Numerical sign problem: Difference between revisions

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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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The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas:

* [[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 journalbook |last=Philipsen |first=O. |yeartitle=2008Proceedings of VIIIth Conference Quark Confinement and the Hadron Spectrum — PoS(ConfinementVIII) |titlechapter=Lattice calculations at non- zero chemical potential: The QCD phase diagram |urlyear=http://pos.sissa.it//archive/conferences/077/011/Confinement8_011.pdf |journal=Proceedings of Science2008 |volume=77 |pages=011 |doi=10.22323/1.077.0011|doi-access=free }}</ref> (In [[lattice field theory]], the problem is also known as the '''complex action problem'''<!--boldface per WP:R#PLA-->.)<ref>{{cite journal |doi=10.1103/PhysRevD.66.106008 |arxiv=hep-th/0108041 |bibcode=2002PhRvD..66j6008A |title=New approach to the complex-action problem and its application to a nonperturbative study of superstring theory |journal=Physical Review D |volume=66 |issue=10 |pages=106008 |year=2002 |last1=Anagnostopoulos |first1=K. N. |last2=Nishimura |first2=J.|s2cid=119384615 }}</ref>

==The sign problem in field theory==

Sources:<ref name='Wiese-cluster'/><ref name='Kieu'/>

{{efn|Sources for this section include Chandrasekharan & Wiese (1999)<ref name='Wiese-cluster'/> and Kieu & Griffin (1994),<ref name='Kieu'/> in addition to those cited.}}In a field -theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion [[chemical potential]] <math>\mu</math>. One evaluates the [[Partition function (quantum field theory)|partition function]] <math>Z</math> by summing over all classical field configurations, weighted by <math>\exp(-S)</math>, where <math>S</math> is the [[Action (physics)|action]] of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the [[boson]]ic fields <math>\sigma</math> (which may have been originally part of the theory, or have been produced by a [[Hubbard–Stratonovich transformation]] to make the fermion action quadratic)

:<math>Z = \int D \sigma \;, \rho[\sigma],</math>

where <math>D \sigma</math> represents the measure for the sum over all configurations <math>\sigma(x)</math> of the bosonic fields, weighted by

:<math>\rho[\sigma] = \det(M(\mu,\sigma)) \exp(-S[\sigma]),</math>

where <math>S</math> is now the action of the bosonic fields, and <math>M(\mu,\sigma)</math> is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable <math>A[\sigma]</math> is therefore an average over all configurations weighted by <math>\rho[\sigma]</math>:

:<math>

\langle A \rangle_\rho = \frac{\int D \sigma \;, A[\sigma] \;, \rho[\sigma]}{\int D \sigma \;, \rho[\sigma]} .

</math>

If <math>\rho[\sigma]</math> is positive, then it can be interpreted as a probability measure, and <math>\langle A \rangle_\rho</math> can be calculated by performing the sum over field configurations numerically, using standard techniques such as [[Monte Carlo integration|Monte Carlo importance sampling]].

The sign problem arises when <math>\rho[\sigma]</math> is non-positive. This typically occurs in theories of fermions when the fermion chemical potential <math>\mu</math> is nonzero, i.e. when there is a nonzero background density of fermions. If <math>\mu \neq 0</math>, there is no particle-antiparticleparticle–antiparticle symmetry, and <math>\det(M(\mu,\sigma))</math>, and hence the weight <math>\rho(\sigma)</math>, is in general a [[complex number]], so Monte Carlo importance sampling cannot be used to evaluate the integral.

=== Reweighting procedure ===

A field theory with a non-positive weight can be transformed to one with a positive weight, by incorporating the non-positive part (sign or complex phase) of the weight into the observable. For example, one could decompose the weighting function into its modulus and phase,:

:<math>\rho[\sigma] = p[\sigma]\, \exp(i\theta[\sigma]),</math>

where <math>p[\sigma]</math> is real and positive, so

:<math> \langle A \rangle_\rho

= \frac{ \int D\sigma A[\sigma] \exp(i\theta[\sigma])\;, p[\sigma]}{\int D\sigma \exp(i\theta[\sigma])\;, p[\sigma]}

