Gluon Field Digitization via Group Space Decimation for Quantum Computers
FFERMILAB-PUB-20-198-T
Gluon Field Digitization via Group Space Decimation for Quantum Computers
Yao Ji,
1, 2, ∗ Henry Lamm, † and Shuchen Zhu ‡ (NuQS Collaboration) Theoretische Physik 1, Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen, D-57068 Siegen, Germany Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA Department of Computer Science, Georgetown University, Washington, DC 20057, USA (Dated: June 2, 2020)Efficient digitization is required for quantum simulations of gauge theories. Schemes based ondiscrete subgroups use fewer qubits at the cost of systematic errors. We systematize this approachby deriving a single plaquette action for approximating general continuous gauge groups throughintegrating out field fluctuations. This provides insight into the effectiveness of these approximations,and how they could be improved.
I. INTRODUCTION
Large-scale quantum computers can simulate nonper-turbative quantum field theories which are intractableclassically [1]. Alas, Noisy Intermediate-Scale Quantum(NISQ) era systems will be limited both in qubits andcircuit depths. Whether any gauge theory simulations inthis period are possible depends upon efficient formula-tions. The situation is similar to the early days of latticefield theory when computer memory was limited and thecost of storing SU (3) elements was prohibitive.For fermionic fields, relatively efficient mappings toquantum registers are known [2–5] as evident in that mostexisting quantum calculations have been fermionic [6–9].The bosonic nature of gauge fields preclude such exactmappings, but many proposals for approximate digiti-zation exist with different costs [10–28]. Digitizing re-duces symmetries – either explicitly or through a requiredfinite-truncation [10]. These breakings mean a priori theoriginal model may not be recovered in the continuumlimit [29–34]. Further, choices of digitization may limitthe use of classical resources for Euclidean simulations orstate preparation [35]. In summary, the understandingof resource costs, systematic errors, and the continuumlimit for these proposals is poorly known today.In this work, we systematize the proposal of replacingcontinuous gauge groups G by their discrete subgroups H [11, 28] by deriving the lattice actions using the groupspace decimation procedure of [36, 37]. After derivingthe general third order action, we will investigate the be-haviour of discretizing three distinct gauge groups U (1), SU (2), and SU (3). We begin with a review of the discretesubgroup approximation in Sec. II. In Sec. III we discussthe general aspects of the group space decimation proce-dure. Following that, in Sec. IV we derive the decimatedaction up to 3rd order. Sec. V the continuous group limitof the procedure is studied, and we conclude in Sec. VI. ∗ [email protected] † [email protected] ‡ [email protected] II. DISCRETE SUBGROUPS
Here, we move toward improving the systematic errorsof one digitization scheme– approximating gauge theo-ries by replacing G → H [11, 28]. The best candidateto replace SU (3) is its largest crystal-like discrete sub-group, the Valentiner group V with 1080 elements . As arough comparison, a SU (3) gauge link represented by 9complex numbers using double-precision format requires9 × ×
64 = 1152 qubits. In contrast, V might need asfew as 11 qubits per link. This digitization scheme wasexplored in the early days of Euclidean lattice field theory.The viability of the Z n subgroups replacing U (1) werestudied in [42, 43]. Further studies of the crystal-likediscrete subgroups of SU ( N ) were performed [40, 44, 45],including with fermions [46, 47]. These studies met withmixed success depending on the group and action tested.The fundamental issue of group discretization can beunderstood by considering the Wilson gauge action S [ U ] = − X p βN Re Tr( U p ) , (1)where U p indicates a plaquette of continuous group gaugelinks U (for discrete groups, we will denote plaquettesby u p and links by u ). As β → ∞ , gauge links near thegroup identity dominate, i.e. U ≈ + ε , where ε can bearbitrarily small. Therefore the gap ∆ S = S [ + ε ] − S [ ]goes to zero smoothly. For discrete groups, ε has a min-imum given by the nearest elements N to , and thus∆ S = S [ N ] − S [ ] >
0. This strongly suggests a phasetransition at some critical β f = c/ ∆ S , where c ≈ O (1)depends on spacetime dimensionality, gauge group, andentropy. For U (1) → Z n in 4d, β f = . − cos(2 π/n ) [45].Above β f , all field configurations but u = are expo-nentially suppressed. Thus, H fails as an approximationfor G for β > β f . Another way to understand this be-havior follows [48], where it was shown that the discrete This name is most common in the mathematical literature [38, 39].It has also referred to as S (1080) [28, 36, 37, 40] or Σ × [41]. a r X i v : . [ h e p - l a t ] J un theories are equivalent to continuous groups coupled to aHiggs field. The Higgs mechanism introduces a new phasemissing from the continuous gauge theory when β → ∞ .Both arguments suggest the discrete group can bethought of as an effective field theory for the continu-ous group with a UV-cutoff at Λ f . Provided the typicalseparation of scales of physics m IR (cid:28) Λ f , the approxi-mation could be reliable up O ( m IR / Λ f ) effects.In lattice calculations, one replaces the UV cutoff by afixed lattice spacing a = a ( β ) which shrinks with increas-ing β for theories with asymptotic freedom. To controlerrors when extrapolating to the continuum a →
0, oneshould simulate in the scaling regime of a (cid:28) m − IR . Wedenote the onset of the scaling regime by a s , and β s . For a f ( β f ) ∼ Λ − f , systematic errors from the discrete groupapproximation may prove controllable if a can be reducedsuch that m − IR (cid:29) a (cid:38) a f . Therefore useful simulationwith discrete groups requires β s ≤ β f .For U (1), the scaling regime is β (cid:38) β s = 1. For Z n> one finds β f > β s suggesting that m − IR (cid:29) a (cid:38) a f isachievable. For non-Abelian groups, only a finite set ofcrystal-like subgroups exist. There are three for SU (2):the binary tetrahedral BT , the binary octahedral BO , andthe binary icosahedral BI . While BT has β f = 2 . BO and BI have β f = 3 . β f = 5 . β (cid:38) β s = 2 . BO and BI appear able to approximate SU (2).In the important case of SU (3) with β s = 6, there arefive crystal-like subgroups with V as the largest. For allsubgroups, β f < β s , with V having β f = 3 . V raises β f ≈ β f , attempts to supplementing the Wilsonaction with additional terms were made [28, 36, 37, 40,43, 52–55], although only in [28, 40] were Monte Carlocalculations undertaken for SU (3). Two reasons suggestthis would help. First, additional terms which have acontinuum limit ∝ Re Tr F µν F µν , but take different valueson the element of H (e.g. Tr( u p ) and | Tr( u p ) | − S and thus a f . Second, new terms canreduce finite- a errors as in Symanzik improvement.The term usually added was the adjoint trace, giving S [ u ] = − X p (cid:18) β { } u p ) + β { , − } | Tr( u p ) | (cid:19) , (2)where u p ∈ V , and the first term is normalized so for β { , − } = 0, the S [ u ] matches the Wilson action (with β { } = β ). In these works, no relationship was assumedbetween β { } and β { , − } . That Eq. (2) improves theviability of V over the Wilson action will be shown in [56]. For a different action, S [ u ] = − X p (cid:18) β u p ) + β Re Tr( u p ) (cid:19) , (3)smaller values of a f were demonstrated in [28]. Comparingto Eq. (2) which, up to a constant, adds an additionalgroup character to the action, the second term in Eq. (3)is only a character for Abelian groups. Later, we willrewrite this action in terms of characters of non-Abeliangroups.With these actions, the dimensionless product T c √ t of the pseudocritical temperature and the Wilson flowparameter were found to agree in the continuum with SU (3), allowing one to set the scale of those calculations. a > .
