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^{ Note on the vocabulary: the word "hamiltonian" is used in this question to speak about hermitian matrices.}

The HHL algorithm seems to be an active subject of research in the field of quantum computing, mostly because it solve a very important problem which is finding the solution of a linear system of equations.

According to the original paper Quantum algorithm for solving linear systems of equations (Harrow, Hassidim & Lloyd, 2009) and some questions asked on this site

- Quantum phase estimation and HHL algorithm - knowledge on eigenvalues required?
- Quantum algorithm for linear systems of equations (HHL09): Step 2 - Preparation of the initial states $\vert \Psi_0 \rangle$ and $\vert b \rangle$

the HHL algorithm is limited to some specific cases. Here is a summary (that may be incomplete!) of the characteristics of the HHL algorithm:

# HHL algorithm

The HHL algorithm solves a linear system of equation $$A \vert x \rangle = \vert b \rangle$$ with the following limitations:

## Limitations on $A$:

- $A$ needs to be Hermitian (and only Hermitian matrix works, see this discussion in the chat).
- $A$'s eigenvalues needs to be in $[0,1)$ (see Quantum phase estimation and HHL algorithm - knowledge on eigenvalues required?)
- $e^{iAt}$ needs to be efficiently implementable. At the moment the only known matrices that satisfy this property are:
- local hamiltonians (see Universal Quantum Simulators (Lloyd, 1996)).
- $s$-sparse hamiltonians (see Adiabatic Quantum State Generation and Statistical Zero Knowledge (Aharonov & Ta-Shma, 2003)).

## Limitations on $\vert b \rangle$:

- $\vert b \rangle$ should be efficiently preparable. This is the case for:
- Specific expressions of $\vert b \rangle$. For example the state $$\vert b \rangle = \bigotimes_{i=0}^{n} \left( \frac{\vert 0 \rangle + \vert 1 \rangle}{\sqrt{2}} \right)$$ is efficiently preparable.
- $\vert b \rangle$ representing the discretisation of an efficiently integrable probability distribution (see Creating superpositions that correspond to efficiently integrable probability distributions (Grover & Rudolph, 2002)).

## Limitations on $\vert x \rangle$ (output):

- $\vert x \rangle$ cannot be recovered fully by measurement. The only information we can recover from $\vert x \rangle$ is a "general information" ("expectation value" is the term employed in the original HHL paper) such as $$\langle x\vert M\vert x \rangle$$

**Question:**
Taking into account **all** of these limitations and imagining we are in 2050 (or maybe in 2025, who knows?) with fault-tolerant large-scale quantum chips (i.e. we are not limited by the hardware), what real-world problems could HHL algorithm solve (including problems where HHL is only used as a subroutine)?

I am aware of the paper Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target (Scherer, Valiron, Mau, Alexander, van den Berg & Chapuran, 2016) and of the corresponding implementation in the Quipper programming language and I am searching for other real-world examples where HHL would be applicable in practice. I do not require a published paper, not even an unpublished paper, I just want to have some examples of *real-world* use-cases.

EDIT:

Even if I am interested in every use-case, I would prefer some examples where HHL is directly used, i.e. not used as a subroutine of an other algorithm.

I am even more interested in examples of linear systems resulting of the discretisation of a differential operator that could be solved with HHL.

But let me emphasise one more time **I'm interested by every use-case (subroutines or not) you know about**.

You mention that you want some examples where HHL is "directly used". I am not very clear on what you mean by that. I do know some algorithms (which can potentially have practical uses) in which HHL is one of the primary steps, but surely not the

onlystep. Would something like recognizing genetic sequences using HHL as one of theprimary steps(subject to all the constraints you mentioned), be a suitable answer? The other primary steps mainly involve Hamiltonian simulation and state preparation. – Sanchayan Dutta – 2018-07-11T15:25:41.250I would

prefersome examples where HHL is directly used. It means that the problem can be directly formulated as a linear system of equation to solve. This is the case when solving differential equations: we discretise the equation and solve the discretised problem which is most of the time a sparse linear system. But other examples are welcomed. – Adrien Suau – 2018-07-11T15:29:09.873