注釈

このページは tutorials/optimization/3_minimum_eigen_optimizer.ipynb から生成されました。

IBM Quantum lab でインタラクティブに実行します。

最小固有値オプティマイザー (Minimum Eigen Optimizer)

はじめに

An interesting class of optimization problems to be addressed by quantum computing are Quadratic Unconstrained Binary Optimization (QUBO) problems. Finding the solution to a QUBO is equivalent to finding the ground state of a corresponding Ising Hamiltonian, which is an important problem not only in optimization, but also in quantum chemistry and physics. For this translation, the binary variables taking values in \(\{0, 1\}\) are replaced by spin variables taking values in \(\{-1, +1\}\), which allows to replace the resulting spin variables by Pauli Z matrices, and thus, an Ising Hamiltonian. For more details on this mapping we refer to [1].

Qiskitは、適切な``QuadraticProgram``からイジング・ハミルトニアンへの自動変換を提供します。これにより、-VQE、-QAOA、または-``NumpyMinimumEigensolver``（クラシカルな厳密法）などのすべての``MinimumEigenSolver``を活用できます。

Qiskit wraps the translation to an Ising Hamiltonian (in Qiskit Aqua also called Operator), the call to an MinimumEigensolver as well as the translation of the results back to OptimizationResult in the MinimumEigenOptimizer.

In the following we first illustrate the conversion from a QuadraticProgram to an Operator and then show how to use the MinimumEigenOptimizer with different MinimumEigensolver to solve a given QuadraticProgram. The algorithms in Qiskit automatically try to convert a given problem to the supported problem class if possible, for instance, the MinimumEigenOptimizer will automatically translate integer variables to binary variables or add a linear equality constraints as a quadratic penalty term to the objective. It should be mentioned that Aqua will through a QiskitOptimizationError if conversion of a quadratic program with integer variable is attempted.

The circuit depth of QAOA potentially has to be increased with the problem size, which might be prohibitive for near-term quantum devices. A possible workaround is Recursive QAOA, as introduced in [2]. Qiskit generalizes this concept to the RecursiveMinimumEigenOptimizer, which is introduced at the end of this tutorial.

参考文献

[1] A. Lucas, Ising formulations of many NP problems, Front. Phys., 12 (2014).

[2] S. Bravyi, A. Kliesch, R. Koenig, E. Tang, Obstacles to State Preparation and Variational Optimization from Symmetry Protection, arXiv preprint arXiv:1910.08980 (2019).

QUBOをオペレーターに変換する

from qiskit import BasicAer
from qiskit.aqua import aqua_globals, QuantumInstance
from qiskit.aqua.algorithms import QAOA, NumPyMinimumEigensolver
from qiskit.optimization.algorithms import MinimumEigenOptimizer, RecursiveMinimumEigenOptimizer
from qiskit.optimization import QuadraticProgram

# create a QUBO
qubo = QuadraticProgram()
qubo.binary_var('x')
qubo.binary_var('y')
qubo.binary_var('z')
qubo.minimize(linear=[1,-2,3], quadratic={('x', 'y'): 1, ('x', 'z'): -1, ('y', 'z'): 2})
print(qubo.export_as_lp_string())

\ This file has been generated by DOcplex
\ ENCODING=ISO-8859-1
\Problem name: CPLEX

Minimize
 obj: x - 2 y + 3 z + [ 2 x*y - 2 x*z + 4 y*z ]/2
Subject To

Bounds
 0 <= x <= 1
 0 <= y <= 1
 0 <= z <= 1

Binaries
 x y z
End

Next we translate this QUBO into an Ising operator. This results not only in an Operator but also in a constant offset to be taking into account to shift the resulting value.

op, offset = qubo.to_ising()
print('offset: {}'.format(offset))
print('operator:')
print(op)

offset: 1.5
operator:
SummedOp([
  -0.5 * IIZ,
  0.25 * IZI,
  -1.75 * ZII,
  0.25 * IZZ,
  -0.25 * ZIZ,
  0.5 * ZZI
])