08 fm was achieved without the effects of a f beingseen. This suggest that V can reproduce SU (3) in thescaling region with a modified action, such that practicalquantum computations of SU (3) could be performed.While promising, the choice of new terms was ad-hoc andleft unclear how to systematically improve it or analyzerelative effectiveness. In the next section, we discuss howone can systematically derive lattice actions, discoveringthat the terms added in these two actions are in fact thefirst terms generated in a cumulant expansion. III. GROUP SPACE DECIMATION
Our ultimate goal is to approximate the path integral ofgroup G faithfully by a discrete subgroup H by replacingintegration over G by a summation over H . Group spacedecimation can be understood in analogy to Wilsonianrenormalization, where we integrate out continuous fieldfluctuations instead of UV modes. The typical methodused with discrete subgroup approximations is to replacethe gauge links U ∈ G by u ∈ H such that the action S [ U ] → S [ u ]. This corresponds to simply regularizing afield theory. For strong coupling, this appears sufficient.As β → ∞ , correlations between gauge links increase andthe average field fluctuation becomes smaller. When theaverage field fluctuations decrease below the distance be-tween and N of the discrete group, freeze-out occurs andthe approximation breaks down–similar to probing a regu-lated theory too close to the cutoff. Therefore, improvingthis approximation and understand the systematics can bedone by considering these discarded continuous field fluc-tuations. To do this, instead performing the replacement U → u , we will integrate out the continuous fluctuations,following the decimation formalism developed by Fly-vbjerg [36, 37]. He derived the second order decimatedaction for U (1), SU (2), and SU (3). An important generalfeature of the decimated action though is missing fromthis second order action – while new terms are generatedat every order, it isn’t until third order that the coefficientof existing terms are modified. One would expect suchterms are critical to understanding deviations from thecontinuous group and therefore we compute them for thefirst time in Sec. IV. FIG. 1. A schematic demonstration of Ω (in green) of G (asphere) around (blue point) of the discrete group (shown aspoints). N for H are given by red points. We have appliedthe S metric to obtain the Ω. In groups representable in twodimensions, this region resembles a polygon while in higherdimensions, it becomes a polytope. It is natural to associate every subgroup element u ∈ H with an unique set, or region, Ω u containing all closestcontinuous group elements U ∈ G :Ω u ≡ (cid:8) U ∈ G (cid:12)(cid:12) d ( U, u ) < d ( U, u ) , ∀ u ∈ H \ { u } (cid:9) , (4)where the distance is defined as d ( U, u ) = Tr (cid:0) ( U − u ) † ( U − u ) (cid:1) . By such a definition. the continuous groupis fully covered, i.e., G = ∪ u ∈ H Ω u and a graphical demon-stration of Ω ≡ Ω can be found in Fig. 1. Note that forany U ∈ G , there exist a unique u ∈ H and (cid:15) ∈ Ω suchthat U = u(cid:15) , where we may treat (cid:15) as the error of u ap-proximating U . In this way, without approximation, theEuclidean path integral integrating over G can be writtenas a summation over H and integration over (cid:15) ∈ Ω: Z = Z G DU e − S [ U ] = X u ∈ H Z Ω D(cid:15) e − S [ u,(cid:15) ] , (5)where Z is a functional integral over all gauge links U onthe lattice, or equivalently a functional integral over (cid:15) anda functional sum over u . In this expression, S [ u, (cid:15) ] = S [ U ]is defined by replacing each gauge link U by u(cid:15) .We then expand the exponential in the path integraland integrate over (cid:15) producing a moment expansion Z = X u ∈ H Z Ω D(cid:15) (cid:18) − βS [ u, (cid:15) ] + β S [ u, (cid:15) ] + · · · (cid:19) = X u ∈ H (cid:18) − β h S [ u, (cid:15) ] i + β h S [ u, (cid:15) ] i + · · · (cid:19) , (6)where we have introduced the notation h f i = R Ω D(cid:15) f with normalization R Ω D(cid:15) = 1. What we are really afteris an expansion for the action S [ U ], writing Z in termsof a cumulant expansion Z = X u ∈ H exp ∞ X n =1 ( − β ) n n ! h S [ u, (cid:15) ] n i c ! , (7) allows us to match Eq. (6) with (7) to obtain an ef-fective action. In this way, after integrating over (cid:15) ,the contributions to the action depend only on the dis-crete group gauge links u i.e. h S [ u, (cid:15) ] n i c ≡ S n [ u ] and S [ u ] ≡ P n ( − β ) n n ! S n [ u ]. Up to O ( β ) one has S [ u ] = h S [ u, (cid:15) ] i , (8) S [ u ] = h S [ u, (cid:15) ] i − h S [ u, (cid:15) ] i , (9) S [ u ] = h S [ u, (cid:15) ] i − h S [ u, (cid:15) ] ih S [ u, (cid:15) ] i + 2 h S [ u, (cid:15) ] i . (10)One may worry about poor convergence in the region ofinterest β ≥ β s ≥ β n terms are suppressed by powers of theaverage field fluctuation. Thus, the size of the discretegroup, which determines the size of field fluctuationsintegrated out, also determines the series convergence.Starting with the second order terms computed inRefs. [36, 37], the decimated action generates multi-plaquette contributions. Their inclusion in quantum simu-lations brings substantial non-locality which requires highqubit connectivity and increases circuit depth. Luckilythese contributions will be shown to be small in Sec. IV.In the following, we calculate Eq. (8) to (10) in terms oflinear combination of the group characters starting fromthe Wilson action of Eq. (1): S [ U ] ≡ − X p