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