Numerical sign problem: Difference between revisions

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{{Short description|Problem in applied mathematics}}

The '''numerical sign problem''' refers to the bad [[statistics|statistical]] convergence

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]].

of a numerical integration procedure, applied in conjunction with a

many-particle theory possessing a complex or non-positive semidefinite

weight function. The difficulty relates to the oscillatory nature

of the weight function in a specific parameter range, induced

by strong repulsive forces. Such weight functions typically

occur in field theories of classical and quantum many-particle systems,

obtained via the [[Hubbard-Stratonovich transformation]] (Baeurle 2002,

Baeurle 2007, Schmid 1998, Baer 1998), as well as in real-time

Feynman-path integral quantum dynamics (Makri 1987, Miller 2005).

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

==Overview<!--'Complex action problem' redirects here-->==

== Sign problem of statistical field theories ==

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 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, 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.

=== Sampling with non-positive semidefinite weight functions ===

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

To explain the numerical sign problem, let us in the following consider

* [[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>

a statistical field theory of interacting particles, whose statistics is

* [[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.

described by the non-positive semidefinite weight function

* [[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>

<math>\rho \left[ \; \sigma \; \right]</math>

(Baeurle 2002a). For such a system, the thermodynamic average of a

physical property <math>A</math> can be expressed in the general form

(Baeurle 2003)

:<math>

< \; A \; > = \frac{\int D \sigma \; A

\left[ \; \sigma \; \right] \; \rho \left[ \;

\sigma \; \right]}{\int D \sigma

\; \rho \left[ \; \sigma \; \right]},

</math>

where <math>D \sigma</math> represents the field integration measure and

<math>A \left[ \; \sigma \; \right]</math> the estimator belonging to the

real and non-positive semidefinite distribution <math>\rho \left[ \; \sigma

\; \right]</math>. It is well-established that an average in the presence

of such a distribution cannot be evaluated with standard numerical

integration techniques like the Metropolis Monte Carlo algorithm, since

they are only valid for real and positive probability distributions

(Salcedo 1997).

==The sign problem in field theory==

=== Reweighting procedure ===

{{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)

The common approach in such cases is to employ a reweighting procedure,

which consists in factorizing the original distribution into a real and

positive definite part, called the reference distribution, and a remainder,

which is included in the estimator. The ensemble average of the property

<math>A</math> can then be calculated with a standard simulation approach,

like Metropolis Monte Carlo, by evaluating the following expression

:<math>

< \; A \; > = \frac{

\int D \sigma \; A \left[ \; \sigma \; \right]

\; \frac{\rho \left[ \; \sigma \; \right]}{\rho^{ref}

\left[ \; \sigma \; \right]}

\; \rho^{ref} \left[ \; \sigma \; \right]}{

\int D \sigma \; \frac{\rho \left[ \; \sigma \; \right]}{\rho^{ref}

\left[ \; \sigma \; \right]} \; \rho^{ref} \left[ \; \sigma \; \right]}

= \frac{< \; A \left[ \; \sigma \; \right]

\; \frac{\rho \left[ \; \sigma \; \right]}{\rho^{ref}

\left[ \; \sigma \; \right]} \; >^{ref}}{

< \; \frac{\rho \left[ \; \sigma \; \right]}{\rho^{ref}

\left[ \; \sigma \; \right]} \; >^{ref}},

</math>

where the brackets <math>< \; \cdots \; >^{ref}</math> denote averaging

with respect to the real and positive definite reference distribution

<math>\rho^{ref} \left[ \; \sigma \; \right]</math>. In practice, the

averages in the numerator and denominator are approximated by their

respective discrete sum (Baeurle 2003)

:<math>

< \; A \; > \approx \lim_{\tau_{run} \longrightarrow \infty}

\frac{\sum\limits_{i=1}^{\tau_{run}}

\; A \left[ \; \sigma_i \; \right] \; \frac{\rho

\left[ \; \sigma_i \; \right]}{\rho^{ref} \left[ \; \sigma_i \; \right]}}{

\sum\limits_{i=1}^{\tau_{run}} \; \frac{\rho \left[ \; \sigma_i \;

\right]}{\rho^{ref} \left[ \; \sigma_i \; \right]}},

</math>

where <math>\tau_{run}</math> defines the total number of simulation steps.

