HHL: Numerical example without using hidden information

Question

I have the following resource request:

I am looking for a numerical example of an implementation of HHL that does not "cheat" by using knowledge that would be hidden in a realistic scenario.

I am working through this paper. It helps. Though, I am surprised that in section III (Numerical Example), they use the knowledge of the eigenvalues and eigenvectors of the matrix. In a realistic scenario they are unknown, right? (Otherwise, inverting the matrix would be classically easy). But if the eigenvalues and eigenvectors are unknown how could a concrete implementation of HHL look like?

Answer 1

First, a note. You need to have at least some knowledge on the eigenvalues of your matrix. This question and its accompanying answer give, I think, a good overview on that issue.

Now, about real implementation that do no "cheat" by hiding behind a non-implemented oracle that is magically given to the implementer, there are not a lot I am aware of. This is due to the fact that implementing correctly an oracle is hard, tedious, and long.

Below is a list of papers I know about that might answer your question:

• Quantum Algorithm for Solving Tri-Diagonal Linear Systems of Equations, the Master thesis of Almudena Carrera Vázquez, provides a nice analysis on how to implement efficiently the HHL algorithm on a specific case: a matrix $A$ that is tri-diagonal.
• Concrete resource analysis of the quantum linear-system algorithm used to compute the electromagnetic scattering cross section of a 2D target by Artur Scherer, Benoît Valiron, Siun-Chuon Mau, Scott Alexander, Eric van den Berg & Thomas E. Chapuran, gives in great details the HHL algorithm implementation and the accompanying oracle implementation for a specific problem: finding the electromagnetic scattering cross section of a 2D target.
• Practical Quantum Computing: Solving the Wave Equation Using a Quantum Approach from Adrien Suau (myself), Gabriel Staffelbach and Henri Calandra might also be interesting. The paper explains how we solved a specific partial differential equation and is not using HHL, but most of its content is about implementing an Hamiltonian simulation algorithm (the basic Trotterization approach) along with the accompanying oracle, which is one of the two non-trivial implementation block of HHL (along with eigenvalue inversion), hence its inclusion here.

An important note: your first paragraph is rather generic, but in the second paragraph you focus on the matrix $A$ and the assumption that the eigenvalues are known beforehand. Note that the right-hand side $b$ is at least as much a problem as $A$ as you need a way to encode it in a quantum state.

Also note that there are scenario in which you know the eigenvalues (e.g. Toeplitz matrices).