HHL Linear Systems Algorithm

HHL Linear Systems Algorithm

The Harrow-Hassidim-Lloyd (HHL) algorithm is a 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, it prepares a quantum state proportional to the inverse of A applied to b, which can then be used to estimate expectation values or other linear functionals of the solution, offering an exponential speedup over classical solvers for suitable sparse, well-conditioned systems. This Cirq example implements the complete algorithm across three qubit registers: a single ancilla, an eigenvalue register, and a memory register that holds the input vector and ultimately the solution. It proceeds in the classic three stages: quantum phase estimation to write the eigenvalues of A into the register, a controlled rotation of the ancilla by an angle proportional to the inverse eigenvalue, and uncomputation via inverse phase estimation to disentangle the registers. The script constructs the circuit for a concrete Hermitian matrix and input vector, simulates it, and reports the expectation values of Pauli operators on the recovered solution state. It is an excellent worked example of how phase estimation and controlled rotations combine to perform linear algebra on a quantum computer.

Run it

pip install -r requirements.txt
python hhl.py

Source and license

Imported from examples/hhl.py in quantumlib/Cirq at v1.6.1, under the Apache License 2.0. Original authors: The Cirq Developers. The upstream LICENSE is included alongside this example.