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

=== Choice of reference system and sign problem ===

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>:

A crucial issue for the effective evaluation of the ensemble average

<math>< \; A \; ></math> is to choose a reference distribution, which

minimizes the standard deviations of the averages of the numerator and

denominator, as well as is independent of the estimator

<math>A \left[ \; \sigma \; \right]</math>. The best possible reference

distribution is obtained through the application of the variational method

(Kieu 1994), which provides

:<math>

\rho^{ref}langle A \left[rangle_\rho = \;frac{\int D \sigma \;, A[\rightsigma] =\, \fracrho[\sigma]}{\left|int D \;sigma \, \rho[\sigma]}.

\rho \left[ \; \sigma \; \right] \; \right|}{\int D \sigma

\left| \; \rho \left[ \; \sigma \; \right] \; \right|},

</math>

with the optimal standard deviation for the denominator average

:<math>

\sigma \left( < \; {\rm sign} \left[ \; \sigma \;

\right] \; >^{ref} \right) = \sqrt{1 \; - \; < \;

{\rm sign} \left[ \; \sigma \; \right] \; >^{ref \; ^2}},

</math>

where

:<math>

< \; {\rm sign} \left[ \; \sigma \;

\right] \; >^{ref} = \frac{\int D \sigma \; \rho

\left[ \; \sigma \; \right]}{\int D \sigma \; \left|

\; \rho \left[ \; \sigma \; \right] \; \right|}

</math>

is the average of the sign function. The sign problem now occurs when

the average of the sign

:<math>

< \; {\rm sign} \left[ \; \sigma \;

\right] \; >^{ref} \longrightarrow 0,

</math>

which causes that, unless a huge number of configurations are sampled,

the large statistical fluctuations of the quantity render the calculation

meaningless.

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]].

== Methods for reducing the sign problem ==

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–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.

=== Analytical techniques ===

=== Reweighting procedure ===

Several analytical and numerical techniques have been developed to

alleviate the numerical sign problem effectively. The analytical techniques

essentially base on the contour-shifting procedure, which makes use of the

Cauchy’s integral theorem to shift the integration contour of the functional

integral into the complex plane in such way that it crosses as many critical

points as possible. Critical points in a mathematical sense (Baeurle 2003a)

define configurations, which provide the main contributions to the integral,

like e.g. the mean field solution. Contour-shifts through the mean field

solution of functional integrals have successfully been employed in

field-theoretic electronic structure calculations (Baer 1998) and in

statistical simulations of classical many-particle systems

(Baeurle 2002, Baeurle 2002a). In a very recent work it has been

demonstrated that better contour shifts with regard to statistical

convergence can be obtained by employing tadpole renormalization

techniques (Baeurle 2002).

=== Numerical techniques ===

Numerical techniques essentially rely on the stationary phase Monte Carlo

technique (Doll 1988), which is a numerical strategy for importance sampling

around the stationary phase points of the functional integral. Such techniques

have sucessfully been used for sampling Feynman-path integrals in real-time

quantum dynamics, as well as statistical field theories of classical

many particle systems (Baeurle 2003a).

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:

== References ==

:<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.

* {{cite journal |

It can be shown<ref name='Wiese-cluster'/> that

|url = http://prola.aps.org/abstract/PRL/v89/i8/e080602

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

|last = Baeurle

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.

|first = S.A.

|title = Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods

|journal = Phys. Rev. Lett.

|volume = 89

|pages = 080602

|year = 2002}}

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

* {{cite journal |

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

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TXW-4NXHCCV-1&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=c46d37a250d912279b63702d2dd8826e

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

|last = Baeurle

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

|first = S.A.

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

|coauthors = Nogovitsin, E.A.

|title = Challenging scaling laws of flexible polyelectrolyte solutions with effective renormalization concepts

|journal = Polymer

|volume = 48

|pages = 4883

|year = 2007}}

==Methods for reducing the sign problem==

* {{cite journal |

|url = http://www.iop.org/EJ/article/0953-8984/10/37/002/c837r1.pdf?request-id=dc51bfac-c186-4d9f-9c07-1e0214fb6c1c

|last = Schmid

|first = F.

|title = Self-consistent-field theories for complex fluids

|journal = J. Phys.: Condens. Matter

|volume = 10

|pages = 8105

|year = 1998}}

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.

