Numerical sign problem: Difference between revisions

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{{Short description|Problem in applied mathematics}}

TheIn [[applied mathematics]], the '''numerical sign problem''' refers tois the difficultyproblem 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]].

The sign problem is one of the major unsolved problems in the physics of [[many-particle systemssystem]]s. It often arises in calculations of the properties of a [[quantum mechanical]] system with large number of strongly- interacting [[fermion|fermions]]s, or in field theories involving a non-zero density of strongly- interacting fermions.

==TheOverview<!--'Complex signaction problem' inredirects physicshere-->==

In physics, the sign problem is typically (but not exclusively) encountered in calculations of the properties of a quantum mechanical system with large number of strongly- interacting [[fermion|fermions]], or in field theories involving a non-zero density of strongly- interacting fermions. Because the particles are strongly interacting, [[perturbation theory]] is inapplicable, and one is forced to use brute-force numerical methods. Because the particles are fermions, their [[wavefunction]] changes sign when any two fermions are interchanged (due to the anti-symmetry of the wave function, see [[Pauli principle]]). So unless there are cancellations arising from some symmetry of the system, the quantum-mechanical sum over all multi-particle states involves an integral over a function that is highly oscillatory, and hence hard to evaluate numerically, particularly in high dimension. Since the dimension of the integral is given by the number of particles, the sign problem becomes severe in the [[thermodynamic limit]]. The field-theoretic manifestation of the sign problem is discussed below.

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>E.{{cite Lohjournal et|doi=10.1103/PhysRevB.41.9301 al|pmid=9993272 |bibcode=1990PhRvB.,.41.9301L "|title=Sign problem in the numerical simulation of many-electron systems" [http://prb|journal=Physical Review B |volume=41 |issue=13 |pages=9301–9307 |year=1990 |last1=Loh |first1=E.aps Y.org/abstract/PRB/v41/i13/p9301_1 Phys|last2=Gubernatis |first2=J. RevE. B|last3=Scalettar 41,|first3=R. 9301–9307T. (1990)]|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|neutron stars]]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 book |last=Philipsen |first=O. |title=Proceedings of VIIIth Conference Quark Confinement and the Hadron Spectrum — PoS(ConfinementVIII) |chapter=Lattice calculations at non zero chemical potential |year=2008 |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>

* Particle physics. It prevents the use of [[Lattice QCD]] to predict the phases and properties of [[quark matter]]. <ref name='Philipsen'>O. Philipsen, "Lattice calculations at non-zero chemical potential: The QCD phase diagram", [http://pos.sissa.it//archive/conferences/077/011/Confinement8_011.pdf PoS Confinement8 011 (2008)], Plenary talk at Quark Confinement and the Hadron Spectrum 8, Mainz, Germany, Sept 2008</ref>

==The sign problem in field theory==

{{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_Partition function (quantum_field_theoryquantum 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|bosonicboson]]ic fields <math>\sigma</math> (which may have been originally part of the theory, or have been produced by a [[Hubbard-StratonovichHubbard–Stratonovich transformation]] to make the fermion action quadratic)

(References for this section: <ref name='Wiese-cluster'/>,<ref name='Kieu'/>).

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

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 of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the [[Boson|bosonic]] 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)

