1. Quantum Behaviors: synthesis and measurement 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

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.  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) Quantum computing basics On of the particular properties of Quantum Computation is the superposition of states: allowing to synthesise quantum probabilistic logic functions  and entanglement (initially known as EPR) Meaning of entanglement in terms of gestures

9. Three Types of Quantum Inductive Learning ab c01 00 01 11 10 0 1 1 1 1 0 0 0 ab c01 00 01 11 10 0 1 1 1 - - - - ab c01 00 01 11 10 0 1 1 1 ab c01 00 01 11 10 0 1 1 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* VIV*V* VX CNOT V*V* * V V

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

12. V*V* V M0M0 0 or nothing 1 or nothing M1M1 Various types of measurement If nothing, previous action is continued

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

14. Symbolic Quantum synthesis  Assume function to be synthesized: ab c01 00 01 11 10 0 - 0 - 1 - 1 - ab c01 00 01 11 10 0 0 0 1 1 1 1 0 ab c01 00 01 11 10 0 0 1 1 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 2 n or 0. When the output is specified by probabilities corresponding to V 0 or V 1 the parity of applying the V-based gates is odd (2 n -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 M 0 is the state of the system after being measured for 0, and m 0 = 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 d g1g1 f1f1 g2g2 f2f2 gngn fnfn...........

18. Simulations and results a b c d g1g1 f1f1 g2g2 f2f2 gngn fnfn........... Functions f i are “simple”: a) Linear b) Affine c) Toffoli-like They can be binary or multiple- valued Functions g i are “square roots of unity”: a) NOT b) Square-root-of-NOT c) Fourth-order-root-of-NOT d) etc They can be for realization of binary or multiple-valued logic Exhaustive Search A* search Genetic Algorithm Iterative Deepening

19. e

20. Simulations and results  Example 1: ab cd0001 00 01 11 10 0 0 1 0 - 1 - 001 -- 01 10 1110 abc 000 001 011 010 110 111 101 100 - VV * VV VV * VV VVVV * VV VV * ddd - I NOT I I NOT *I 0 0 1 0 1 1 1 0 Symbolic synthesis Method Classical Synthesis of reversible functions applied as a classical Machine Learning

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

22. Simulations and results (contd.)  Example 2: ab c01 00 01 11 10 - 0 - 1 0 - 1 - ab c01 00 01 11 10 V VV V - V - V ab c01 00 01 11 10 V 0 NOT - V 1 - V 0 ab c01 00 01 11 10 V 0 01 0 V 1 1 V 0 Synthesized function matches all required cares VV a b c Multi-valued (quaternary) Synthesis of quantum functions applied as a new Quantum Machine Learning

23. Simulations and results (contd.) Another solution (completely deterministic) is just slightly more complicated: ab c01 00 01 11 10 - 0 - 1 0 - 1 - VV a b c 0  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: Measurement dependent synthesis  Example 2 (contd.): Multi-valued (quaternary) Synthesis of quantum functions applied as a new Quantum Machine Learning

24. Learning error

26. Conclusion Quantum CircuitMeasurement Environment Measurement dependent Learning Symbolic quantum learning Boolean Inputs Probabilities Symbolic method assumes known hidden states, predicts probabilistically the output events  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)

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 2 n (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  implementation and verification of these mechanisms in the Cynthea robotic framework

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

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