Variational Quantum Linear Solver

Variational Quantum Linear Solver

This PennyLane demo implements the Variational Quantum Linear Solver (VQLS), a near-term algorithm for solving systems of linear equations of the form Ax = b on a quantum computer without the deep circuits that the textbook HHL algorithm requires. Solving linear systems is ubiquitous across science and engineering, and VQLS recasts it as a variational optimization: a parameterized circuit prepares a candidate solution state, and a cost function measures how far applying the matrix A to that state is from the (normalized) target vector b, so minimizing the cost drives the circuit toward the solution. The tutorial expresses the matrix A as a linear combination of unitaries and the vector b as a state-preparation circuit, builds the cost function from Hadamard-test-style overlap measurements in PennyLane, and optimizes the parameterized solution circuit with gradient-based optimization. It walks through constructing the problem, evaluating the variational cost, and verifying the recovered solution against the classical answer. By trading deep coherent circuits for a shallow variational loop, VQLS shows a practical, near-term route to quantum linear algebra, presented hands-on in PennyLane.

Run it

pip install -r requirements.txt
python demo.py

Source and license

Imported from demonstrations_v2/tutorial_vqls/demo.py in PennyLaneAI/demos at c52c0abeb5122218aa96b38eea848864cce7323f, under the Apache License 2.0. Original authors: Xanadu and the PennyLane community. The upstream LICENSE is included alongside this example.