Quantum Behaviors: synthesis and measurement

1. Martin Lukac Normen Giesecke Sazzad Hossain and Marek Perkowski Department of Electrical Engineering Portland State University 1900 SW Fourth Avenue Oregon, USA E-mail: {lukacm,mperkows}@ece.pdx.edu Dong Hwa Kim Dept. of Instrumentation and Control Engn. Hanbat National University, 16-1 San Duckmyong-Dong Yuseong-Gu, Daejon, Korea, 305-719. E-mail: kimdh@hanbat.ac.kr Quantum Behaviors:synthesis and measurement

2. Overview • Motivations and Problem Definition • Quantum computing basics • Quantum Inductive Learning • Controlled [V/V*] gate synthesis • Measurement dependent synthesis • Simulations and results • Conclusion and future work

3. Motivations • Human-Human interaction is highly variable, individual, unique, non-repeating, etc. • Emotional Robot, Humanoid Robot • Quantum emotional state machine • Control logic for robotic quantum controllers in order to increase interactivity and quality of communication • Logic synthesis of such circuits is in the middle of this paper

4. Synthesis from examples • Quantum mappings – Quantum Braitenberg Vehicles – Arushi ISMVL 2007 • Quantum Oracles such as Grover – Yale ISMVL 2007 • Emotional State Machines – Lukac ISMVL 2007 • Quantum Automata and Cellular Quantum Automata – Lukac ULSI 2007 • Motion – Quay and Scott

5. Quantum Robots for Teenagers

6. Quantum Emotional Facial Gestures

7. Quantum computing basics • Units are qubits, quantum bits, represented by wave function, on real (observable bases) in the complex Vector Space H. • Unitary transformations on single and two qubits (rotations in the Complex Hilbert Space), example rotation around X axis : • Because quantum states are complex, they are measured (or observed) before they can be recorded in the real world. The measurement operation describe this fact: Difference of complete measurement and expected measurement in a robot.

8. Quantum computing basics • On of the particular properties of Quantum Computation is the superposition of states: allowing to synthesise quantum probabilistic logic functions • Because the coefficients of the states are complex positive and negative), interference occurs allowing to sum or subtract probabilities of observation of each state. Gates such as CV can be used to synthesize permutative functions with real state transition coefficients (boolean reversible functions) • and entanglement (initially known as EPR) Meaning of entanglement in terms of gestures

9. Three Types of Quantum Inductive Learning c 0 1 ab 00 0 - 01 1 - 11 - 1 c 0 1 ab 10 - 1 c 0 1 00 0 1 ab c 0 1 01 1 0 ab 00 0 11 0 1 00 0 01 1 10 0 1 01 1 11 1 11 1 10 1 10 1 Classical Deterministic Learning Probabilistic and Quantum Probabilistic Learning Quantum Probabilistic and Measurement Dependent Learning

10. Controlled [V/V*] gates • V, V*, C-V, C-V*, are well know elements of quantum logic synthesis for pseudo-boolean (permutative) functions. V V* V* V* X * V* V I V* V X V CNOT

11. Various types of measurement • V, V*, C-V, C-V*, are well know elements of quantum logic synthesis for pseudo-boolean (permutative) functions. 0 or 1 M V V*

12. Various types of measurement 0 or nothing M0 V V* 1 or nothing M1 If nothing, previous action is continued

13. Various types of measurement Measurement here is deterministic 0 or 1 M V V V* V0 or V1 Operator built-in the measurement Measurement here would be non-deterministic

14. Symbolic Quantum synthesis c 0 1 ab c 0 1 ab 00 0 1 00 0 1 01 - - c 0 1 ab 01 0 1 11 1 0 00 11 1 0 10 - - 10 0 1 01 0 11 1 0 10 1 • Assume function to be synthesized: In the case when all outputs are deterministic (using only CNOT, CV, CV*, V and V*, the parity of application of each V-type gate on the output must be of order 2n or 0. When the output is specified by probabilities corresponding to V0 or V1 the parity of applying the V-based gates is odd (2n-1, for n > 0).

