qiskit.algorithms.HHL¶

class HHL(epsilon=0.01, expectation=None, quantum_instance=None)[source]¶

Systems of linear equations arise naturally in many real-life 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 \(s\) non-zero 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[QuantumInstance, BaseBackend, Backend, None]) – Quantum Instance or Backend. If None, a Statevector calculation is done.

__init__(epsilon=0.01, expectation=None, quantum_instance=None)[source]¶

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[QuantumInstance, BaseBackend, Backend, None]) – Quantum Instance or Backend. If None, a Statevector calculation is done.

Methods

__init__([epsilon, expectation, …])

type epsilon: float

construct_circuit(matrix, vector)

Construct the HHL circuit.

solve(matrix, vector[, observable, …])

Tries to solve the given linear system of equations.

Attributes

`expectation`	The expectation value algorithm used to construct the expectation measurement from the observable.
`quantum_instance`	Get the quantum instance.
`scaling`	The scaling of the solution vector.

construct_circuit(matrix, vector)[source]¶

Construct the HHL circuit.

Parameters

matrix (Union[ndarray, QuantumCircuit]) – The matrix specifying the system, i.e. A in Ax=b.
vector (Union[ndarray, QuantumCircuit]) – The vector specifying the right hand side of the equation in Ax=b.

Return type

QuantumCircuit

Returns

The HHL circuit.

Raises

ValueError – If the input is not in the correct format.
ValueError – If the type of the input matrix is not supported.

property expectation¶

The expectation value algorithm used to construct the expectation measurement from the observable.

Return type: ExpectationBase

property quantum_instance¶

Get the quantum instance.

Return type: Optional[QuantumInstance]
Returns: The quantum instance used to run this algorithm.

property scaling¶

The scaling of the solution vector.

Return type: float

solve(matrix, vector, observable=None, observable_circuit=None, post_processing=None)[source]¶

Tries to solve the given linear system of equations.

Parameters

matrix (Union[ndarray, QuantumCircuit]) – The matrix specifying the system, i.e. A in Ax=b.
vector (Union[ndarray, QuantumCircuit]) – The vector specifying the right hand side of the equation in Ax=b.
observable (Union[LinearSystemObservable, BaseOperator, List[LinearSystemObservable], List[BaseOperator], None]) – Optional information to be extracted from the solution. Default is the probability of success of the algorithm.
observable_circuit (Union[QuantumCircuit, List[QuantumCircuit], None]) – Optional circuit to be applied to the solution to extract information. Default is None.
post_processing (Optional[Callable[[Union[float, List[float]]], Union[float, List[float]]]]) – Optional function to compute the value of the observable. Default is the raw value of measuring the observable.

Raises

ValueError – If an invalid combination of observable, observable_circuit and post_processing is passed.

Return type

LinearSolverResult

Returns

The result object containing information about the solution vector of the linear system.