Fast Signal Transforms for Quantum ComputersMartin

We present the discrete Fourier transform as a basic primitive in the treatment of controlled quantum systems. Based on the complexity model for quantum circuits the Fourier transform of size 2 n surprisingly can be realised with O(n 2) elementary operations which is an exponential speedup compared to the classical case. This is the reason for its presence in almost all known quantum algorithms among which Shor's algorithm for factoring is the most prominent example. We show how recent results in the theory of signal processing (for a classical computer) can be applied to obtain fast quantum algorithms for various discrete signal transforms. As an example we derive a quantum circuit implementing the discrete cosine transform DCTIV(8) eeciently.

[1]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[2]  R. Feynman.  Simulating physics with computers , 1999 .

[3]  Charles M. Rader,et al.  Fast transforms: Algorithms, analyses, applications , 1984 .

[4]  Henrique S. Malvar,et al.  Signal processing with lapped transforms , 1992 .

[5]  Umesh V. Vazirani,et al.  Quantum complexity theory , 1993, STOC.

[6]  Torsten Minkwitz.  Algorithmensynthese für lineare Systeme mit Symmetrie , 1993 .

[7]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[8]  Reck,et al.  Experimental realization of any discrete unitary operator. , 1994, Physical review letters.

[9]  Barenco,et al.  Elementary gates for quantum computation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[10]  J. Cirac,et al.  Quantum Computations with Cold Trapped Ions. , 1995, Physical review letters.

[11]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[12]  Sebastian Egner,et al.  Zur algorithmischen Zerlegungstheorie linearer Transformationen mit Symmetrie , 1997 .

[13]  Michele Mosca,et al.  The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer , 1998, QCQC.

[14]  Markus Püschel.  Konstruktive Darstellungstheorie und Algorithmengenerierung , 1998 .

[15]  Martin Rötteler,et al.  Fast Quantum Fourier Transforms for a Class of Non-Abelian Groups , 1999, AAECC.

[16]  W. Kester.  Fast Fourier Transforms , 2003 .

[17]  Shmuel Friedland,et al.  On the graph isomorphism problem , 2008, ArXiv.