βN Re Tr( U p ) = − X p βN Re χ { } . (11)Here we introduced χ r , the character of the group repre-sentation r . This is the natural basis for the decimatedaction. All characters required for our β calculation arein Table I. In the interest of deriving a decimated actionfor general gauge groups, we have chosen a nonstandardbasis for U (1) and SU (2). This allows for one generalscheme for U ( N ) and SU ( N ) groups. This basis is notlinearly independent and relations between representa-tions exist. This dependence is typically used to write U (1) and SU (2) in reduced sets of representations. For U (1), the resulting identities are χ { } = − χ { , , − } ,χ { , ± } = χ { , ± } = χ { , , } = 0 . (12)For SU (2), one finds that χ { } = χ { , } , χ { } = χ { , − } , χ { , } = 1 ,χ { , , } = χ { , , − } = 0 , χ { } = χ { , − } , (13)and for SU (3), the set of dependent representationsneeded up to third order in the cumulant expansion are χ { } = χ { , } , χ { } = χ { , , − } χ { , − } = χ { , } , χ { , , } = 1 . (14) Through out this work we suppress the argument of χ r , but itcan only be U p or u p and context makes clear which is meant. In deriving the decimated action, integrating out thefield fluctuations require us to reduce expressions of the form h (cid:15) i j · · · (cid:15) i n j n i . To simplify these, we use an identityderived in [57] for SU ( N ) and U ( N ) groups that for anyinteger n ≤ N h (cid:15) i j · · · (cid:15) i n j n i = c ε i A ...A N − ε j A ...A N − × · · · × ε i n A n ...A nN − ε j n A n ...A nN − + c ε i i A ...A N − ε j j A ...A N − × · · · × ε i n A n − ...A n − N − ε j n A n − ...A n − N − + · · · + c B n ε i i ...i n A ...A N − n ε j j ...j n A ...A N − , (15)where ε is Levi-Civita symbol, A ji are the contracteddummy indices, and B n is the Bell number accounting forthe number of ways that one can put the open indices i k , j l on ε such that no i k and j l appear in the same ε . In [57],Eq. (15) was derived for integrating over the entire group G . Hence in our case, we need to determine the constants c i ’s when integrating only over Ω for h (cid:15) ij i , h (cid:15) ij (cid:15) k‘ i , h (cid:15) ij (cid:15) † k‘ i , h (cid:15) ij (cid:15) k‘ (cid:15) mn i , h (cid:15) ij (cid:15) k‘ (cid:15) † mn i , with i, j, k, l, m, n ∈ [ N ]. This isdone by contracting the tensor structure on each side ofEq. (15) with products of Kronecker delta’s and solvingthe resulting linear equations. What we are left with areexpectation values of χ r over Ω V r ≡ d r h Re χ r i , (16)where d r is the dimension of representation r . At first order, only one integral is needed: h (cid:15) ij i = V { } δ ij . (17)At second order, there are two relations h (cid:15) ij (cid:15) kl i = 12 (cid:0) V { } + V { , } (cid:1) δ ij δ kl + 12 (cid:0) V { } − V { , } (cid:1) δ il δ jk , (18)and h (cid:15) ij (cid:15) † kl i = V { , − } δ ij δ kl + 1 N (1 − V { , − } ) δ il δ jk . (19)At third order, there are four structures, but by complexconjugation one can reduce this to two unique ones: h (cid:15) ij (cid:15) kl (cid:15) mn i = 16 (cid:0) V { } + 4 V { , } + V { , , } (cid:1) δ ij δ kl δ mn + 16 (cid:0) V { } − V { , , } (cid:1) ( δ il δ jk δ mn + δ in δ jm δ kl + δ ij δ kn δ lm )+ 16 (cid:0) V { } − V { , } + V { , , } (cid:1) ( δ in δ jk δ lm + δ il δ kn δ jm ) , (20) h (cid:15) ij (cid:15) kl (cid:15) † mn i = 12 (cid:0) V { , − } + V { , , − } (cid:1) δ ij δ kl δ mn + 12 (cid:0) V { , − } − V { , , − } (cid:1) δ il δ jk δ mn + (cid:18) N ( N − N + 1) V { } − N + 1) V { , − } − N − V { , , − } (cid:19) ( δ in δ jm δ kl + δ ij δ kn δ lm )+ (cid:18) − N − N + 1) V { } − N + 1) V { , − } + 12( N − V { , , − } (cid:19) ( δ in δ jk δ lm + δ il δ kn δ jm ) . (21)For U (1) → Z n , there is only one representation ateach order of the cumulant expansion, V { h } = h (cid:15) h i . Theseterms can be computed analytically by a change of vari-ables (cid:15) = e iφ [37]: V { h } = 1 V Z πn − πn dφ e iφh = nπh sin (cid:18) πhn (cid:19) (22)with h = 1 , , . . . being integers and the normalization constant V = R Ω d(cid:15) (cid:15) = πn .Extending this to non-abelian groups, e.g. SU ( N ), Ωbecomes a high-dimensional polytope. In [36], the V r for BI and V were computed up to second order by approxi-mating these polytopes with hyperspheres to two signifi-cant figures. It is crucial to remove these approximationsfor our purpose because the uncertainty δV r ∼ O (1%)is magnified in the coupling constants of the decimatedaction. These couplings are combinations of powers of V r with extreme cancellations making the fraction errorsgrow rapidly. Hence we avoid making the hypersphereapproximation and numerically compute all the V r nec-essary for the 3rd order actions to 4 significant figures.(Results found in Table I.) IV. ORDER-BY-ORDER DECIMATION
