Unbalanced penalization: A new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms
Quantum Science and Technology(2022)
摘要
Solving combinatorial optimization problems of the kind that can be codified
by quadratic unconstrained binary optimization (QUBO) is a promising
application of quantum computation. Some problems of this class suitable for
practical applications such as the traveling salesman problem (TSP), the bin
packing problem (BPP), or the knapsack problem (KP) have inequality constraints
that require a particular cost function encoding. The common approach is the
use of slack variables to represent the inequality constraints in the cost
function. However, the use of slack variables considerably increases the number
of qubits and operations required to solve these problems using quantum
devices. In this work, we present an alternative method that does not require
extra slack variables and consists of using an unbalanced penalization function
to represent the inequality constraints in the QUBO. This function is
characterized by larger penalization when the inequality constraint is not
achieved than when it is. We evaluate our approach on the TSP, BPP, and KP,
successfully encoding the optimal solution of the original optimization problem
near the ground state cost Hamiltonian. Additionally, we employ D-Wave
Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the
performance of the slack variables approach by achieving solutions for up to 29
items, whereas the slack variables approach only handles up to 11 items. This
new approach can be used to solve combinatorial problems with inequality
constraints with a reduced number of resources compared to the slack variables
approach using quantum annealing or variational quantum algorithms.
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关键词
combinatorial optimization,QUBO,inequality constraints,quantum optimization,QAOA,quantum annealing,D-Wave
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