HHL¶

class
HHL
(epsilon=0.01, expectation=None, quantum_instance=None)[source]¶ Bases:
qiskit.algorithms.linear_solvers.linear_solver.LinearSolver
Systems of linear equations arise naturally in many reallife applications in a wide range of areas, such as in the solution of Partial Differential Equations, the calibration of financial models, fluid simulation or numerical field calculation. The problem can be defined as, given a matrix \(A\in\mathbb{C}^{N\times N}\) and a vector \(\vec{b}\in\mathbb{C}^{N}\), find \(\vec{x}\in\mathbb{C}^{N}\) satisfying \(A\vec{x}=\vec{b}\).
A system of linear equations is called \(s\)sparse if \(A\) has at most \(s\) nonzero entries per row or column. Solving an \(s\)sparse system of size \(N\) with a classical computer requires \(\mathcal{ O }(Ns\kappa\log(1/\epsilon))\) running time using the conjugate gradient method. Here \(\kappa\) denotes the condition number of the system and \(\epsilon\) the accuracy of the approximation.
The HHL is a quantum algorithm to estimate a function of the solution with running time complexity of \(\mathcal{ O }(\log(N)s^{2}\kappa^{2}/\epsilon)\) when \(A\) is a Hermitian matrix under the assumptions of efficient oracles for loading the data, Hamiltonian simulation and computing a function of the solution. This is an exponential speed up in the size of the system, however one crucial remark to keep in mind is that the classical algorithm returns the full solution, while the HHL can only approximate functions of the solution vector.
Examples
import numpy as np from qiskit import QuantumCircuit from qiskit.algorithms.linear_solvers.hhl import HHL from qiskit.algorithms.linear_solvers.matrices import TridiagonalToeplitz from qiskit.algorithms.linear_solvers.observables import MatrixFunctional matrix = TridiagonalToeplitz(2, 1, 1 / 3, trotter_steps=2) right_hand_side = [1.0, 2.1, 3.2, 4.3] observable = MatrixFunctional(1, 1 / 2) rhs = right_hand_side / np.linalg.norm(right_hand_side) # Initial state circuit num_qubits = matrix.num_state_qubits qc = QuantumCircuit(num_qubits) qc.isometry(rhs, list(range(num_qubits)), None) hhl = HHL() solution = hhl.solve(matrix, qc, observable) approx_result = solution.observable
References
[1]: Harrow, A. W., Hassidim, A., Lloyd, S. (2009). Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 15 (2009), 1–15.
[2]: Carrera Vazquez, A., Hiptmair, R., & Woerner, S. (2020). Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation. arXiv:2009.04484
 Parameters
epsilon (
float
) – Error tolerance of the approximation to the solution, i.e. if \(x\) is the exact solution and \(\tilde{x}\) the one calculated by the algorithm, then \(x  \tilde{x} \le epsilon\).expectation (
Optional
[ExpectationBase
]) – The expectation converter applied to the expectation values before evaluation. If None then PauliExpectation is used.quantum_instance (
Union
[Backend
,BaseBackend
,QuantumInstance
,None
]) – Quantum Instance or Backend. If None, a Statevector calculation is done.
Methods
Construct the HHL circuit.
Tries to solve the given linear system of equations.
Attributes

expectation
¶ The expectation value algorithm used to construct the expectation measurement from the observable.
 Return type
ExpectationBase

quantum_instance
¶ Get the quantum instance.
 Return type
Optional
[QuantumInstance
] Returns
The quantum instance used to run this algorithm.

scaling
¶ The scaling of the solution vector.
 Return type
float