Sometimes an QuadraticProgram might also directly be given in the form of an Operator. For such cases, Qiskit also provides a converter from an Operator back to a QuadraticProgram, which we illustrate in the following.

qp=QuadraticProgram()
qp.from_ising(op, offset, linear=True)
print(qp.export_as_lp_string())

\ This file has been generated by DOcplex
\ ENCODING=ISO-8859-1
\Problem name: CPLEX

Minimize
 obj: x_0 - 2 x_1 + 3 x_2 + [ 2 x_0*x_1 - 2 x_0*x_2 + 4 x_1*x_2 ]/2
Subject To

Bounds
 0 <= x_0 <= 1
 0 <= x_1 <= 1
 0 <= x_2 <= 1

Binaries
 x_0 x_1 x_2
End

This converter allows, for instance, to translate an Operator to a QuadraticProgram and then solve the problem with other algorithms that are not based on the Ising Hamiltonian representation, such as the GroverOptimizer.

Solving a QUBO with the MinimumEigenOptimizer

まず、使用する MinimumEigensolver を初期化します。

aqua_globals.random_seed = 10598
quantum_instance = QuantumInstance(BasicAer.get_backend('statevector_simulator'),
                                   seed_simulator=aqua_globals.random_seed,
                                   seed_transpiler=aqua_globals.random_seed)
qaoa_mes = QAOA(quantum_instance=quantum_instance, initial_point=[0., 0.])
exact_mes = NumPyMinimumEigensolver()

次に、MinimumEigensolver を使用して MinimumEigenOptimizer を作成します。

qaoa = MinimumEigenOptimizer(qaoa_mes)   # using QAOA
exact = MinimumEigenOptimizer(exact_mes)  # using the exact classical numpy minimum eigen solver

We first use the MinimumEigenOptimizer based on the classical exact NumPyMinimumEigensolver to get the optimal benchmark solution for this small example.

exact_result = exact.solve(qubo)
print(exact_result)

optimal function value: -2.0
optimal value: [0. 1. 0.]
status: SUCCESS

Next we apply the MinimumEigenOptimizer based on QAOA to the same problem.

qaoa_result = qaoa.solve(qubo)
print(qaoa_result)

optimal function value: -2.0
optimal value: [0. 1. 0.]
status: SUCCESS

RecursiveMinimumEigenOptimizer

The RecursiveMinimumEigenOptimizer takes a MinimumEigenOptimizer as input and applies the recursive optimization scheme to reduce the size of the problem one variable at a time. Once the size of the generated intermediate problem is below a given threshold (min_num_vars), the RecursiveMinimumEigenOptimizer uses another solver (min_num_vars_optimizer), e.g., an exact classical solver such as CPLEX or the MinimumEigenOptimizer based on the NumPyMinimumEigensolver.

以下では、以前に導入した 2 つの MinimumEigenOptimizer を使用して RecursiveMinimumEigenOptimizer を使用する方法を示します。

最初に RecursiveMinimumEigenOptimizer を構築し、問題サイズを3つの変数から1つの変数に縮小し、最後の変数に正確なソルバーを使用します。 次に、検討された問題を最適化するために solve を呼び出します。

rqaoa = RecursiveMinimumEigenOptimizer(min_eigen_optimizer=qaoa, min_num_vars=1, min_num_vars_optimizer=exact)

rqaoa_result = rqaoa.solve(qubo)
print(rqaoa_result)

optimal function value: -2.0
optimal value: [0. 1. 0.]
status: SUCCESS

import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
Qiskit	None
Terra	0.16.0
Aer	0.7.0
Ignis	0.5.0
Aqua	0.8.1
IBM Q Provider	0.10.0
System information
Python	3.7.4 (default, Aug 13 2019, 15:17:50) [Clang 4.0.1 (tags/RELEASE_401/final)]
OS	Darwin
CPUs	2
Memory (Gb)	12.0
Wed Nov 11 11:22:40 2020 EST

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