Improved Complexity of Quantum Oracles for Ternary Grover Algorithm for Graph Coloring

The paper presents a generalization of the well-known Grover Algorithm to operate on ternary quantum circuits. We compare complexity of oracles and some of their commonly used components for binary and ternary cases and various sizes and densities of colored graphs. We show that ternary encoding leads to quantum circuits that have significantly less qud its and lower quantum costs. In case of serial realization of quantum computers, our ternary algorithms and circuits are also faster.

[1]  Yale Fan,et al.  A Generalization of the Deutsch-Jozsa Algorithm to Multi-Valued Quantum Logic , 2007, 37th International Symposium on Multiple-Valued Logic (ISMVL'07).

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

[3]  C. Lavor,et al.  Grover's Algorithm: Quantum Database Search ∗ , 2003 .

[4]  Lun Li,et al.  A Quantum CAD Accelerator Based on Grover’s Algorithm for Finding the Minimum Fixed Polarity Reed-Muller Form , 2006, 36th International Symposium on Multiple-Valued Logic (ISMVL'06).

[5]  D. Michael Miller,et al.  QMDD: A Decision Diagram Structure for Reversible and Quantum Circuits , 2006, 36th International Symposium on Multiple-Valued Logic (ISMVL'06).

[6]  Richard Jozsa.  Searching in Grover's Algorithm , 1999 .

[7]  Sazzad Hossain,et al.  Classical search and quantum search algorithms for synthesis of quantum circuits and optimization of quantum oracles , 2009 .

[8]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[9]  Tad Hogg,et al.  Graph coloring with quantum heuristics , 2002, AAAI/IAAI.

[10]  Anas N. Al-Rabadi,et al.  Multiple-Valued Quantum Logic , 2002 .

[11]  Jr.,et al.  Multivalued logic gates for quantum computation , 2000, quant-ph/0002033.

[12]  Gerhard W. Dueck,et al.  Improved quantum cost for n-bit Toffoli gates , 2003 .

[13]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[14]  U. Vazirani,et al.  Quantum algorithms and the fourier transform , 2004 .

[15]  Katarzyna Radecka,et al.  Scaling and Better Approximating Quantum Fourier Transform by Higher Radices , 2007, IEEE Transactions on Computers.