In this section, we will undertake the task of computingthe decimated action order-by-order. The first order isrelatively straight-forward, and only contains a singleplaquette term. Working from Eq. (8) βS [ u ] = βN h Re Tr (cid:0) u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † (cid:1) i (23)= βN Re( u ab u cd u † ef u † gh h (cid:15) bc ih (cid:15) de ih (cid:15) † fg ih (cid:15) † ha i ) . After applying Eq. (17), the remaining S [ u ] is found todepend only on u : βS [ u ] = V { } βN Re( u ab u cd u † ef u † gh ) δ bc δ de δ fg δ ha = V { } βN Re χ { } ≡ β (1) { } N Re χ { } , (24) where β ( n ) r is the n -th order contribution to the coefficientof d r Re χ r .It is comforting that at first order, no new terms aregenerated in the decimated action. This allows for rescal-ing β (1) { } → β , recovering the regularization procedure ofsimply replacing U → u in the Wilson action. Althoughthis rescaling is permitted, V { } contains content aboutthe approximation of G by a given H . As the number ofelements of H increases, Ω should shrink and V { } → V { } quantifies how densely H covers G and thus how small field fluctuations can bebefore freeze-out occurs. Since β (1) { } = V { } β , these de-creases in V { } signals the poorness of approximatingusing Eq. (24) alone. This is discussed further in Sec. V.We now proceed to calculate the second order deci-mated action while fixing a few typos in [36] along theway. The second order decimated action h S [ u, (cid:15) ] i c = h S [ u, (cid:15) ] i − h S [ u, (cid:15) ] i depends upon two plaquettes U p = U U U † U † and U q = U U U † U † . A natural decomposi-tion of h S [ u, (cid:15) ] i c can be made into three terms based onhow the two plaquettes p and q are related: p = q (one-plaquette contribution), p ∩ q = 1-link (two-plaquettecontribution), and p ∩ q = 0-links. To all orders, the p ∩ q = 0 contributions to the decimated action vanish.The first term in Eq. (9) for case p = q reads: β h S [ u, (cid:15) ] i = β N h Re (cid:0) Tr (cid:0) u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † (cid:1)(cid:1) Re (cid:0) Tr (cid:0) u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † (cid:1)(cid:1) i = β N (cid:16) | V { , } | Re χ { , } + | V { } | Re χ { } + V { , − } χ { , − } + 1 (cid:17) , (25)where we have utilized Eqs. (18) and (19) to contractthe u ’s after integration. The second term of Eq. (9) isobtained from first order action of Eq. (24) which reads, β h S [ u, (cid:15) ] i = (cid:18) N β V { } Re χ { } (cid:19) (26)= 12 N β V { } (cid:0) Re χ { } + Re χ { , } + χ { , − } + 1 (cid:1) , where we have used the identity(Re χ { } ) = 12 Re( χ { } + χ { , } + χ { , − } + 1) . (27)Hence we conclude that the second order effective actionterm for two identical plaquettes is12! β h S [ u, (cid:15) ] i c, p = β (2) { } + β (2) { } N ( N + 1) Re χ { } + β (2) { , } N ( N −
1) Re χ { , } + β (2) { , − } N − χ { , − } , (28) where the β (2) r can be found in Table II.Next, we calculate the case of p ∩ q = 1-link for thesecond order decimation. Contracting the δ ’s in Eqs. (18)and (19), where unlike Eq. (25), we only identify one linkas the same between the two plaquettes. This leads tothe following expression,12! β h S [ u, (cid:15) ] i c, p (29)= β { r } N Re (cid:2) χ { } ( u p ) (cid:3) N Re (cid:2) χ { } ( u q ) (cid:3) + β { i } N Im (cid:2) χ { } ( u p ) (cid:3) N Im (cid:2) χ { } ( u q ) (cid:3) + β { t } N Re (cid:2) χ { } ( u p ∗ q † ) (cid:3) + β { u } N Re (cid:2) χ { } ( u p ∗ q ) (cid:3) where we have used the fact that all the V r ’s are realdue to our choice of the integration region. The explicitexpressions for the couplings are found in Table II. Notethat this expression is also applicable to U (1).Having derived the two-plaquette term, we should com-ment upon how it – and general multiplaquette terms – TABLE I. The dimension, d r , the character χ r , and V r [ G → H ] = d − r h Re χ r i of character r for the decimations U (1) → Z n , SU (2) → BI , and SU (3) → V . We have followed the normalizations in Table 14 of [58]. r d r χ r V r [ U (1) → Z n ] V r [ SU (2) → BI ] V r [ SU (3) → V ] { } N Tr( U ) nπ sin (cid:0) πn (cid:1) { } N ( N +1)2 12 (cid:0) Tr ( U ) + Tr( U ) (cid:1) n π sin (cid:0) πn (cid:1) { , } N ( N − (cid:0) Tr ( U ) − Tr( U ) (cid:1) — 1 0.8342 { , − } N − | Tr( U ) | − { } N ( N +1)( N +2)6 16 (cid:0) Tr ( U ) + 2 Tr( U ) + 3 Tr( U ) Tr( U ) (cid:1) n π sin (cid:0) πn (cid:1) { , } N ( N − (cid:0) Tr ( U ) − Tr( U ) (cid:1) — 0.9648 0.6599 { , , } N ( N − N − (cid:0) Tr ( U ) + 2 Tr( U ) − U ) Tr( U ) (cid:1) — — 1 { , − } N ( N − N +2)2 12 (cid:0) Tr ( U ) Tr( U † ) + Tr( U ) Tr( U † ) (cid:1) − Tr( U ) — 0.8325 0.4679 { , , − } | N ( N +1)( N − | (cid:0) Tr ( U ) Tr( U † ) − Tr( U ) Tr( U † ) (cid:1) − Tr( U ) − nπ sin (cid:0) πn (cid:1) — 0.6299TABLE II. β r [ G → H ] of character r for a general group decimation. For completeness, we have included the 4 two-plaquetteterms derived in [36, 37] at second order labeled as 2 r, i, t and 2 u . r β r { } N [1 − V { } ] β { } V { } β + N V { } [4 V { } − V { , } − V { , − } − V { } ] β { } N +18 N [ V { } − V { } ] β { , } N − N [ V { , } − V { } ] β { , − } N − N [ V { , − } − V { } ] β { } ( N +1)( N +2)6 N [ V { } + V { } − V { } V { } ] β { , } ( N − N [ V { , } + V { } − V { } V { , } − V { } V { } ] β { , , } ( N − N − N [ V { , , } + V { } − V { } V { , } ] β { , − } ( N − N +2)16 N [ V { , − } + 2 V { } − V { } (2 V { , − } + V { } )] β { , , − } | ( N +1)( N − N | [ V { , , − } + 2 V { } − V { } (2 V { , − } + V { , } )] β { r } V { } [ V { } + V { , } + V { , − } − V { } ] β { i } V { } [ V { } + V { , } − V { , − } ] β { t } N V { } [ V { } − V { , } ] β { u } N V { } [1 − V { , − } ] β u u u u u u u u u u u u u u u u FIG. 2. Example of two plaquettes u p and u q where p ∩ q = u = u . The second order contributions de-pend on (top) u p ∗ q = u u u u † u † u † and (bottom) u p ∗ q † = u u u u † u † u u † u † . contributes to the action. It would be desirable if theseterms could be neglected, because they require substantialquantum resources to implement. By inspecting Table III,one observes that the two-plaquette β r are O (0 .
