# Difference between revisions of "Afternoon Session: Quantum Monte Carlo"

From ALPS

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+ | == Talks == | ||

+ | * Topic, more specifically: Path Integral Quantum Monte Carlo | ||

+ | |||

* Chair: Boris Svistunov and Synge Todo | * Chair: Boris Svistunov and Synge Todo | ||

* Speakers | * Speakers | ||

Line 11: | Line 14: | ||

**** Ising: energy allone suffices, cannot get better than N^2 behaviour | **** Ising: energy allone suffices, cannot get better than N^2 behaviour | ||

** Anatoly Kuklov: ''Flowgram Method in QM'' | ** Anatoly Kuklov: ''Flowgram Method in QM'' | ||

+ | *** Apply to known problems! | ||

** Matthias Troyer: ''The Sign Problem'' | ** Matthias Troyer: ''The Sign Problem'' | ||

*** Where does it appear? | *** Where does it appear? | ||

Line 20: | Line 24: | ||

**** Basis-dependent! | **** Basis-dependent! | ||

**** In general: NP-hard & QMA-hard | **** In general: NP-hard & QMA-hard | ||

− | **** Scaling with system size | + | **** Scaling with system size: using an appropriate method away from criticality, the scaling should not be exponential |

+ | **** In some cases, symmetries help to solve sign problem completely: Meron cluster algorithm | ||

+ | **** In these cases, phases are close to classical phases | ||

*** Where does it come from? | *** Where does it come from? | ||

**** Sampling with absolute value of the weights is equivalent to sampling bosons to learn about fermions | **** Sampling with absolute value of the weights is equivalent to sampling bosons to learn about fermions | ||

** Boris Svistunov | ** Boris Svistunov | ||

+ | |||

+ | == Discussion == | ||

+ | * What models? What system sizes/temperatures? | ||

+ | * QMC: unfrustrated spin systems | ||

+ | * Why spin chains and Haldane conjecture? | ||

+ | ** Because I can! | ||

+ | ** Because Petaflop computers are there and this is an application: do what you can do | ||

+ | ** First purpose of the petaflop machine is '''to be used''' | ||

+ | * Create a list of open problems in physics? | ||

+ | ** Quantum magnets: beta * Vol = 10^8 on standard clusters | ||

+ | *** 10^7 spins or 10^5 lattice bosons or 10^4 continuous bosons on a single node | ||

+ | *** 10^8 on a MPP: memory constraints | ||

+ | *** Disorder etc: up to a few 100,000 CPUs | ||

+ | ** Bosonic models without frustration (hopping matrix cannot trivially be mapped to positive definite matrix) | ||

+ | ** No real time dynamics (only equilibrium statistics) (short time with diagrammatic MC) | ||

+ | * Challenges | ||

+ | ** Better representations, better analytical/conceptual ideas are needed | ||

+ | ** General challenges: solve the sign problem | ||

+ | ** More specific ones: find a good representation for specific problems -> diagrammatic Monte Carlo | ||

+ | ** Finite-size analysis | ||

+ | ** First-order phase transitions | ||

+ | ** Disrodered systems | ||

+ | ** Correctly distinguishing second order from weakly first order transitions: flowgram technique | ||

+ | * Methods | ||

+ | ** Loop algorithm | ||

+ | ** Worm algorithm | ||

+ | ** Directed loop | ||

+ | ** Worm spin-offs | ||

+ | ** Determinant Monte Carlo | ||

+ | *** Under control? | ||

+ | ** All of the above: path integral representation (and SSE) | ||

+ | *** Discrete time (cont. space), continuous time (discr. space), SSE | ||

+ | ** Treat ''representation'' and ''update strategy'' separately | ||

+ | ** Are there more or less reliable methods? | ||

+ | *** Some people call variational/fixed-node/... Monte Carlo QMC | ||

+ | *** Reliable == controllable error (no systematic errors) | ||

+ | *** Who's doing it?: Sufficient equilibration | ||

+ | *** Diagrammatic MC: is it clear ow controlled it is? | ||

+ | *** Path Integral MC is well-established | ||

+ | *** Defining well-established | ||

+ | **** Depend on mapping to classical system: d -> d+1 | ||

+ | **** Everything depends on having sufficiently good statistics | ||

+ | **** Everything said about classical statistics maps into some quantum counterpart | ||

