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{{Short description|Problem in applied mathematics}}
In [[applied mathematics]], the '''numerical sign problem''' is the problem of numerically evaluating the [[integral]] of a highly [[Oscillation|oscillatory]] [[Function (mathematics)|function]] of a large number of variables. [[Numerical methods]] fail because of the near-cancellation of the positive and negative contributions to the integral. Each has to be integrated to very high [[Accuracy and precision|precision]] in order for their difference to be obtained with useful [[Accuracy and precision|accuracy]].
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* [[Condensed matter physics]] — It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the [[Hubbard model]].<ref>{{cite journal |doi=10.1103/PhysRevB.41.9301 |pmid=9993272 |bibcode=1990PhRvB..41.9301L |title=Sign problem in the numerical simulation of many-electron systems |journal=Physical Review B |volume=41 |issue=13 |pages=9301–9307 |year=1990 |last1=Loh |first1=E. Y. |last2=Gubernatis |first2=J. E. |last3=Scalettar |first3=R. T. |last4=White |first4=S. R. |last5=Scalapino |first5=D. J. |last6=Sugar |first6=R. L.}}</ref>
* [[Nuclear physics]] — It prevents the ''[[ab initio]]'' calculation of properties of [[nuclear matter]] and hence limits our understanding of [[atomic nucleus|nuclei]] and [[neutron star]]s.
* [[Quantum field theory]] — It prevents the use of [[lattice QCD]]<ref>{{Cite journal |author=de Forcrand, Philippe |title=Simulating QCD at finite density |journal=Pos Lat |volume=010 |pages=010 |year=2010 |arxiv=1005.0539 |bibcode=2010arXiv1005.0539D}}</ref> to predict the phases and properties of [[quark matter]].<ref name='Philipsen'>{{cite
==The sign problem in field theory==
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The sign problem is [[NP-hard]], implying that a full and generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time.<ref>{{Cite journal |arxiv=cond-mat/0408370 |doi=10.1103/PhysRevLett.94.170201 |pmid=15904269 |bibcode=2005PhRvL..94q0201T |title=Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations |journal=Physical Review Letters |volume=94 |issue=17 |pages=170201 |year=2005 |last1=Troyer |first1=Matthias |last2=Wiese |first2=Uwe-Jens |s2cid=11394699 }}</ref> If (as is generally suspected) there are no polynomial-time solutions to NP problems (see [[P versus NP problem]]), then there is no ''generic'' solution to the sign problem. This leaves open the possibility that there may be solutions that work in specific cases, where the oscillations of the integrand have a structure that can be exploited to reduce the numerical errors.
In systems with a moderate sign problem, such as field theories at a sufficiently high temperature or in a sufficiently small volume, the sign problem is not too severe and useful results can be obtained by various methods, such as more carefully tuned reweighting, [[analytic continuation]] from imaginary <math>\mu</math> to real <math>\mu</math>, or [[Taylor series|Taylor expansion]] in powers of <math>\mu</math>.<ref name='Philipsen'/><ref>{{Cite
=== List: current approaches ===
There are various proposals for solving systems with a severe sign problem:
* ''Contour deformation
* ''[[Meron (physics)|Meron]]-cluster algorithms:'' These achieve an exponential speed-up by decomposing the fermion world lines into clusters that contribute independently. Cluster algorithms have been developed for certain theories,<ref name='Wiese-cluster'>{{cite journal |doi=10.1103/PhysRevLett.83.3116 |arxiv=cond-mat/9902128 |bibcode=1999PhRvL..83.3116C |title=Meron-Cluster Solution of Fermion Sign Problems |journal=Physical Review Letters |volume=83 |issue=16 |pages=3116–3119 |year=1999 |last1=Chandrasekharan |first1=Shailesh |last2=Wiese |first2=Uwe-Jens|s2cid=119061060 }}</ref> but not for the [[Hubbard model]] of electrons, nor for [[Quantum chromodynamics|QCD]] ''i.e.'' the theory of quarks.
* ''[[
* ''Majorana algorithms
* Fixed-node method. One fixes the ___location of nodes (zeros) of the multiparticle wavefunction, and uses Monte Carlo methods to obtain an estimate of the energy of the ground state, subject to that constraint.<ref>{{cite journal |doi=10.1103/PhysRevLett.72.2442 |pmid=10055881 |bibcode=1994PhRvL..72.2442V |title=Fixed-Node Quantum Monte Carlo Method for Lattice Fermions |journal=Physical Review Letters |volume=72 |issue=15 |pages=2442–2445 |year=1994 |last1=Van Bemmel |first1=H. J. M. |last2=Ten Haaf |first2=D. F. B. |last3=Van Saarloos |first3=W. |last4=Van Leeuwen |first4=J. M. J. |author-link4=Hans van Leeuwen (physicist)▼
▲* ''Fixed-node
|last5=An |first5=G. |hdl=1887/5478|url=https://openaccess.leidenuniv.nl/bitstream/handle/1887/5478/850_066.pdf?sequence=1 |hdl-access=free }}</ref>
▲* Majorana algorithms. Using Majorana fermion representation to perform Hubbard-Stratonovich transformations can help to solve the fermion sign problem of a class of fermionic many-body models.<ref>{{cite journal |doi=10.1103/PhysRevB.91.241117 |arxiv=1408.2269 |bibcode=2015PhRvB..91x1117L |title=Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation |journal=Physical Review B |volume=91 |issue=24 |pages=241117 |year=2015 |last1=Li |first1=Zi-Xiang |last2=Jiang |first2=Yi-Fan |last3=Yao |first3=Hong|s2cid=86865851 }}</ref><ref>{{Cite journal |doi=10.1103/PhysRevLett.117.267002 |pmid=28059531 |arxiv=1601.05780 |bibcode=2016PhRvL.117z7002L |title=Majorana-Time-Reversal Symmetries: A Fundamental Principle for Sign-Problem-Free Quantum Monte Carlo Simulations |journal=Physical Review Letters |volume=117 |issue=26 |pages=267002 |year=2016 |last1=Li |first1=Zi-Xiang |last2=Jiang |first2=Yi-Fan |last3=Yao |first3=Hong|s2cid=24661656 }}</ref>
* ''[[Diagrammatic Monte Carlo]]:''
==See also==
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