1) or smaller than the single-plaquette terms. The largest cou-pling, β i , multiples a term Im χ Im χ ≈
0. Strongcancellations are expected from correlations between theremaining terms (shown in Fig. (2)) as evident by theobservation β t ≈ − β u .It is reasonable to expect these individual reasons topersist at higher orders, suggesting that at a fixed orderall multi-plaquette terms can be neglected compared totheir 1 − plaquette counterpart. But can we argue thatthe multi-plaquette terms generated at order O ( β n − ) arestill negligible when the O ( β n ) contribution is introduced?To do this, we look at the continuum limit of each termbeing introduced. In this way, we recognize that thetwo-plaquette terms are related to the L¨uscher-Weiszaction [59]. k -plaquette terms corresponds to applying2 k − a h Tr F F i and are thus O ( a k +2 ).Here F is the field strength tensor projected onto thelattice directions. Combining this with the observationthat for a coupling β j generated at O ( β n ) has the scaling β j ≈ − n β n , we estimate that h S mk − plaq [ u ] ih S n [ u ] i ≈ (cid:18) β (cid:19) n − m a k +2 h D k − (Tr F F ) i a h Tr F F i , (30)where D is a covariant derivative projected onto the lattice directions. The combination of higher powers of a andtheir associated expectation values of higher-dimensionaloperators of slow-varying fields should be sufficient tosuppress the mild enhancement from the couplings, atleast for third-order decimated actions. For these reasons,we will neglect higher order multi-plaquette terms in thiswork.For the third-order terms of Eq. (10), we, therefore,only focus on the case where three plaquettes are identical.This will be done term by term, where the first term is: β h S [ u, (cid:15) ] i = β N h Re Tr( u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † ) Re Tr( u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † ) Re Tr( u (cid:15) u (cid:15) ( u (cid:15) ) † ( u (cid:15) ) † ) i = β N (cid:18) V { } χ { } + V { , } Re χ { , } + V { , , } χ { , , } + 3 V { , − } χ { , − } + 3 V { , , − } χ { , , − } + 3 V { } Re χ { } (cid:19) . (31)For the mixed-order term in Eq. (10): − β h S [ u, (cid:15) ] ih S [ u, (cid:15) ] i = − β V { } N Re χ { } (cid:2) V { } Re χ { } + V { , } Re χ { , } + V { , − } χ { , − } + 1 (cid:3) = − β V { } N (cid:2) ( V { , } + 2 V { , − } + V { } + 2) Re χ { } + ( V { , } + 2 V { , − } ) Re χ { , , − } + ( V { , } + V { } ) Re χ { , } + V { , } Re χ { , , } + (2 V { , − } + V { } ) Re χ { , − } + V { } Re χ { } (cid:3) , (32)where the second line was simplified with the identities:Re χ { } Re χ { } = 12 Re( χ { } + χ { , } + χ { , − } + χ { } ) , (33)Re χ { } Re χ { , } = 12 Re( χ { } + χ { , , } + χ { , , − } + χ { , } ) , (34)Re χ { } Re χ { , − } = Re( χ { } + χ { , − } + χ { , , − } ) . (35)The final term in Eq. (10) follows from another identity:2 β h S [ u, (cid:15) ] i = 2 β V { } N (Re χ { } ) = β V { } N (cid:0) Re χ { } + 2 Re χ { , } + Re χ { , , } + 6 Re χ { } + 3 Re χ { , − } + 3 Re χ { , , − } (cid:1) . (36)Combining Eqs. (31), (32), and (36) we arrive at thethird order contribution to the single-plaquette decimated action13! S [ u ] = β (3) { } d { } Re χ { } + β (3) { , } d { , } Re χ { , } (37)+ β (3) { , , } d { , , } Re χ { , , } + β (3) { , − } d { , − } Re χ { , − } + β (3) { , , − } d { , , − } Re χ { , , − } + β (3) { } d { } Re χ { } , where the overall factor of 1 /
3! has been absorbed intothe definition of β (3) r . Note that, unlike the second orderresults where only certain decimation programs gener-ate renormalization for existing terms, the third order S [ u ] introduces corrections to Re χ for all G → H . Ad-ditionally, a number of the specific group identities inEqs. (12)-(14) also lead to renormalization.Putting together Eqs. (24), (28), and (37), the deci-mated action of Eq. (7) to the third order for a generalgauge group including only the single plaquette termsread, S [ u ] = X p (cid:18) β (1) { } + 13! β (3) { } (cid:19) N Re( χ { } ) + 12 β (2) { } + 12 β (2) { } N ( N + 1) Re χ { } + 12 β (2) { , } N ( N −
1) Re χ { , } + 12 β (2) { , − } N − χ { , − } + 13! β (3) { } N ( N + 1)( N + 2) Re χ { } + 13! β (3) { , } N ( N −
1) Re χ { , } + 13! β (3) { , , } N ( N − N −
2) Re χ { , , } + 13! β (3) { , − } N ( N − N + 2) Re χ { , − } + 13! β (3) { , , − } N ( N + 1)( N −
2) Re χ { , , − } , (38)where the β r are collected in Table II. Note that thisdecimated action is correct for any G → H with itsassociated V r . Referring back to Eqs. (12), (13), and (14),for a given gauge group simplifications occur. We write S [ u ] for 3 groups of prime interest, U (1) , SU (2) , and SU (3), with β r ≡ P n n ! β ( n ) r . For U (1): S [ u ] = X p ( β { } − β { , , − } ) Re χ { } + β { } + β { } Re χ { } + β { } Re χ { } . (39)For SU (2): S [ u ] = X p (cid:0) β { } + β { , } (cid:1)
12 Re χ { } + (cid:0) β { } + β { , } (cid:1) + (cid:0) β { } + β { , − } (cid:1)
13 Re χ { } + ( β { } + β { , − } ) 14 Re χ { } . (40)For SU (3): S [ u ] = X p (cid:0) β { } + β { , } (cid:1)
13 Re χ { } + (cid:0) β { } + β { , , } (cid:1) + (cid:0) β { } + β { , , − } (cid:1)
16 Re χ { } + (cid:0) β { , − } + β { , } (cid:1) χ { , − } + β { }
10 Re χ { } + β { , − }
15 Re χ { , − } . (41) V. FINITE GROUP EFFECTS
With Eq. (38), it is possible for us to investigate sys-tematically the effect of replacing the continuous groupby its finite subgroup. In order to proceed, it is usefulto introduce a new parameter which approximately rep-resents the field fluctuations. To do this, consider therepresentation of a continuous group lattice gauge link interms of the corresponding generators λ a in the adjointrepresentation, U = e iλ a A a , where a summation over color indices a is implied. In this form, we see that the gaugefields correspond to amplitudes in each of the genera-tors. For (cid:15) ∈ Ω, inserting its small parameter expansion (cid:15) ≈ + iλ a A a − ( λ a A a ) + . . . into Eq. (16) gives V r ≈ − Z Ω DA c (2) r X a A a + . . . ! , (42)where DA is a measure over all A a which respectsgauge symmetry and c ( n ) r are representation and group-dependent constants. From this, we see that as the sub-group H incorporates more elements, Ω → V r → A a in Ω is an indicator for deviations from G of H . Flyvbjerg defines a parameter R as the radius of ahypersphere with equal volume to Ω to get a handle ondomain of A a . This allows him to approximate V r ana-lytically [36, 37]. Here, we can use this idea to roughlyunderstand the scaling of V r .For U (1) → Z n , the hypersphere is exactly Ω and R cleanly defines (cid:15) ≤ R = π/n . Beyond U (1), the connec-tion between Ω and a single value of R is complicatedbecause the Ω of H form polytopes in the hypervolumeof their continuous partner. In this case, while Ω is con-tained by a hypersphere centered at whose boundaryincorporates elements in Ω, some element of the hyper-sphere are not included in Ω. On the other hand, thereexists a largest hypersphere centered at that only con-tains elements in Ω. In this way, we define an upper andlower bound for R . Note, this is different from [36, 37]where the polytopes of H were always approximated byhyperspheres with definite radii. For SU(2) with BI ,we find 0 . ≤ R ≤ .
