An Adaptive Quantum-inspired Differential Evolution Algorithm
for 0-1 Knapsack Problem
Ashish Ranjan Hota
Department of Electrical Engineering
Indian Institute of Technology
Department of Mathematics
Indian Institute of Technology
Abstract — Differential evolution (DE) is a population based
evolutionary algorithm widely
multidimensional global optimization
continuous spaces. However, the design of its operators makes
it unsuitable for many real-life constrained combinatorial
optimization problems which operate on binary space. On the
other hand, the quantum inspired evolutionary algorithm
(QEA) is very well suitable for handling such problems by
applying several quantum computing techniques such as Q-bit
representation and rotation gate operator, etc. This paper
extends the concept of differential operators with adaptive
parameter control to the quantum paradigm and proposes the
adaptive quantum-inspired differential evolution algorithm
(AQDE). The performance of AQDE is found to be
significantly superior as compared to QEA and a discrete
version of DE on the standard 0-1 knapsack problem for all the
considered test cases.
used for solving
evolutionary algoithm; 0-1 knapsack problem; quantum
differential evolution; quantum inspired
Differential Algorithm (DE), introduced by Storn and
Price [1,2] has been shown to give significantly better
performance in terms of efficiency and robustness on many
benchmark multimodal continuous functions than other
population based evolutionary algorithms. For exploration of
the search space and to introduce diversity, it employs two
simple mutation and crossover operators respectively
followed by a greedy replacement strategy. The performance
is found to be very sensitive to the mutation and crossover
parameters chosen and the best combination of both the
parameters changes from one function to another. Thus, a
large number of modifications have been proposed to make
the selection of control parameters adaptive and free from
function dependency [3-7].
Because of its superior performance on continuous
optimization problems, several modifications have been
introduced in the past, so that it operates on binary space.
Pampara, Engelbrecht and Franken  proposed an angle
modulation scheme (AMDE) to map the continuous space to
binary. On similar lines, binary differential evolution
(binDE) and normalization DE (normDE) were proposed
based on sigmoid function mapping and normalization of
continuous space respectively  giving better results as
compared to AMDE. A discrete binary version of differential
evolution (DBDE) for solving 0-1 knapsack problem was
also proposed .
To solve various optimization problems better than the
conventional evolutionary algorithms, a broad class of
algorithms have been proposed by applying several concepts
of quantum computing in the past decade. Quantum
computing uses the quantum mechanical phenomena like
superposition, entanglement, interference, de-coherence, etc
to develop quantum algorithms. Many quantum algorithms
have been shown to be exponentially faster and massively
parallel as compared to classical algorithms [11, 12]. Thus
quantum inspired genetic algorithms with interference as
crossover operator , quantum inspired evolutionary
algorithms (QEA) , quantum behaved particle swarm
optimization  etc has been developed for both continuous
and binary spaces.
QEA uses superposition of binary bits known as Q-bit for
representation of individuals and updates the individuals
depending on their values with respect to the global best
solution by suitably deciding the parameter of the rotation
gate operator. Broadly it comes under the class of estimation
of distribution algorithms (EDA) . QEA has
demonstrated quite significant results on binary optimization
problems and some improvements on QEA have also been
proposed [16, 17]. QEAs have been extended by differential
operators to solve flow shop scheduling problems , N-
queen’s problem , for classification rule discovery 
and some benchmark functions .
In this paper, an adaptive quantum-inspired differential
evolution algorithm (AQDE) is proposed with adaptive
control of mutation and crossover parameters and the
operators acting directly on the superposition states of the
individual. The proposed AQDE outperforms QEA and
DBDE under different conditions of population size and item
size of the 0-1 knapsack problem.
The rest of this paper is organized as follows: Section II
gives a brief introduction of knapsack problem, DE, DBDE
and QEA. The proposed AQDE is explained in detail in
section III. Experimental settings and the results obtained are
mentioned under section IV. Finally, section V concludes the
Figure 3-8 show the progress of the convergence by
depicting the average of best profits over 30 runs for the
previously mentioned population sizes and item sizes. For
all the cases considered, the curve of mean best profit for
AQDE lies slightly below the curves of QEA and DBDE for
the initial 50 generations, but soon after that, it goes above
the curves of QEA and DBDE, thereby showing
significantly better results. The plots suggest a premature
convergence of both QEA and DBDE as compared to
In this paper, we have proposed a novel AQDE
algorithm for solving the 0-1 Knapsack problem. The
proposed algorithm is a hybrid of QEA and DE along with a
novel adaptive parameter control method. The experimental
results have proved the superior performance of AQDE
compared to QEA and DBDE. Here, the performance of
AQDE was tested only on the 0-1 Knapsack problem. With
some modifications, the concept of the algorithm may be
extended to other discrete combinatorial optimization
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