Quantum Information Processing, Vol. 5, No. 1, February 2006 (© 2006)
On the Complexity of Searching for a Maximum
of a Function on a Quantum Computer
Received October 16, 2005; accepted January 14, 2006
We deal with the problem of ﬁnding a maximum of a function from the H
class on a quantum computer. We show matching lower and upper bounds on
the complexity of this problem. We prove upper bounds by constructing an algo-
rithm that uses a pre-existing quantum algorithm for ﬁnding maximum of a dis-
crete sequence. To prove lower bounds we use results for ﬁnding the logical OR of
sequence of bits. We show that quantum computation yields a quadratic speed-up
over deterministic and randomized algorithms.
KEY WORDS: Numerical optimization; optimal algorithm; quantum comput-
ing; query complexity.
PACS: 03.67.Lx; 02.60.Pn.
Quantum algorithms yield a speed-up over deterministic and Monte–Carlo
algorithms for many problems. Many papers deal with quantum algo-
rithms for discrete problems, starting from the work of Ref. 20, followed
by database search algorithm of Ref. 7. Usefulness of the Grover’s search
algorithm for an actual database was investigated by Zalka.
crete problems, such as discrete summation, computation of the mean,
median and kth-smallest element were also studied (see e.g. Refs. 4–6, 8
and 15). There is also a progress in studying the quantum complexity of
numerical problems. Comparing the complexity of numerical problems in
the quantum and the classical deterministic and randomized settings is
a challenging task. The ﬁrst paper dealing with the quantum complexity
Department of Applied Mathematics, AGH University of Science and Technology, Al.
Mickiewicza 30, paw. B7, II p., pok. 24, 30-059 Cracow, Poland. E-mail: email@example.com.
1570-0755/06/0200-0031/0 © 2006 Springer Science+Business Media, Inc.