Editing Lattice QCD (section)

==Techniques==

===Monte-Carlo simulations===

After [[Wick rotation]], the [[Path integral formulation|path integral]] for the [[Partition function (quantum field theory)|partition function]] of QCD takes the form

<math> Z = \int \mathcal{D} U \, e^{-S[U]} 
= \int \prod_{x, \mu} dU_\mu(x) \, e^{-S[U]} </math>

where the gauge links <math>U_\mu(x) \in \mathrm{SU}(3)</math> range over all the sites <math>x</math> and space-time directions <math>\mu</math> in a 4-dimensional space-time lattice, <math>S[U]</math> denotes the (Euclidean) [[Action (physics)|action]] and <math>dU_\mu(x)</math>  denotes the [[Haar measure]] on <math>\mathrm{SU}(3)</math>. Physical information is obtained by computing observables

<math> \left\langle \mathcal{O} \right\rangle
= \frac{1}{Z} \int \mathcal{D} U \, \mathcal{O}(U)
e^{-S[U]} </math>

For cases where evaluating observables pertubatively is difficult or impossible, a [[Monte Carlo method|Monte Carlo]] approach can be used, computing an observable <math> \mathcal{O} </math> as

<math> \left\langle \mathcal{O} \right\rangle
\approx \sum_{i=1}^{N} \mathcal{O}(U_i)
</math>

where <math>U_1, \dots, U_{N}</math> are [[Independent and identically distributed random variables|i.i.d random variables]] distributed according to the [[Boltzmann distribution|Boltzman distribution]] <math> U_i \sim e^{-S[U_i]}/Z </math>. For practical calculations, the samples <math>\{U_i\}</math> are typically obtained using [[Markov chain Monte Carlo]] methods, in particular [[Hybrid Monte Carlo]], which was invented for this purpose.<ref>{{cite journal | url=https://doi.org/10.1016/0370-2693(87)91197-X | doi=10.1016/0370-2693(87)91197-X | title=Hybrid Monte Carlo | date=1987 | last1=Duane | first1=Simon | last2=Kennedy | first2=A.D. | last3=Pendleton | first3=Brian J. | last4=Roweth | first4=Duncan | journal=Physics Letters B | volume=195 | issue=2 | pages=216–222 }}</ref>

===Fermions on the lattice===
Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use approximate physical constants which are different from their true physical values:
* The lattice discretization means approximating continuous and infinite space-time by a finite lattice spacing and size. The smaller the lattice, and the bigger the gap between nodes, the bigger the error. Limited resources commonly force the use of smaller physical lattices and larger lattice spacing than wanted, leading to larger errors than wanted.
* The quark masses are also approximated. Quark masses are larger than experimentally measured. These have been steadily approaching their physical values, and within the past few years a few collaborations have used nearly physical values to extrapolate down to physical values.<ref name="Bazavov" />

In order to compensate for the errors one improves the lattice action in various ways, to minimize mainly finite spacing errors.

===Lattice perturbation theory===
In lattice perturbation theory physical quantities (such as the [[scattering matrix]]) are [[taylor expansion|expanded]] in powers of the lattice spacing, ''a''. The results are used primarily to [[renormalization|renormalize]] Lattice QCD Monte-Carlo calculations. In perturbative calculations both the operators of the action and the propagators are calculated on the lattice and expanded in powers of ''a''. When renormalizing a calculation, the coefficients of the expansion need to be matched with a common continuum scheme, such as the [[MS-bar scheme]], otherwise the results cannot be compared. The expansion has to be carried out to the same order in the continuum scheme and the lattice one.

The lattice regularization was initially introduced by [[Kenneth G. Wilson|Wilson]] as a framework for studying strongly coupled theories non-perturbatively. However, it was found to be a regularization suitable also for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant is small, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function, are necessary.

Lattice perturbation theory can also provide results for [[condensed matter]] theory. One can use the lattice to represent the real atomic [[crystal]]. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed (a UV regulator), and a quantum field theory can be formulated and solved on the physical lattice.

===Quantum computing===
The U(1), SU(2), and SU(3) lattice gauge theories can be reformulated into a form that can be simulated using "spin qubit manipulations" on a [[universal quantum computer]].<ref>{{Cite journal|last1=Byrnes|first1=Tim|last2=Yamamoto|first2=Yoshihisa|title=Simulating lattice gauge theories on a quantum computer|journal=Physical Review A|volume=73|issue=2|pages=022328|doi=10.1103/PhysRevA.73.022328|date=17 February 2006|arxiv=quant-ph/0510027|bibcode=2006PhRvA..73b2328B|s2cid=6105195}}</ref>