15. Measurement dependent synthesis • The measurement synthesis is interesting from the behavioral point of view: when a robotic controller generates commands all signals going to classical actuators must be completely deterministic. • Because measurement is considered as action the robot must do to generate output, the function can be minimized with respect to M (measurement) Assume completely defined reversible function: With respect to the expected result after the measurement on the output qubits, the function can be written as:

16. Measurement dependent synthesis (contd.) • Further introduction of entanglement into the output in the form of Bell bases states: • Allows to rewrite the measurement based definition to a single qubit dependent form. Also note that M0 is the state of the system after being measured for 0, and m0 = 1 is the actual output (value 0) after this measurement : • Using, the fact that we have to measure only a single qubit to obtain a completely specified (not probabilistic) result. The output specification of the function requires in this case a real (not quantum) register holding the values of the measured qubit, allowing to determine whether the measurement operation yielded a correct result

17. Simulations and results • All methods have been simulated using Genetic Algorithm to test this approach. • In this case we tested specifically single qubit quantum functions. • These are functions in which only one bit is truly quantum, other bits are permutative functions • The quantum symbolic synthesis is based on a circuit-type generator of the form: a b c f1 f2 fn ........... gn d g1 g2

18. Simulations and results • Functions fi are “simple”: • Linear • Affine • Toffoli-like • They can be binary or multiple-valued • Functions gi are “square roots of unity”: • NOT • Square-root-of-NOT • Fourth-order-root-of-NOT • etc • They can be for realization of binary or multiple-valued logic a b c f1 f2 fn Exhaustive Search A* search Genetic Algorithm Iterative Deepening ........... gn d g1 g2

19. e

20. Simulations and results cd 00 01 11 10 ab 00 - - 1 0 01 0 1 0 1 abc d d d 11 0 0 - - 000 - - 0 10 0 1 0 1 001 VV* I 0 011 VV NOT 1 010 VV* I 0 110 VV NOT 1 111 VVVV* NOT *I 1 101 VV NOT 1 100 VV* I 0 • Example 1: Classical Synthesis of reversible functions applied as a classical Machine Learning Symbolic synthesis Method

21. Simulations and results (contd.) Observe that this is a generalization of the well-known realization of Toffoli invented by Barenco et al • Circuit for the function from previous slide, realizing a symmetric function on the output (D) qubit: a b c d V V V V* We can create this type of functions for any number of variables They are inexpensive in quantum but complex in Reed-Muller • Observe: • All controls are linear only • All targets are square roots and their adjoints only

22. Simulations and results (contd.) c 0 1 ab 00 0 1 01 - - 11 1 0 10 - - c 0 1 c 0 1 c 0 1 ab ab ab 00 - - 00 - - 00 0 1 01 V V 01 V0 V1 01 V0 V1 11 VV VV 11 NOT NOT 11 1 0 10 V V V0 V1 V0 V1 10 10 Multi-valued (quaternary) Synthesis of quantum functions applied as a new Quantum Machine Learning • Example 2: a b c V V Synthesized function matches all required cares

23. Measurement dependent synthesis Simulations and results (contd.) c 0 1 ab 00 0 1 01 - - 11 1 0 10 - - • Example 2 (contd.): Multi-valued (quaternary) Synthesis of quantum functions applied as a new Quantum Machine Learning • Another solution (completely deterministic) is just slightly more complicated: a b c 0 V V • This function can also be easily synthesized using entanglement and measurement. Simply generate an entanglement circuit creating these bases states: And define measurement criteria satisfying for each care in the K-map the desired output value:

24. Learning error

26. Conclusion • We proposed two complementary mechanisms for learning: Symbolic and Measurement Dependent. Measurement Dependent method assumes known output events and their probabilities – there are several unitary matrices for the same input-output probabilistic behavior (H or V) Symbolic method assumes known hidden states, predicts probabilistically the output events Symbolic quantum learning Measurement dependent Learning Quantum Circuit Measurement Boolean Inputs Probabilities Environment

27. Conclusion (contd.) • The quantum symbolic method is ideal for single output reversible functions, heuristics and AI search methods can be easily applied • The measurement dependent method requires the external register of size 2n (or of size of the desired input-output set of data), however the synthesis part is trivial. • Any entanglement circuit can be automatically build for any reversible function using only gates H, CNOT and X. • Future work: • extensions to multi-qubit quantum functions, d-level functions • implementation and verification of these mechanisms in the Cynthea robotic framework

28. Future work Grover Loop Constants Hadamards Measurements Grover Search Oracle or Quantum Circuit Quantum Braitenberg Vehicle Inputs- sensors Outputs - actuators Measurements New Concept of Real-time Quantum Search Grover Loop Constants Controlled Hadamards Outputs - actuators Measurements Inputs- sensors LEARNING Control

29. Additional Slides

30. Toffoli Gate as an example of composition of affine control gates and rotation target gates

32. Controlled gates

40. Synthesis of Majority