1. Quantum Circuit Simplification Using Templates D. Maslov - University of Victoria, Canada G. W. Dueck - UNB, Canada C. Young - University of Victoria, Canada D. M. Miller - University of Victoria, Canada

2. Outline - Background - The templates (definition, application) - Results - Future work page 1/11 DATE, Munich, Germany March 10, 2005

3. Background page 2/11 DATE, Munich, Germany March 10, 2005 NOT CNOT (Feynman) InOut 01 10 InOut 0 0 1 1 01 1 0 A quantum circuit is composed of gates arranged in a cascade: …

4. Background page 3/11 DATE, Munich, Germany March 10, 2005 V Iff control value is 1, the target is changed according to. In practice, controlled-V gates are used w/Boolean controls. InOut 0 0 1 1 01 v0 1 1 v1 InOut 0 v0 0 v1 1 v01 1 v11 0 and

5. Background page 4/11 DATE, Munich, Germany March 10, 2005 V+V+ Controlled-V+ is the inverse of controlled- V. - Set of gates: NOT, CNOT, controlled-V and controlled-V+ is complete. - Set of gates NOT and controlled-V is complete. - Controlled-V is complete on its own. With the addition of a constant line:

6. The Templates: Definition page 5/11 DATE, Munich, Germany March 10, 2005 Observation 1. If one has a circuit equality then the gates satisfy the equation Observation 3. If, then Observation 2. For each circuit and for any parameter p,.

7. The Templates: Definition page 6/11 DATE, Munich, Germany March 10, 2005 A size m template is a cascade of m gates that realizes the identity function. Any template T of size m should be independent of smaller size templates, i.e. application of smaller templates does not decrease the number of gates in T or make it equal to another template. Given G 0 G 1 …G m-1, a template of size m, its application for parameter p, is: - for

8. The Templates: Definition page 7/11 DATE, Munich, Germany March 10, 2005 A B C D E F G Example. Template ABCDEFG. p=4. Starting gate B. Direction: backward. B -1 B A -1 A G -1 G F -1 F C C D D E E

9. Quantum Gate Templates page 8/11 DATE, Munich, Germany March 10, 2005 Gate-inverse rule. Moving rule. Gate G 1 with control c 1 and target t 1 passes gate G 2 with c 2, t 2 iff Other templates:

10. The Templates: Application page 9/11 DATE, Munich, Germany March 10, 2005 Similar to the string matching methods in bioinformatics… V V+V+ V+V+V V+V+ V

11. The Templates: Application page 9/11 DATE, Munich, Germany March 10, 2005 Similar to the string matching methods in bioinformatics… V V+V+ V+V+ VV+V+ V

12. The Templates: Application page 9/11 DATE, Munich, Germany March 10, 2005 Similar to the string matching methods in bioinformatics… V+V+ V V V+V+

13. The Templates: Application page 9/11 DATE, Munich, Germany March 10, 2005 Similar to the string matching methods in bioinformatics… V+V+ V V V+V+

14. The Templates: Application page 9/11 DATE, Munich, Germany March 10, 2005 Similar to the string matching methods in bioinformatics… V+V+ V V V+V+

15. Results page 10/11 DATE, Munich, Germany March 10, 2005 Size m+1 Toffoli gates in simulation requiring m-2 auxiliary bits simplified from 16m-32 to 12m-22. Size m+1 Toffoli gates in simulation requiring a single auxiliary bit simplified from 32m-64 to 24m-64. NameCost beforeOptimized 4mod5139 5mod58045 full adder12(8)6 mod1024adder19751521 rd538668

16. Future Work DATE, Munich, Germany March 10, 2005 Find more templates and construct complete sets of templates of small size. Extend our work to support other gate bases. page 11/11 Optimize critical path in the circuit using the templates (gate level compaction). Find better algorithms for large Toffoli gate decomposition into the elementary quantum.

17. END Thank you for your attention! Quantum Circuit Simplification Using Templates