15 which can be compared to R sphere = 0 .
12 of [36, 37]. In the case of SU(3) with V ,0 . ≤ R ≤ .
93 compared to R sphere = 0 . β , the actualbehavior is controlled by both β and R with R control-ling V r . As pointed out in [36, 37], the leading orderbehavior for small R for a given power of β α is actually O ([ βR ] α R − ). Therefore one would predict that therelative smallness of R for BI compare to V signals that β f should be larger for BI which is indeed the case.For the subgroups of SU (3), this scaling behavior be- TABLE III. Numerical values of β r [ G → H ] of character r for the decimations U (1) → Z , SU (2) → BI , and SU (3) → V . Forcompleteness, we have included the 4 two-plaquette terms derived in [36, 37] at second order. r β r [ U (1) → Z ] β r [ SU (2) → BI ] β r [ SU (3) → V ] { } . β . β . β + 0 . β { } . β + 0 . β . β − . β . β + 0 . β − . β { } − . β − . β − . β − . β { , } — — — { , − } — — − . β − . β { } . β . β . β { , } — — — { , , } — — — { , − } — — 0 . β { , , − } — — — { r } − . β . β . β { i } . β . β . β { t } . β − . β − . β { u } . β . β . β comes unsatisfactory because R ∼
1. It is possible tostudy this breakdown in U (1) → Z n where the systematiceffect of decimation can be studied in detail both becauseerrors can be made arbitrarily small for large n and be-cause V r and β r are known analytically. In terms of R ,one can expand the β r for the U (1) action of Eq. (39) tofind: β { } ≈ (cid:18) R − R
90 + . . . (cid:19) β , (43) β { } + β { , , − } ≈ (cid:18) − R R . . . (cid:19) β + (cid:18) − R
90 + 311 R
945 + . . . (cid:19) β , (44) β { } ≈ (cid:18) − R R
90 + . . . (cid:19) β , (45) β { } ≈ (cid:18) R − R R − R . . . (cid:19) β . (46)The first thing to note is that the O ([ βR ] α R − ) scalingfound in [36, 37] continues to the third order. One mightbe tempted to use this leading behavior to estimate the β f or the radius of convergence of this series, but thiswould be incorrect. Instead, it would behoove one tonotice that for both 2nd and 3rd order contributions, theexpansion coefficients to the subleading terms [ R ] k R − with k > α initially grow until a 1 /k ! factor dominatesover all the other factors. But what is the origin of this behavior? For simplicity,we can understand this behavior by considering the ex-pansion of V mj which form β r . The specific combinationof V mj dictated by the cumulant expansion ensures thatorders lower than O ([ βR ] α R − ) cancel in β r . The j representation contributes to β { r , ··· ,r k } in the form of V m { j , ··· ,j l } · · · V m k { j k , ··· ,j kn } under the constraint | r | = m i e j i where | r | = | r | + · · · + | r k | and e j i = | j i | + · · · + | j il | . Onemight worry that studying the expansion of V mj isn’trepresentative, but one can verify that the scaling behav-ior observed below persists in β r , although the numericalfactors become cumbersome. For these V mj , we have V mj ≈ − m ( e jR ) + 145 (10 m − m )( e jR ) + O ( m [ e jR ] )(47)from which, we see that the coefficients of the [ R ] k R − contributions to β r are accompanied by a factor ∝ k ! m k − e j k − . While the factorials ensure the series con-verges, β r for higher representations r have larger m , j , or both leading to higher order terms in the expan-sion being large for moderate R . This helps explainingwhy Z with the Wilson action fails to replicate U (1)substantially above β = 1 – while the naive scalingwould suggest R (cid:46) √ β k − k would be enough to suppresshigher representations, in reality a stronger bound ofmax { k ! m k − ( e jR ) k − } ≤ e j ≤| r | (cid:46) ∀ β r is required forall subleading terms to be small. Considering the rangeof m with fixed | r | , e j , the bound is strictest when | r | = e j yielding R (cid:46) / e j in order for the lowest order contributionto dominate such that R (cid:46) √ β α − α provides a reasonableestimate for the range of β where the decimated actionprovides a reasonable approximation for its continuouspartner. While these conditions are satisfied for BI , they0 TABLE IV. Parameters of a discrete subgroups necessary tostudy the behavior of β f . G H ∆ S N C V { } U (1) Z Z Z
10 3 −√ SU (2) BT BO −√ BI −√
12 10 0.9648 SU (3) S (108)
18 4 0.7138 S (216)
54 4 0.7557 S (648) 1 − (cid:0) cos π + cos π (cid:1)
24 9 0.7855 V −√
72 5 0.8342 are violated for V in which case the dominant term in the R expansion isn’t clear.Another feature observed in the R expansion of the Z n group is that because V r ∝ sin rR , the sign of the O ([ rR ] k ) terms oscillate, and therefore the sign of β ( n ) r can depend sensitively on R . Since Re χ r ( ) > Re χ r ( N ),the overall sign of β ( n ) r determines whether or not the r -thterm in the action enters the frozen phase in the limitof β → ∞ . This behavior is observed in Fig. 5 where β (1 , { } > β (3) { } < U (1), we can improvethe quantitative understanding of how well H can approxi-mate G , even when β r are not known analytically. Clearly, V r → R →