* {{cite journal |

|url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000109000015006219000001&idtype=cvips&gifs=yes

|last = Baer

|first = R.

|coauthors = Head-Gordon, M.; Neuhauser, D.

|title = Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: Straddling the sign problem

|journal = J. Chem. Phys.

|volume = 109

|pages = 6219

|year = 1998}}

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 book |arxiv=hep-lat/0610116 |last1=Schmidt |first1=Christian |title=Proceedings of XXIVth 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>

* {{cite journal |

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TFN-44KWPF1-9B&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=d6d674e2397e693f3167aa955fde9f31

|last = Makri

|first = N.

|coauthors = Miller, W.H.

|title = Monte carlo integration with oscillatory integrands: implications for feynman path integration in real time

|journal = Chemical Physics Letters

|volume = 139

|pages = 10

|year = 1987}}

=== List: current approaches ===

* {{cite journal |

|url = http://www.pnas.org/cgi/content/abstract/102/19/6660

|last = Miller

|first = W.H.

|title = Quantum dynamics of complex molecular systems

|journal = PNAS

|volume = 102

|pages = 6660

|year = 2005}}

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

* {{cite journal |

|url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000117000007003027000001&idtype=cvips&gifs=yes

|last = Baeurle

|first = S.A.

|coauthors = Martonak, R.; Parrinello, M.

|title = A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble

|journal = J. Chem. Phys.

|volume = 117

|pages = 3027

|year = 2002a}}

* ''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>

* {{cite journal |

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-47MKK8M-2&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=60e4b8712970cf7e07a14b05c5f0e609

|last = Baeurle

|first = S.A.

|title = Computation within the auxiliary field approach

|journal = J. Comput. Phys.

|volume = 184

|pages = 540

|year = 2003}}

* ''[[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.

* {{cite journal |

|url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000038000003001710000001&idtype=cvips&gifs=yes

|last = Salcedo

|first = L.L.

|title = Representation of complex probabilities

|journal = J. Math. Phys.

|volume = 38

|pages = 1710

|year = 1997}}

* ''[[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>

* {{cite journal |

|url = http://prola.aps.org/abstract/PRE/v49/i5/p3855_1

|last = Kieu

|first = T.D.

|coauthors = Griffin, C.J.

|title = Monte Carlo simulations with indefinite and complex-valued measures

|journal = Phys. Rev. E

|volume = 49

|pages = 3855

|year = 1994}}

* ''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>

* {{cite journal |

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-4BHK1JN-4&_user=616165&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=fe7a7f52aadb5375994112ee03cebefc

|last = Baeurle

|first = S.A.

|title = Grand canonical auxiliary field Monte Carlo: a new technique for simulating open systems at high density

|journal = Comput. Phys. Commun.

|volume = 157

|pages = 201

|year = 2004}}

* ''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)

* {{cite journal |

|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>

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ5-48XCNNY-3&_user=616165&_coverDate=08%2F01%2F2003&_alid=745709877&_rdoc=6&_fmt=high&_orig=search&_cdi=5301&_sort=d&_docanchor=&view=c&_ct=8&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=c5a0e9b03b74325c0dc7bf7d88406ecf

|last = Baeurle

|first = S.A.

|title = The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies

|journal = Comput. Phys. Commun.

|volume = 154

|pages = 111

|year = 2003a}}

* ''[[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>

* {{cite journal |

|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TFN-44K9504-F&_user=616165&_coverDate=01%2F15%2F1988&_fmt=abstract&_orig=search&_cdi=5231&view=c&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=6930fc8a82d9efc28d2ddef3a53e4d78&ref=full

|last = Doll

|first = J.D.

|coauthors = Freeman, D.L.; Gillan, M.J.

|title = Stationary phase Monte Carlo methods: An exact formulation

|journal = Chem. Phys. Lett.

|volume = 143

|pages = 277

|year = 1988}}

== See also ==

* [[Method of stationary phase]]

* [[Oscillatory integral]]

==Footnotes==

* [[Auxiliary field Monte Carlo]]

{{notelist|1}}

==References==

== External links ==

{{reflist}}

*[http://www-dick.chemie.uni-regensburg.de/group/stephan_baeurle/index.html Particle and Polymer Field Theory Group]

{{DEFAULTSORT:Numerical Sign Problem}}

[[Category:Statistical mechanics]]

[[Category:Numerical artifacts]]

[[Category:Unsolved problems in physics]]