classifying simulators canada 10 iqst, dongsheng wang calgary, 12/06/2015
TRANSCRIPT
CLASSIFYING SIMULATORS
Canada 10
iQST, Dongsheng Wang
Calgary, 12/06/2015
Contents
• Motivations: simulators, classifying simulators
• Model of Simulation
• Quantum Simulation
• Classifying Quantum Simulators
• Quantum Channel Simulator
• Conclusion
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
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.
Model of Simulation
Simulator
Simulatee
User
R1
R2
R3
S
O
U
Model of Simulation (O, S, U, R1, R2, R3)• 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 R1, R2, R3
R1
R2R3
SO
U
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, experiment2. Quantum fields in many-body physics3. Computer simulation (run simulation program)4. Quantum simulation5. Classical simulation of quantum processes
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
• What kind of simulation?
• What physics? • What computers?
Computation as a branch of physics
Quantum
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!
Classifying Quantum Simulators
SO
U
R1
R2
R3
• Six-variable classification scheme
Examples: Analog vs. Digital quantum simulators
Analog simulator Digital simulator
S. Lloyd, Science, 1996.
SO
U
SO
U
mapping Encode &compute
control inputlearn compute
Examples: Analog vs. Digital quantum simulators
Analog Digital
S: simulator is a well-controlled systemU: Active user (S is white-box to U)R(U,S): user can control the simulator R(S,O): mapping of parametersO: an object the user is interested inR(U,O): user want to learn something about O
S: fault-tolerant quantum circuitU: Passive user (S is black-box to U)R(U,S): user provides initial value for the simulatorR(S,O): S encodes & computes OO: an object the user is interested inR(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
SO
U
SO
U
Examples: Strong vs. Weak quantum simulators
Weak simulator
D. Wang, PRA, 2015.
SO
U
Property of
inputlearn
SO
U
Compute
inputcompute
Strong simulator
Examples: Strong vs. Weak quantum simulators
Weak
D. Wang, PRA, 2015.
Strong
O: an object or its propertyU: user wants to know partial information of OR(U,O): O is black-box to UR(S,O): S simulates effects of O instead of OR(U,S): user provides initial value for the simulator S: fault-tolerant quantum circuit
O: an object the user is interested inU: user wants to know complete information of OR(U,O): O is white-box to UR(S,O): S simulates OR(U,S): user provides initial value for the simulator S: fault-tolerant quantum circuit
• More flexible • Yield more direct information while not always possible
SO
U
SO
U
Formal definitions
From operator topology. Can be generalized to mixed state case.
Weak quantum simulation problem
• There could be many different algorithms as long as it approximates <O>;
• If strong, one needs to simulate the channel itself.
Weak quantum simulation circuit
D. Wang, PRA, 2015.
Quantum channel strong simulation
• Stinespring dilation & Kraus operator-sum representation
• Circuit complexity O( N 6 )
ℰ→𝑈 ,ℰ (𝜌 )=∑𝑖
𝐾 𝑖 𝜌 𝐾 𝑖+¿ , h𝑤𝑖𝑡 𝐾 𝑖= ⟨𝑖|𝑈|0 ⟩ .¿
US
E
N
N2 O(N2)?
The problem is Kraus operators only occupy the first block column of U
Other methods?
• The set of channel, S, is convex.
Convex polytope Convex body Concave polytope
• One element ℰ in the set S is convex combination of extremes ℰ=∑𝑖
𝑝𝑖ℰ𝑖𝑒 ,∑
𝑖
𝑝𝑖=1 ,𝑝𝑖≥0.
Geometry of channel set
Trading classical and quantum computational resources
US
E
N
N2
USE
NN
USE
NN
USE
NN
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 ℰg
Wang & Sanders, NJP, 2015.
• Quantum circuit cost• Two qudits instead of three;• O(d2) instead of O(d6).
• Classical cost:• A classical dit.
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
R1
R2R3
SO
U