qcr:2606.09747.1

HHL Algorithm for Linear Systems (Qiskit)

The Harrow-Hassidim-Lloyd (HHL) algorithm is the foundational quantum routine for solving systems of linear equations of the form Ax = b, where A is a Hermitian matrix and b a known vector. Rather than returning the full solution vector classically (which would already cost time linear in the system size and erase any speedup), it prepares a quantum state proportional to the inverse of A applied to b and then estimates functions (observables) of that solution, giving an exponential speedup in the system dimension for sparse, well-conditioned matrices under suitable oracle assumptions. This is the clean, modern Qiskit implementation maintained as the quantum_linear_solvers package, the continuation of the HHL code that originally shipped inside Qiskit (IBM, 2021-2022) before it was removed from the core library. It runs on Qiskit 1.x and exposes a well-structured HHL solver alongside the building blocks the algorithm needs: efficient Hamiltonian-evolution matrices such as the tridiagonal Toeplitz matrix, observables such as MatrixFunctional and AbsoluteAverage that read out linear functionals of the solution, and a NumPy reference solver for verification. Internally it performs the canonical three stages: quantum phase estimation to write the eigenvalues of A into a register, an eigenvalue-inversion rotation of an ancilla via an exact reciprocal or a Chebyshev approximation, and uncomputation through inverse phase estimation to disentangle the registers, with optional Richardson extrapolation to reduce the error introduced by Trotterized Hamiltonian simulation. It is an excellent, production-quality reference for how phase estimation and controlled rotations combine to perform linear algebra on a quantum computer.

Linear Algebra

Qubit

Circuit-based

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anedumla/quantum_linear_solvers

README.md

Quantum linear solvers

Contains classical and quantum algorithms to solve systems of linear equations such as the HHL algorithm.

Installation

git clone https://github.com/anedumla/quantum_linear_solvers.git
cd quantum_linear_solvers
pip install .

Documentation

Tutorial: https://learn.qiskit.org/course/ch-applications/solving-linear-systems-of-equations-using-hhl-and-its-qiskit-implementation

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Publication

doi:10.48550/arxiv.0811.3171

Quantum algorithm for solving linear systems of equations

Aram W. Harrow, Avinatan Hassidim, Seth Lloyd

Versions

v1 Latest

Jun 17, 2026

qcr:2606.09747.1

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Tools used

Qiskit

Keywords

hhl

linear-systems

phase-estimation

linear-algebra

qiskit