Abstract
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D (4) transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, S. Lloyd, “Experimental realization of a quantum algorithm„, Nature, 393, p.143, 1998.
J. A. Jones, M. Mosca, R. H. Hansen, “Implementation of a Quantum Search Algorithm on a Nuclear Magnetic Resonance Quantum Computer„, Nature, 393, p.344, 1998.
I. Chuang and Y. Yamamoto, “A Simple Quantum Computer„, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9505011, 1995.
D. Deutsch and R. Jozsa,“Rapid solution of problems by quantum computation„, Proc. Royal Society London, Series A, Vol. 439, p. 553, 1992.
P. Shor, “Algorithms for quantum computation: discrete logarithms and factoring„, Proc. 35th Annual Symposium on Foundations of Computer Science, p. 124, 1994.
L. K. Grover, “A Fast Quantum Mechanical Algorithm for Database Search„ Proc. 28th Annual ACM Symposium on the Theory of Computing, Philadelphia, p. 212, 1996.
Brassard, P. Hoyer, A. Tapp, “Quantum Counting„, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9805082, 1998.
N.J. Cerf, L. K. Grover and C. P. Williams, “Nested quantum search and NP-complete problems„, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9806078, 1998.
W. van Dam, P. Hoyer, A. Tapp, “Multiparty Quantum Communication Complexity„, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9710054, 1997.
C. Zalka,“Grover’s quantum searching algorithm is optimal„, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9711070, 1998.
D. Aharonov, A. Kitaev, N. Nisan,“Quantum circuits with mixed states,„ Proc. 13th Annual ACM Symposium on Theory of Computation, p. 20, 1997. Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits 33
R. Jozsa,“Quantum algorithms and the Fourier transform,„ Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9707033, 1997.
M. Reck, A. Zeilinger, H.J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator„, Physical Review Letters, 73, p. 58, 1994.
E. Knill,“Approximation by quantum circuits,„Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9508006, 1995.
A. Barenco, A. Ekert, K-A Suominen, and P. Torma,“Approximate quantum Fourier transform and decoherence,„ Physical Review A, 54, p. 139, 1996.
C. Van Loan, Computational Frameworks for the Fast Fourier Transform. SIAM Publications, Philadelphia, 1992.
B.J. Fino and R. Alghazi, “A unified treatment of discrete unitary transforms,„ SIAM J. Comput., 6(4), p. 700, 1977.
V. Vedral, A. Barenco, A. Ekert,“Quantum networks for elementary arithmetic operations,„ Physical Review A, 54, p. 147, 1996.
I. Daubechies, “Orthonormal bases of compactly supported wavelets,„ Comm Pure Appl. Math., 41, p. 909, 1988.
P. Hoyer,“Efficient quantum Transforms,„ Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9702028, Feb. 1997.
D. Beckman, A.N. Chari, S. Devabhatuni, and J. Preskill, “Efficient networks for quantum factoring,„ Physical Review A, 54, p. 1034, 1996.
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. 2nd Edition, Cambridge Univ. Press, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fijany, A., Williams, C.P. (1999). Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits. In: Williams, C.P. (eds) Quantum Computing and Quantum Communications. QCQC 1998. Lecture Notes in Computer Science, vol 1509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49208-9_2
Download citation
DOI: https://doi.org/10.1007/3-540-49208-9_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65514-5
Online ISBN: 978-3-540-49208-5
eBook Packages: Springer Book Archive