braket.experimental.algorithms.hhl.hhl module

Harrow-Hassidim-Lloyd (HHL) Algorithm for Solving Linear Systems of Equations.

The HHL algorithm is a quantum algorithm for solving systems of linear equations of the form Ax = b. Given an N x N Hermitian matrix A and a unit vector b, the algorithm produces a quantum state |x> proportional to A^{-1}|b>.

For certain classes of problems (sparse, well-conditioned matrices), and when only summary statistics of the solution are needed (e.g., <x|M|x> for some operator M), HHL can offer a speedup over classical methods. However, the overall advantage depends on the efficiency of state preparation and readout.

This implementation provides a simplified version of HHL suitable for small systems (2x2 matrices), illustrating the core concepts: 1. State preparation: encode |b> into a quantum state 2. Quantum Phase Estimation (QPE): decompose |b> in the eigenbasis of A 3. Controlled rotation: apply the eigenvalue inversion C/lambda 4. Inverse QPE: uncompute the eigenvalue register 5. Measurement: post-select on the ancilla qubit

References

[1] A. W. Harrow, A. Hassidim, S. Lloyd, “Quantum algorithm for linear systems

of equations”, Phys. Rev. Lett. 103, 150502 (2009). arXiv:0811.3171

[2] Wikipedia: https://en.wikipedia.org/wiki/HHL_algorithm

braket.experimental.algorithms.hhl.hhl.hhl_circuit(matrix: ndarray, b_vector: ndarray, num_clock_qubits: int = 2, scaling_factor: float | None = None) → Circuit[source]

Construct the full HHL circuit for solving Ax = b.

The circuit uses: - 1 input qubit for encoding |b> - num_clock_qubits clock qubits for QPE - 1 ancilla qubit for eigenvalue inversion (post-selection)

Qubit layout: - Clock qubits: 0 to num_clock_qubits - 1 - Input qubit: num_clock_qubits - Ancilla qubit: num_clock_qubits + 1

Parameters:

  • matrix (np.ndarray) – A 2x2 Hermitian matrix A.

  • b_vector (np.ndarray) – A normalized 2-element vector b.

  • num_clock_qubits (int) – Number of clock qubits for QPE (default: 2).

  • scaling_factor (Optional[float]) – Scaling factor for Hamiltonian simulation. If None, automatically computed from the eigenvalues.

Returns:

Circuit – The complete HHL circuit.

Raises:

ValueError – If matrix is not 2x2 Hermitian or b_vector is invalid.

braket.experimental.algorithms.hhl.hhl.run_hhl(circuit: Circuit, device: Device, shots: int = 1000) → QuantumTask[source]

Run the HHL circuit on the specified device.

Parameters:

  • circuit (Circuit) – The HHL circuit to run.

  • device (Device) – Braket device backend.

  • shots (int) – Number of measurement shots (default: 1000).

Returns:

QuantumTask – Task from running HHL.

braket.experimental.algorithms.hhl.hhl.get_hhl_results(task: QuantumTask, matrix: ndarray, b_vector: ndarray, num_clock_qubits: int = 2, verbose: bool = False) → Dict[str, Any][source]

Post-process results from an HHL run.

Extracts the solution state by post-selecting on the ancilla qubit measuring |1>. The solution |x> is proportional to A^{-1}|b>.

Parameters:

  • task (QuantumTask) – The task containing HHL results.

  • matrix (np.ndarray) – The original 2x2 matrix A.

  • b_vector (np.ndarray) – The original vector b.

  • num_clock_qubits (int) – Number of clock qubits used (default: 2).

  • verbose (bool) – If True, prints detailed results (default: False).

Returns:

Dict[str, Any] –

Dictionary containing:

  • measurement_counts: Raw measurement counts

  • post_selected_counts: Counts post-selected on ancilla=1

  • solution_state_probabilities: Probabilities of solution components

  • classical_solution: The exact classical solution for comparison

  • fidelity: Fidelity between quantum and classical solutions

  • success_probability: Probability of ancilla measuring |1>