DeepAI
Log In Sign Up

Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices

03/19/2022
by   Daan Camps, et al.
0

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 encoding and quantum eigenvalue / singular value transformations. Block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U 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 gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, 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 are particularly interested in sparse stochastic matrices that represent random walks on a graph, and show how the block encodings of such matrices yield efficient quantum walks.

READ FULL TEXT

page 1

page 2

page 3

page 4

01/31/2022

Quantum machine learning with subspace states

We introduce a new approach for quantum linear algebra based on quantum ...
06/07/2020

Random circuit block-encoded matrix and a proposal of quantum LINPACK benchmark

The LINPACK benchmark reports the performance of a computer for solving ...
06/27/2022

Quantum Regularized Least Squares

Linear regression is a widely used technique to fit linear models and fi...
07/22/2021

Nonlinear transformation of complex amplitudes via quantum singular value transformation

Due to the linearity of quantum operations, it is not straightforward to...
02/08/2021

Learning with Density Matrices and Random Features

A density matrix describes the statistical state of a quantum system. It...
04/05/2018

The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation

We apply the framework of block-encodings, introduced by Low and Chuang ...