1. CLASSIFYING SIMULATORS Canada 10 iQST, Dongsheng Wang Calgary, 12/06/2015

2. Contents Motivations: simulators, classifying simulators Model of Simulation Quantum Simulation Classifying Quantum Simulators Quantum Channel Simulator Conclusion

3. Why Simulators? To solve problems, functioning as a computer For the benefit of users: training, fun Many other purposes: compare different theories, such as quantum- classical distinction

4. Why Classifying Simulators? Periodic Table Phase Diagram Put all kinds of simulators in a table. Once there is an empty seat, there is a chance to make a new discovery! Especially for design of quantum simulators.

5. Model of Simulation Simulator Simulatee User R1R1 R2R2 R3R3 S O U

6. Model of Simulation (O, S, U, R 1, R 2, R 3 ) Simulatee O: Physical objects (process, structure, matter, etc) in reality; mathematical objects (model, theory, equation, etc). Simulator S: Computer; well-designed physical systems. User U: Single user (black box); multipartite interactive users; a controller (computer/simulator) Relations R 1, R 2, R 3 R1R1 R2R2 R3R3 S O U

7. Examples of Simulation in Physics Simulation is common in real life and engineering, but also in physics 1.Electric simulators, lots of devices for display, experiment 2.Quantum fields in many-body physics 3.Computer simulation (run simulation program) 4.Quantum simulation 5.Classical simulation of quantum processes

8. Classical simulation of quantum processes Gottesman-Knill theorem (see Nielsen & Chuang book) A quantum system dynamics with initial state |0> and discrete-time dynamics including H gates, phase gates, CNOT gates, and Pauli gates, and finally Pauli observable measurements can be efficiently simulated classically. Methods: Keep the information of the states after each gate operation; States information can be efficiently recorded: stabilizer formalism. Benefits: Stabilizer formalism, power of quantum computer, q-computing models

9. What kind of simulation? What physics? What computers? Computation as a branch of physics Quantum

10. Quantum Simulators Simulators made of quantum objects and run according to non-trivial quantum rules (superposition, interference, entangling, etc…). 1.Solve some problems faster than classical computers. 2.Display quantum processes, effects, phenomena. (e.g., quantum simulator of tunneling) 3.Learn/train quantum physics. Note: does not forbid classical components!

11. Classifying Quantum Simulators S O U R1R1 R2R2 R3R3 Six-variable classification scheme

12. Examples: Analog vs. Digital quantum simulators Analog simulatorDigital simulator S. Lloyd, Science, 1996. S O U S O U mapping Encode & compute control input learncompute

13. Examples: Analog vs. Digital quantum simulators AnalogDigital S: simulator is a well-controlled system U: Active user (S is white-box to U) R(U,S): user can control the simulator R(S,O): mapping of parameters O: an object the user is interested in R(U,O): user want to learn something about O S: fault-tolerant quantum circuit U: Passive user (S is black-box to U) R(U,S): user provides initial value for the simulator R(S,O): S encodes & computes O O: an object the user is interested in R(U,O): user want to compute something about O S. Lloyd, Science, 1996. Large-scale simulation but limited or no computation power May have computation power yet expensive to build S O U S O U

14. Examples: Strong vs. Weak quantum simulators Weak simulator D. Wang, PRA, 2015. S O U Property of input learn S O U Compute input compute Strong simulator

15. Examples: Strong vs. Weak quantum simulators Weak D. Wang, PRA, 2015. Strong O: an object or its property U: user wants to know partial information of O R(U,O): O is black-box to U R(S,O): S simulates effects of O instead of O R(U,S): user provides initial value for the simulator S: fault-tolerant quantum circuit O: an object the user is interested in U: user wants to know complete information of O R(U,O): O is white-box to U R(S,O): S simulates O R(U,S): user provides initial value for the simulator S: fault-tolerant quantum circuit More flexible Yield more direct information while not always possible S O U S O U

16. Formal definitions From operator topology. Can be generalized to mixed state case.

17. Weak quantum simulation problem There could be many different algorithms as long as it approximates ; If strong, one needs to simulate the channel itself.

18. Weak quantum simulation circuit D. Wang, PRA, 2015.

19. Quantum channel strong simulation Stinespring dilation & Kraus operator-sum representation Circuit complexity O( N 6 ) U S E NN2NN2 O(N 2 )? The problem is Kraus operators only occupy the first block column of U Other methods?

20. The set of channel, S, is convex. Convex polytope Convex bodyConcave polytope Geometry of channel set

21. Trading classical and quantum computational resources U S E NN2NN2 U S E N N U S E N N U S E N N

22. Quantum channel simulation algorithm Input: arbitrary qudit quantum channel Output: quantum channel simulator Procedure: Optimization for decomposition Such that diamond distance Quantum circuit design for each channel ℰ g Wang & Sanders, NJP, 2015. Quantum circuit cost Two qudits instead of three; O(d 2 ) instead of O(d 6 ). Classical cost: A classical dit.

23. Conclusion Classifying Simulators Establish simulator and simulation as subject in Physics Classifying Quantum Simulators Design quantum devices and machines Search for quantum simulation algorithms Strong & Weak Quantum Channel Simulator Simulate quantum open-system dynamics Generator of: noise, quantum states Dissipative quantum computing R1R1 R2R2 R3R3 S O U