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

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

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>\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 in tointo the observable. For example, one could decompose the weighting function in tointo 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 in tointo 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'>T.{{cite Djournal |doi=10.1103/PhysRevE.49.3855 Kieu|pmid=9961673 and|arxiv=hep-lat/9311072 C|bibcode=1994PhRvE. J.49.3855K Griffin, "|title=Monte Carlo simulations with indefinite and complex-valued measures", [http://pre.aps.org/abstract/PRE/v49/i5/p3855_1|journal=Physical Phys. Rev.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>I.{{Cite journal |arxiv=hep-lat/9705042 |last1=Barbour et|first1=I. al,M. "|title=Results on finiteFinite densityDensity QCD", |journal=Nuclear Nucl.Physics Phys.B Proc.- Suppl.Proceedings 60ASupplements 220-234|volume=60 (|issue=1998), [http://arxiv|pages=220–233 |last2=Morrison |first2=S.org/abs/hep-lat/9705042 arXiv:hep-lat/9705042],E. presented|last3=Klepfish at|first3=E. InternationalG. Workshop|last4=Kogut on|first4=J. LatticeB. QCD|last5=Lombardo on|first5=M.-P. Parallel|doi=10.1016/S0920-5632(97)00484-2 Computers,|year=1998|bibcode=1998NuPhS..60..220B Tsukuba,|s2cid=16172956 Japan}}</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>M{{Cite journal |arxiv=cond-mat/0408370 |doi=10.1103/PhysRevLett.94.170201 Troyer,|pmid=15904269 U|bibcode=2005PhRvL.-J.94q0201T Wiese, "|title=Computational complexityComplexity and fundamentalFundamental limitationsLimitations to fermionicFermionic quantumQuantum Monte Carlo simulations",Simulations [http://prl.aps.org/abstract/PRL/v94/i17/e170201|journal=Physical Phys.Review Rev. Lett.Letters |volume=94, |issue=17 |pages=170201 (|year=2005)], [http://arxiv.org/abs/cond|last1=Troyer |first1=Matthias |last2=Wiese |first2=Uwe-mat/0408370Jens |s2cid=11394699 arXiv:cond-mat/0408370]}}</ref>. If (as is generally suspected) there are no polynomial-time solutions to NP-hard problems (see [[P versus NP problem]]), then there is no ''generic'' solution to the sign problem (on non-quantum computers). 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>C.{{Cite Schmidt, "Lattice QCD at Finite Density",book PoS LAT2006 021 (2006) [http://|arxiv.org/abs/=hep-lat/0610116 arXiv:/hep-lat/0610116],|last1=Schmidt plenary|first1=Christian talk|title=Proceedings atof 24thXXIVth International Symposium on Lattice Field Theory — PoS(LAT2006) |chapter=Lattice QCD 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-cluster algorithms. These achieve an exponential speed-up by decomposing the fermion world lines in to clusters that contribute independently. Cluster algorithms have been developed for certain theories <ref name='Wiese-cluster'>S. Chandrasekharan and U.-J. Wiese, "Meron-Cluster Solution of Fermion Sign Problems", [http://prl.aps.org/abstract/PRL/v83/i16/p3116_1 Phys. Rev. Lett. 83, 3116–3119 (1999)] [http://arxiv.org/abs/cond-mat/9902128 arXiv:cond-mat/9902128]</ref>, but not for the Hubbard model of electrons, nor for [[QCD]], the theory of quarks.

* ''[[Meron (physics)|Meron]]-cluster algorithms.:'' These achieve an exponential speed-up by decomposing the fermion world lines in tointo clusters that contribute independently. Cluster algorithms have been developed for certain theories ,<ref name='Wiese-cluster'>S.{{cite Chandrasekharanjournal and U|doi=10.-J1103/PhysRevLett.83.3116 Wiese,|arxiv=cond-mat/9902128 "|bibcode=1999PhRvL..83.3116C |title=Meron-Cluster Solution of Fermion Sign Problems", [http://prl.aps.org/abstract/PRL/v83/i16/p3116_1|journal=Physical Phys.Review Rev. Lett.Letters |volume=83, |issue=16 |pages=3116–3119 (|year=1999)] [http://arxiv.org/abs/cond-mat/9902128|last1=Chandrasekharan arXiv:cond|first1=Shailesh |last2=Wiese |first2=Uwe-mat/9902128]Jens|s2cid=119061060 }}</ref>, but not for the [[Hubbard model]] of electrons, nor for [[Quantum chromodynamics|QCD]], ''i.e.'' 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>G. Aarts, "Can stochastic quantization evade the sign problem? The relativistic Bose gas at finite chemical potential", [http://prl.aps.org/abstract/PRL/v102/i13/e131601 Phys. Rev. Lett. 102, 131601 (2009)], [http://arxiv.org/abs/0810.2089 arXiv:0810.2089]</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>G{{cite journal |doi=10.1103/PhysRevLett.102.131601 Aarts,|pmid=19392346 "|arxiv=0810.2089 |bibcode=2009PhRvL.102m1601A |title=Can stochasticStochastic quantizationQuantization evadeEvade the signSign problemProblem? The relativisticRelativistic Bose gasGas at finiteFinite chemicalChemical potential",Potential [http://prl.aps.org/abstract/PRL/v102/i13/e131601|journal=Physical Phys.Review Rev. Lett.Letters |volume=102, |issue=13 |pages=131601 (|year=2009)], |last1=Aarts [http://arxiv.org/abs/0810.2089|first1=Gert|s2cid=12719451 arXiv:0810.2089]}}</ref>.

* 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>H. J. M. van Bemmel et al, "Fixed-node quantum Monte Carlo method for lattice fermions", [http://prl.aps.org/abstract/PRL/v72/i15/p2442_1 Phys. Rev. Lett. 72, 2442–2445 (1994)]</ref>

* ''Majorana algorithms:'' Using [[Majorana fermion]] representation to perform [[Hubbard-Stratonovich transformation]]s can help to solve the fermion sign problem in 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)

==References==

|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>

<references/>

* ''[[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}}

{{DEFAULTSORT:Numerical Sign Problem}}

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