= \frac{ \langle A[\sigma] \exp(i\theta[\sigma]) \rangle_p}{ \langle \exp(i\theta[\sigma]) \rangle_p} .</math>

Note that the desired expectation value is now a ratio where the numerator and denominator are expectation values that both use a positive weighting function, <math>p[\sigma]</math>. However, the phase <math>\exp(i\theta[\sigma])</math> is a highly oscillatory function in the configuration space, so if one uses Monte Carlo methods to evaluate the numerator and denominator, each of them will evaluate to a very small number, whose exact value is swamped by the noise inherent in the Monte Carlo sampling process. The "badness" of the sign problem is measured by the smallness of the denominator <math>\langle \exp(i\theta[\sigma]) \rangle_p</math>: if it is much less than 1, then the sign problem is severe.

It can be shown (e.g.<ref name='Wiese-cluster'/>) that

:<math>\langle \exp(i\theta[\sigma]) \rangle_p \propto \exp(-f V/T),</math>

where <math>V</math> is the volume of the system, <math>T</math> is the temperature, and <math>f</math> is an energy density. The number of Monte Carlo sampling points needed to obtain an accurate result therefore rises exponentially as the volume of the system becomes large, and as the temperature goes to zero.

The decomposition of the weighting function into modulus and phase is just one example (although it has been advocated as the optimal choice since it minimizes the variance of the denominator <ref name='Kieu'>{{cite journal |doi=10.1103/PhysRevE.49.3855 |pmid=9961673 |arxiv=hep-lat/9311072 |bibcode=1994PhRvE..49.3855K |title=Monte Carlo simulations with indefinite and complex-valued measures |journal=Physical Review E |volume=49 |issue=5 |pages=3855–3859 |year=1994 |last1=Kieu |first1=T. D. |last2=Griffin |first2=C. J.|s2cid=46652412 }}</ref>). In general one could write

:<math>\rho[\sigma] = p[\sigma] \frac{\rho[\sigma]}{p[\sigma]},</math>

where <math>p[\sigma]</math> can be any positive weighting function (for example, the weighting function of the <math>\mu = 0</math> theory.).<ref>{{Cite journal |arxiv=hep-lat/9705042 |last1=Barbour |first1=I. M. |title=Results on Finite Density QCD |journal=Nuclear Physics B - Proceedings Supplements |volume=60 |issue=1998 |pages=220–233 |last2=Morrison |first2=S. E. |last3=Klepfish |first3=E. G. |last4=Kogut |first4=J. B. |last5=Lombardo |first5=M.-P. |doi=10.1016/S0920-5632(97)00484-2 |year=1998|bibcode=1998NuPhS..60..220B |s2cid=16172956 }}</ref> The badness of the sign problem is then measured by

:<math>\left\langle \frac{\rho[\sigma]}{p[\sigma]}\right\rangle_p \propto \exp(-f V/T),</math>

which again goes to zero exponentially in the large-volume limit.

==Methods for reducing the sign problem==

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 journalbook |arxiv=hep-lat/0610116 |last1=Schmidt |first1=Christian |title=Proceedings of XXIVth International Symposium on Lattice QCDField atTheory Finite— DensityPoS(LAT2006) |journalchapter=PosLattice LatQCD at finite density |volume=021 |pages=21.1 |year=2006|doi=10.22323/1.032.0021 |bibcode=2006slft.confE..21S |s2cid=14890549 |doi-access=free }}</ref>

=== List: current approaches ===

There are various proposals for solving systems with a severe sign problem:

* ''Contour deformation:'' The field space is complexified and the [[Contour integration|path integral contour]] is deformed from <math>R^N</math> to another <math>N</math>-dimensional manifold embedded in complex <math>C^N</math> space.<ref>{{Cite journal |last1=Alexandru |first1=Andrei |last2=Basar |first2=Gokce |last3=Bedaque |first3=Paulo |last4=Warrington |first4=Neill |year=2022 |title=Complex paths around the sign problem |journal=Reviews of Modern Physics |volume=94 |issue=1 |pages=015006 |arxiv=2007.05436 |doi=10.1103/RevModPhys.94.015006|bibcode=2022RvMP...94a5006A }}</ref>

* [[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}}</ref> but not for the Hubbard model of electrons, nor for [[Quantum chromodynamics|QCD]], the theory of quarks.