0, and in that limit the twoactions would agree. Therefore, the difference betweenthe two actions S G − S H ≈ βχ { } ( U ) − β { } χ { } ( u ) ≈ (1 − V { } ) βχ { } ( u ) serve as an indicator of β f .This proxy can be compared to others in the litera-ture. The simplest estimate is β − f ∝ ∆ S = Re Tr( ) − Re Tr( N ) [45]. While this estimate finds monotonic be-havior for discrete groups of U (1) and SU (2), different O (1) factors are needed. It also fails completely for SU (3),as seen in the left panel of Fig. 3.Observing the differing O (1) factors, [45] suggested adifferent estimate. For discrete Non-abelian subgroupsnear β f , S [ u ] is dominated by contributions from u p = N .From duality arguments , the action near β f could beapproximately rewritten as a Z C action where C is theminimal cycle such that u C = for all u ⊂ N . Since C = n for Z n , these arguments predict a single curve β f ≈ . / (1 − cos(2 π/C )) directly from the study of β f in Z n for all discrete subgroups. The discrepancy between SU (2) and U (1) was reduced from ∼ ∼ SU (3) albeit without numerical evidence.Since then β f for the subgroups of SU (3) have beenfound and as anticipated, this estimator proves to bepoor as presented in the center of Fig. 3. In the plot on the right of Fig. 3, β f is plotted as a function of(1 − V { } ) − . We find that monotonic, linear behavioris observed within the uncertainties for each continuousgroup. Best fit lines have been included for each group toguide the eye. This suggests that our estimator capturessome of the non-perturbative behavior near the freezingtransition better than ∆ S − or C . Physics of the differentgroups differ, as signaled by their different scaling regimes.If we divide β f by a rough estimate of β s = [1 , . , U (1) , SU (2) , SU (3) respectively, we might expect tofurther remove some of this group dependence. Doing soin Fig. 4, we find that SU (2) and SU (3) collapse onto asingle line and U (1) within 25%.Using our higher order results, one can then gain insightinto the effectiveness of the ad-hoc actions of V . Eachof these actions corresponds to terms that are generatedat 2nd order in the decimated action. The first ad-hocaction used in [28] can be rewritten as S [ u ] = − X p (cid:18) β u p ) + β Re Tr( u p ) (cid:19) , = − X p (cid:18) ( β − β ) 13 Re χ { } + (6 β ) 16 Re χ { } (cid:19) , (48)where we have used β = aβ + b with a = − . b = 0 . S [ u ] = − X p ˜ β { } χ { } + ˜ β { , } χ { , − } ! (49)where ˜ β { , } = a ˜ β { } + b with a = − .
587 and b = 1 . β by eye. From Fig. 5, wesee that in both ad-hoc actions, reasonably agreement isfound for intermediate β for the 3rd order action. Here β is the coefficient in front of Re χ { } for the ad-hoc actions.The ad-hoc trajectories are known to poorly reflect G at low β , because they lack curvature to fix the knownrequirements at β = 0. At large β , we expect higher orderterms in the cumulant expansion to become relevant andthus disagreement is expected to occur. This surprisingagreement in the intermediate region of β suggests thatactions formed by neglecting terms in the cumulant ex-pansion are optimized in their character basis by settingthe couplings to results given by the resummed cumulantexpansion. VI. CONCLUSION
In this work, we used the cumulant expansion to developa systematic method for studying and improving latticeactions that replace continuous gauge groups by theirdiscrete subgroups. This is a step in the ongoing trektoward developing accurate and efficient digitization on1 β f ∆ S − β f C β f (1 − V { } ) − U (1) SU (2) SU (3) FIG. 3. β f as a function of (left) ∆ S − , (center) the cycle C of N , (right) (1 − V { } ) − . Note that for the subgroups of U (1) and SU (2), monotonic behavior is observed for all three variables, but only for (1 − V { } ) − are the subgroups of SU (3) monotonic. . . . β f / β s (1 − V { } ) − U (1) SU (2) SU (3) FIG. 4. β f /β s as a function of (1 − V { } ) − . − − − − − β { } β { } Ad-hoc S [ u ] S (2) [ u ] S (3) [ u ] − − − − − β { , − } β { } Ad-hoc S [ u ] S (2) [ u ] S (3) [ u ] FIG. 5. β { r } trajectories of S n [ u ] to the ad-hoc actions withadditional terms (left) Re χ { } of [28] and (right) Re χ { , − } of [56]. The open circles indicate the boundary between thefrozen and unfrozen phases obtained on 2 lattices. quantum computers. These decimated actions, throughthe factor V { } , have superior predictive power for findingthe freezing transition compared to prior estimators.We further computed the third-order, single-plaquettecontribution for the general group. These higher-orderterms are necessary for systematizing the decimationprocedure of SU (3) → V where it has been observedthat the inclusion of terms generated in the second-ordercumulant expansion with ad-hoc couplings improve theapproximation of SU (3). The most immediate work inthese directions would be a classical simulation of the fulldecimated action of Eq. 38 and compare it’s effectivenessto the results of [28, 56]. Given the large corrections fromsecond to third order for V , additional work should bedevoted to computing the fourth-order contributions.Another important step in studying the feasibility ofthis procedure is to explicitly construct the quantum reg-isters and primitive gates `a la [60] where smaller discretegroups were investigated. Together with classical latticeresults, this would allows for resource counts. ACKNOWLEDGMENTS
The authors would like to thank Scott Lawrence, JustinThaler, and Yukari Yamauchi for helpful comments onthis work. Y.J. is grateful for the support of DFG, grantsBR 2021/7-2 and SFB TRR 257. H.L. is supported bya Department of Energy QuantiSED grant. Fermilab isoperated by Fermi Research Alliance, LLC under contractnumber DE-AC02-07CH11359 with the United States De-partment of Energy. S.Z. is supported by the National Sci-ence Foundation CAREER award (grant CCF-1845125). [1] R. P. Feynman, Int. J. Theor. Phys. , 467 (1982). [2] P. Jordan and E. P. Wigner, Z. Phys. , 631 (1928). [3] F. Verstraete and J. I. Cirac, J. Stat. Mech. , P09012(2005), arXiv:cond-mat/0508353 [cond-mat].[4] E. Zohar and J. I. Cirac, Phys. Rev. B98 , 075119 (2018),arXiv:1805.05347 [quant-ph].[5] J. D. Whitfield, V. Havl´ıˇcek, and M. Troyer, PhysicalReview A , 030301 (2016), arXiv:1605.09789 [quant-ph].[6] E. A. Martinez et al. , Nature , 516 (2016),arXiv:1605.04570 [quant-ph].[7] N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris,R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M. J.Savage, Phys. Rev. A98 , 032331 (2018), arXiv:1803.03326[quant-ph].[8] H. Lamm and S. Lawrence, Phys. Rev. Lett. , 170501(2018), arXiv:1806.06649 [quant-ph].[9] O. Shehab, K. A. Landsman, Y. Nam, D. Zhu, N. M.Linke, M. J. Keesan, R. C. Pooser, and C. R. Monroe,(2019), arXiv:1904.04338 [quant-ph].[10] E. Zohar, J. I. Cirac, and B. Reznik, Phys. Rev.
