• DOI:10.1007/s10208-002-0074-6
• Corpus ID: 10519614

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

@article{Novak2004TractabilityOA,
  title={Tractability of Approximation 
for Weighted Korobov Spaces
on Classical and Quantum Computers},
  author={Erich Novak and Ian H. Sloan and Henryk Wozniakowski},
  journal={Foundations of Computational Mathematics},
  year={2004},
  volume={4},
  pages={121-156}
}

• E. Novak, I. Sloan, H. Wozniakowski
• Published 4 June 2002
• Mathematics, Computer Science
• Foundations of Computational Mathematics

Abstract
We study the approximation problem (or problem of optimal recovery in the
$L_2$-norm) for weighted Korobov spaces with smoothness
parameter $\a$. The weights $\gamma_j$ of the Korobov spaces moderate
the behavior of periodic functions with respect to successive variables.
The nonnegative smoothness parameter $\a$ measures the decay
of Fourier coefficients. For $\a=0$, the Korobov space is the
$L_2$ space, whereas for positive $\a$, the Korobov space
is a space of periodic functions with some smoothness
and the approximation problem
corresponds to a compact operator. The periodic functions are defined on
$[0,1]^d$ and our main interest is when the dimension $d$ varies and
may be large. We consider algorithms using two different
classes of information.
The first class $\lall$ consists of arbitrary linear functionals.
The second class $\lstd$ consists of only function values
and this class is more realistic in practical computations.

We want to know when the approximation problem is
tractable. Tractability means that there exists an algorithm whose error
is at most $\e$ and whose information cost is bounded by a polynomial
in the dimension $d$ and in $\e^{-1}$. Strong tractability means that
the bound does not depend on $d$ and is polynomial in $\e^{-1}$.
In this paper we consider the worst case, randomized, and quantum
settings. In each setting, the concepts of error and cost are defined
differently and, therefore, tractability and strong tractability
depend on the setting and on the class of information.

In the worst case setting, we apply known results to prove
that strong tractability and tractability in the class $\lall$
are equivalent. This holds
if and only if $\a>0$ and the sum-exponent $s_{\g}$ of weights is finite, where
$s_{\g}= \inf\{s>0 : \xxsum_{j=1}^\infty\g_j^s\,<\,\infty\}$.

In the worst case setting for the class $\lstd$ we must assume
that $\a>1$ to guarantee that
functionals from $\lstd$ are continuous. The notions of strong
tractability and tractability are not equivalent. In particular,
strong tractability holds if and only if $\a>1$ and
$\xxsum_{j=1}^\infty\g_j<\infty$.

In the randomized setting, it is known that randomization does not
help over the worst case setting in the class $\lall$. For the class
$\lstd$, we prove that strong tractability and tractability
are equivalent and this holds under the same assumption
as for the class $\lall$ in the worst case setting, that is,
if and only if $\a>0$ and $s_{\g} < \infty$.

In the quantum setting, we consider only upper bounds for the class
$\lstd$ with $\a>1$. We prove that $s_{\g}<\infty$ implies strong
tractability.

Hence for $s_{\g}>1$, the randomized and quantum settings
both break worst case intractability of approximation for
the class $\lstd$.

We indicate cost bounds on algorithms with error at
most $\e$. Let $\cc(d)$ denote the cost of computing $L(f)$ for
$L\in \lall$ or $L\in \lstd$, and let the cost of one arithmetic
operation be taken as unity.
The information cost bound in the worst case setting for the
class $\lall$ is of order $\cc (d) \cdot \e^{-p}$
with $p$ being roughly equal to $2\max(s_\g,\a^{-1})$.
Then for the class $\lstd$
in the randomized setting,
we present an algorithm with error at most $\e$ and whose total cost is
of order $\cc(d)\e^{-p-2} + d\e^{-2p-2}$, which for small $\e$ is roughly
$$
d\e^{-2p-2}.
$$
In the quantum setting, we present a quantum algorithm
with error at most $\e$ that
uses about only $d + \log \e^{-1}$ qubits
and whose total cost is of order
$$
(\cc(d) +d) \e^{-1-3p/2}.
$$
The ratio of the costs of the algorithms in the quantum setting and
the randomized setting is of order
$$
\frac{d}{\cc(d)+d}\,\left(\frac1{\e}\right)^{1+p/2}.
$$
Hence, we have a polynomial speedup of order $\e^{-(1+p/2)}$.
We stress that $p$ can be arbitrarily large, and in this case
the speedup is huge.