+ | **** Many quantum problems map into well-behaved classical problems | ||

+ | ***** Slowly equilibrating problems are rarely considered in QMC calculations | ||

+ | ***** Counterexample: melting of solid Helium |

## Latest revision as of 04:50, 4 May 2009

## Talks

- Topic, more specifically: Path Integral Quantum Monte Carlo

- Chair: Boris Svistunov and Synge Todo
- Speakers
- Synge Todo:
*Parallel Monte Carlo*- Why large load imbalance for parallel tempering?
- Dependence of CPU time on temperature requires fine-tuning

- Multi-canonical ensemble
- Weight depends on global properties of the system

- Why large load imbalance for parallel tempering?
- Simon Trebst:
*Extended statistical ensembles*- More than one variable? Possible, but no examples where it helps significantly
- Energy/temperature are good choices for thermal phase transitions
- Ising: energy allone suffices, cannot get better than N^2 behaviour

- More than one variable? Possible, but no examples where it helps significantly
- Anatoly Kuklov:
*Flowgram Method in QM*- Apply to known problems!

- Matthias Troyer:
*The Sign Problem*- Where does it appear?
- Bosonic systems: no
- Fermions: exchanging fermions leads to sign problems
- Bosons in a gauge field: phases
- Frustrated magnets: exchange leads to sign problem

- Can it be solved?
- Basis-dependent!
- In general: NP-hard & QMA-hard
- Scaling with system size: using an appropriate method away from criticality, the scaling should not be exponential
- In some cases, symmetries help to solve sign problem completely: Meron cluster algorithm
- In these cases, phases are close to classical phases

- Where does it come from?
- Sampling with absolute value of the weights is equivalent to sampling bosons to learn about fermions

- Where does it appear?
- Boris Svistunov

- Synge Todo:

## Discussion

- What models? What system sizes/temperatures?
- QMC: unfrustrated spin systems
- Why spin chains and Haldane conjecture?
- Because I can!
- Because Petaflop computers are there and this is an application: do what you can do
- First purpose of the petaflop machine is
**to be used**

- Create a list of open problems in physics?
- Quantum magnets: beta * Vol = 10^8 on standard clusters
- 10^7 spins or 10^5 lattice bosons or 10^4 continuous bosons on a single node
- 10^8 on a MPP: memory constraints
- Disorder etc: up to a few 100,000 CPUs

- Bosonic models without frustration (hopping matrix cannot trivially be mapped to positive definite matrix)
- No real time dynamics (only equilibrium statistics) (short time with diagrammatic MC)

- Quantum magnets: beta * Vol = 10^8 on standard clusters
- Challenges
- Better representations, better analytical/conceptual ideas are needed
- General challenges: solve the sign problem
- More specific ones: find a good representation for specific problems -> diagrammatic Monte Carlo
- Finite-size analysis
- First-order phase transitions
- Disrodered systems
- Correctly distinguishing second order from weakly first order transitions: flowgram technique

- Methods
- Loop algorithm
- Worm algorithm
- Directed loop
- Worm spin-offs
- Determinant Monte Carlo
- Under control?

- All of the above: path integral representation (and SSE)
- Discrete time (cont. space), continuous time (discr. space), SSE

- Treat
*representation*and*update strategy*separately - Are there more or less reliable methods?
- Some people call variational/fixed-node/... Monte Carlo QMC
- Reliable == controllable error (no systematic errors)
- Who's doing it?: Sufficient equilibration
- Diagrammatic MC: is it clear ow controlled it is?
- Path Integral MC is well-established
- Defining well-established
- Depend on mapping to classical system: d -> d+1
- Everything depends on having sufficiently good statistics
- Everything said about classical statistics maps into some quantum counterpart
- Many quantum problems map into well-behaved classical problems
- Slowly equilibrating problems are rarely considered in QMC calculations
- Counterexample: melting of solid Helium