210 research outputs found
Wavelets and Wavelet Packets on Quantum Computers
We show how periodized wavelet packet transforms and periodized wavelet
transforms can be implemented on a quantum computer. Surprisingly, we find that
the implementation of wavelet packet transforms is less costly than the
implementation of wavelet transforms on a quantum computer.Comment: 11 pages, 10 postscript figure, to appear in Proc. of Wavelet
Applications in Signal and Image Processing VI
On the Irresistible Efficiency of Signal Processing Methods in Quantum Computing
We show that many well-known signal transforms allow highly efficient
realizations on a quantum computer. We explain some elementary quantum circuits
and review the construction of the Quantum Fourier Transform. We derive quantum
circuits for the Discrete Cosine and Sine Transforms, and for the Discrete
Hartley transform. We show that at most O(log^2 N) elementary quantum gates are
necessary to implement any of those transforms for input sequences of length N.Comment: 15 pages, LaTeX 2e. Expanded version of quant-ph/0111038. SPECLOG
2000, Tampere, Finlan
On the Monomiality of Nice Error Bases
Unitary error bases generalize the Pauli matrices to higher dimensional
systems. Two basic constructions of unitary error bases are known: An algebraic
construction by Knill, which yields nice error bases, and a combinatorial
construction by Werner, which yields shift-and-multiply bases. An open problem
posed by Schlingemann and Werner (see
http://www.imaph.tu-bs.de/qi/problems/6.html) relates these two constructions
and asks whether each nice error basis is equivalent to a shift-and-multiply
basis. We solve this problem and show that the answer is negative. However, we
also show that it is always possible to find a fairly sparse representation of
a nice error basis.Comment: 6 page
Beyond Stabilizer Codes II: Clifford Codes
Knill introduced a generalization of stabilizer codes, in this note called
Clifford codes. It remained unclear whether or not Clifford codes can be
superior to stabilizer codes. We show that Clifford codes are stabilizer codes
provided that the abstract error group has an abelian index group. In
particular, if the errors are modelled by tensor products of Pauli matrices,
then the associated Clifford codes are necessarily stabilizer codes.Comment: 9 pages, LaTeX2e. Minor changes. Title changed by request of IEEE
Trans. I
Hybrid Codes
A hybrid code can simultaneously encode classical and quantum information
into quantum digits such that the information is protected against errors when
transmitted through a quantum channel. It is shown that a hybrid code has the
remarkable feature that it can detect more errors than a comparable quantum
code that is able to encode the classical and quantum information. Weight
enumerators are introduced for hybrid codes that allow to characterize the
minimum distance of hybrid codes. Surprisingly, the weight enumerators for
hybrid codes do not obey the usual MacWilliams identity.Comment: 5 page
Discrete Cosine Transforms on Quantum Computers
A classical computer does not allow to calculate a discrete cosine transform
on N points in less than linear time. This trivial lower bound is no longer
valid for a computer that takes advantage of quantum mechanical superposition,
entanglement, and interference principles. In fact, we show that it is possible
to realize the discrete cosine transforms and the discrete sine transforms of
size NxN and types I,II,III, and IV with as little as O(log^2 N) operations on
a quantum computer, whereas the known fast algorithms on a classical computer
need O(N log N) operations.Comment: 5 pages, LaTeX 2e, IEEE ISPA01, Pula, Croatia, 200
Beyond Stabilizer Codes I: Nice Error Bases
Nice error bases have been introduced by Knill as a generalization of the
Pauli basis. These bases are shown to be projective representations of finite
groups. We classify all nice error bases of small degree, and all nice error
bases with abelian index groups. We show that in general an index group of a
nice error basis is necessarily solvable.Comment: 12 pages, LaTeX2e. Minor changes. Title changed by request of IEEE
Trans. I
Constructions of Mutually Unbiased Bases
Two orthonormal bases B and B' of a d-dimensional complex inner-product space
are called mutually unbiased if and only if ||^2=1/d holds for all b in B
and b' in B'. The size of any set containing (pairwise) mutually unbiased bases
of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets
containing d+1 mutually unbiased bases are known to exist. We give a simplified
proof of this fact based on the estimation of exponential sums. We discuss
conjectures and open problems concerning the maximal number of mutually
unbiased bases for arbitrary dimensions.Comment: 8 pages late
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