non-square matrices. Towards this kind of problems defined by $ Ax = b $ where $ A $$ \in\mathbb{R}^{m \times n} $, we propose a quantum algorithm inspired by the classical multi-row iteration method and provide an explicit quantum circuit based on the quantum comparator and Quantum Random Access Memory (QRAM). The time complexity of our quantum multi-row iteration algorithm is $ O(K \log m) $, with $ K $ representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version. Based on the convergence of the classical multi-row iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.

Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices

TL;DR

It is proved that the quantum algorithm inspired by the classical multi-row iteration method converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)].

Abstract

In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing non-square matrices. Towards this kind of problems defined by where , we propose a quantum algorithm inspired by the classical multi-row iteration method and provide an explicit quantum circuit based on the quantum comparator and Quantum Random Access Memory (QRAM). The time complexity of our quantum multi-row iteration algorithm is , with representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version. Based on the convergence of the classical multi-row iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.
Paper Structure (25 sections, 8 theorems, 71 equations, 7 figures)

This paper contains 25 sections, 8 theorems, 71 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical iteration method
  4. Quantum one-row iteration method
  5. Block-encoding
  6. Quantum multi-row iteration algorithm
  7. Key points of the algorithm implementation
  8. Non-unitarity of the matrix
  9. Overcome the non-unitarity
  10. Equivalent implementation
  11. Data structure
  12. The sketch of the algorithm
  13. Analysis for the resource requirement
  14. Analysis for $k-th$ iteration step
  15. Analysis for the number of iteration steps
  16. ...and 10 more sections

Key Result

Lemma 1

Given a linear system of equations $A\bm x=\bm b$, where $A\in\mathbb{R}^{m\times n},m\ge n$ and $\bm b\in\mathbb{R}^m$. Then, there exists a multi-row iteration protocol where $\tau_k$ is a random set of $q$ row indices sampled with replacement and $\omega_i$ represents the weight corresponding to the $i$th row.

Figures (7)

  • Figure 1: Schematic diagram of data structure. The root gets an address as an input and finds the routes to the corresponding leaves based on each qubit of the address. Each leaf points to a data tree, which stores the row vector of the matrix $A$.
  • Figure 2: An example of the data tree with $n=4$. The leaves hold the individual amplitudes of the elements of the vector, and each internal node holds the square root of the sum of the squares of the norm for the value in children nodes.
  • Figure 3: Quantum circuit implementation of the operator $U_k$.
  • Figure 4: Comparison of the quantum one-row iteration and the quantum multi-row iteration with different choices of the number of iteration rows. $q$ is the number of rows selected in each iteration. $e^k=x^k-x^*$ is the error between the iteration result and the solution or least-square solution. Smaller $\|e^k\|$ indicates a smaller convergence radius.
  • Figure 5: Comparison of the effect of different choices of $\alpha_A$. The classic algorithm achieves the results with $\alpha_A>1$. We choose uniform weights, and the number of iteration rows satisfies $q=10$. $e^k=x^k-x^*$ is the error between the iteration result and the solution or least-square solution.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1: moorman2021randomized
  • Definition 2
  • Lemma 3: moorman2021randomized Theorem 1
  • Definition 4: gilyen2019quantum Block-encoding
  • Definition 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Lemma 9: Convergence rate in the quantum setting
  • Corollary 10
  • ...and 2 more