* [[Stochastic quantization]]. The sum over configurations is obtained as the equilibrium distribution of states explored by a complex [[Langevin equation]]. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.<ref>{{cite journal |doi=10.1103/PhysRevLett.102.131601 |pmid=19392346 |arxiv=0810.2089 |bibcode=2009PhRvL.102m1601A |title=Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential |journal=Physical Review Letters |volume=102 |issue=13 |pages=131601 |year=2009 |last1=Aarts |first1=Gert}}</ref>

* Fixed''[[Meron (physics)|Meron]]-nodecluster method.algorithms:'' OneThese fixesachieve thean ___locationexponential ofspeed-up nodesby (zeros) ofdecomposing the multiparticlefermion wavefunction,world andlines usesinto Monteclusters Carlothat methodscontribute toindependently. obtainCluster analgorithms estimatehave ofbeen thedeveloped energyfor of the groundcertain statetheories, subject to that constraint.<ref name='Wiese-cluster'>{{cite journal |doi=10.1103/PhysRevLett.7283.24423116 |pmidarxiv=10055881cond-mat/9902128 |bibcode=1994PhRvL1999PhRvL..7283.2442V3116C |title=FixedMeron-NodeCluster QuantumSolution Monteof CarloFermion MethodSign for Lattice FermionsProblems |journal=Physical Review Letters |volume=7283 |issue=1516 |pages=2442–24453116–3119 |year=19941999 |last1=Van BemmelChandrasekharan |first1=H. J. M.Shailesh |last2=Ten HaafWiese |first2=D.Uwe-Jens|s2cid=119061060 F.}}</ref> B.but |last3=Vannot Saarloosfor |first3=W.the |last4=Van[[Hubbard Leeuwenmodel]] of electrons, nor for [[Quantum chromodynamics|first4=J.QCD]] M''i. Je.'' |last5=Anthe |first5=Gtheory of quarks. |hdl=1887/5478}}</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 |year=2015 |last1=Li |first1=Zi-Xiang |last2=Jiang |first2=Yi-Fan |last3=Yao |first3=Hong}}</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}}</ref>

* ''[[Stochastic quantization]].:'' The sum over configurations is obtained as the equilibrium distribution of states explored by a complex [[Langevin equation]]. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.<ref>{{cite journal |doi=10.1103/PhysRevLett.102.131601 |pmid=19392346 |arxiv=0810.2089 |bibcode=2009PhRvL.102m1601A |title=Can Stochastic Quantization Evade the Sign Problem? The Relativistic Bose Gas at Finite Chemical Potential |journal=Physical Review Letters |volume=102 |issue=13 |pages=131601 |year=2009 |last1=Aarts |first1=Gert|s2cid=12719451 }}</ref>

* ''Majorana algorithms.:'' Using [[Majorana fermion]] representation to perform [[Hubbard-Stratonovich transformationstransformation]]s can help to solve the fermion sign problem ofin 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>

* ''Fixed-node Monte Carlo:'' 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)

|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>

* ''[[Diagrammatic Monte Carlo]]:'' Stochastically and strategically sampling Feynman diagrams can also render the sign problem more tractable for a Monte Carlo approach which would otherwise be computationally unworkable.<ref>{{Cite journal|last1=Houcke|first1=Kris Van|last2=Kozik|first2=Evgeny|last3=Prokof'ev|first3=Nikolay V.|last4=Svistunov|first4=Boris Vladimirovich|date=2010-01-01|title=Diagrammatic Monte Carlo|journal=Physics Procedia|language=en|volume=6|pages=95–105|doi=10.1016/j.phpro.2010.09.034|arxiv=0802.2923|bibcode=2010PhPro...6...95V |issn=1875-3892|hdl=1854/LU-3234513|s2cid=16490610|hdl-access=free}}</ref>

==See also==

* [[Method of stationary phase]]

* [[Oscillatory integral]]

==Footnotes==

{{notelist|1}}

==References==

{{reflist}}

<references/>

{{DEFAULTSORT:Numerical Sign Problem}}

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[[Category:Unsolved problems in physics]]