CSE 599d  Quantum Computing
The Recursive and Nonrecursive BernsteinVazirani Algorithm
Dave Bacon
Department of Computer Science & Engineering, University of Washington
We have seen in the DeutschJozsa problem that a quantum computer could be used to solve a problem in a single
query which required an exponential number of queries on a classical computer in an
exact
model where no probability
of failure was allowed. However, when we allowed a bounded error in our algorithm, then this advantage disappeared.
After Deutsch and Jozsa introduced this problem, Bernstein and Vazirani took a deep look at quantum computers and
produced a problem which showed a superpolynomial separation between quantum and classical query complexity.
While Deutsch and Jozsa’s problem gave us tantalizing hints that there would be a difference between quantum and
classical query complexity, Bernstein and Vazirani made this hint real and showed, for the first time, that quantum
computers could do things that their classical brethren could not!
The BernsteinVazirani paper is also important for many other reasons, among them showing that the notion of
accuracy we used in our discussions of universal quantum computer adds and showing that bounded error quantum
computing is in the complexity class PSPACE. But we will focus on the query complexity results of this important
paper.
I.
NONRECURSIVE BERNSTEINVAZIRANI ALGORITHM
In the BernsteinVazirani problem, we are given a
n
bit function
f
:
{
0
,
1
}
n
→ {
0
,
1
}
which outputs a single
bit.
This function is guaranteed to be of the form
f
s
(
x
) =
x
·
s
where
s
is an unknown
n
bit string and
x
·
s
=
x
1
s
1
+
x
2
s
2
+
· · ·
+
x
n
s
n
mod 2. The goal of the BernsteinVazirani problem is to find the unknown string
s
.
What is the classical query complexity of this problem? First suppose we are working in the exact query complexity
model. Then a single query to the function can retrieve at most one bit of information about
s
(because
f
outputs
only one bit.) Thus the exact query complexity must be at least
n
. In fact we can easily see that there is an algorithm
which does in exactly
n
queries by querying with bitstrings which have the
i
th 1 and the other digits 0: this reveals
the values of
s
i
.
What about if we allow bounded error? Here again we can use the fact that the function only outputs a single bit.
Suppose that we could query this function in a probabilistic fashion and learn all of the bits of
s
in less that
n
bits
with some probability of failure that is bounded below 1
/
2. Then we would be able to use this function to compress an
n
bit string into less than
n
bits and use this to communicate these
n
bits, with high probability, through a channel.
If we use the rigorous definition of information, we then essentially obtain that a bounded error algorithm will also
need Ω(
n
) queries. Another way to prove this is to use Yao principle. We’ll talk about this approach later.
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 Fall '08
 Staff
 Computer Science, Computational complexity theory, Quantum computer, Qubit, Vazirani, BernsteinVazirani problem

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