A88 ,023617 (2013), arXiv:1303.5040 [quant-ph].[11] D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T.Neil, and J. N. Simone, Phys. Rev. A , 062341 (2019),arXiv:1811.03629 [quant-ph].[12] A. Macridin, P. Spentzouris, J. Amundson, and R. Harnik,Phys. Rev. Lett. , 110504 (2018), arXiv:1802.07347[quant-ph].[13] K. Yeter-Aydeniz, E. F. Dumitrescu, A. J. McCaskey,R. S. Bennink, R. C. Pooser, and G. Siopsis, (2018),arXiv:1811.12332 [quant-ph].[14] N. Klco and M. J. Savage, (2018), arXiv:1808.10378[quant-ph].[15] A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, Phys. Rev. D92 , 076003 (2015),arXiv:1503.08354 [hep-lat].[16] J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S. W.Tsai, and Y. Meurice, Phys. Rev. Lett. , 223201(2018), arXiv:1803.11166 [hep-lat].[17] J. F. Unmuth-Yockey, Phys. Rev. D , 074502 (2019),arXiv:1811.05884 [hep-lat].[18] J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice,and S.-W. Tsai, Phys. Rev. D98 , 094511 (2018),arXiv:1807.09186 [hep-lat].[19] T. V. Zache, F. Hebenstreit, F. Jendrzejewski, M. K.Oberthaler, J. Berges, and P. Hauke, Sci. Technol. ,034010 (2018), arXiv:1802.06704 [cond-mat.quant-gas].[20] I. Raychowdhury and J. R. Stryker, (2018),arXiv:1812.07554 [hep-lat].[21] D. B. Kaplan and J. R. Stryker, (2018), arXiv:1806.08797[hep-lat].[22] J. R. Stryker, Phys. Rev. A99 , 042301 (2019),arXiv:1812.01617 [quant-ph].[23] A. Alexandru, P. F. Bedaque, H. Lamm, andS. Lawrence (NuQS), Phys. Rev. Lett. , 090501 (2019),arXiv:1903.06577 [hep-lat].[24] S. Chandrasekharan and U. J. Wiese, Nucl. Phys.
B492 ,455 (1997), arXiv:hep-lat/9609042 [hep-lat].[25] B. Schlittgen and U. J. Wiese, Phys. Rev.
D63 , 085007(2001), arXiv:hep-lat/0012014 [hep-lat].[26] R. Brower, S. Chandrasekharan, and U. J. Wiese, Phys.Rev.
D60 , 094502 (1999), arXiv:hep-th/9704106 [hep-th].[27] N. Klco, J. R. Stryker, and M. J. Savage,arXiv:1908.06935 [quant-ph].[28] A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm,S. Lawrence, and N. C. Warrington (NuQS), Phys.Rev.D , 114501 (2019), arXiv:1906.11213 [hep-lat]. [29] P. Hasenfratz and F. Niedermayer,
Proceedings, 2001Europhysics Conference on High Energy Physics (EPS-HEP 2001): Budapest, Hungary, July 12-18, 2001 , PoS
HEP2001 , 229 (2001), arXiv:hep-lat/0112003 [hep-lat].[30] S. Caracciolo, A. Montanari, and A. Pelissetto, Phys.Lett.
B513 , 223 (2001), arXiv:hep-lat/0103017 [hep-lat].[31] P. Hasenfratz and F. Niedermayer,
Lattice field theory.Proceedings, 18th International Symposium, Lattice 2000,Bangalore, India, August 17-22, 2000 , Nucl. Phys. Proc.Suppl. , 575 (2001), arXiv:hep-lat/0011056 [hep-lat].[32] A. Patrascioiu and E. Seiler, Phys. Rev. E , 111 (1998).[33] R. Krcmar, A. Gendiar, and T. Nishino, Phys. Rev. E , 022134 (2016).[34] S. Caracciolo, A. Montanari, and A. Pelissetto, PhysicsLetters B , 223 (2001).[35] S. Harmalkar, H. Lamm, and S. Lawrence (NuQS),(2020), arXiv:2001.11490 [hep-lat].[36] H. Flyvbjerg, Nucl. Phys. B243 , 350 (1984).[37] H. Flyvbjerg, Nucl. Phys.
B240 , 481 (1984).[38] H. Valentiner,
De endelige transformations-gruppers the-ori: avec un r´esum´e en fran¸cais , Vol. 2 (Bianco Lunos,1889).[39] S. Crass, Experimental Mathematics , 209 (1999).[40] G. Bhanot, Phys. Lett. , 337 (1982).[41] C. Hagedorn, A. Meroni, and L. Vitale, J. Phys. A ,055201 (2014), arXiv:1307.5308 [hep-ph].[42] M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rev. D20 ,1915 (1979).[43] M. Creutz and M. Okawa, Nucl. Phys.
B220 , 149 (1983).[44] G. Bhanot and C. Rebbi, Phys. Rev.
D24 , 3319 (1981).[45] D. Petcher and D. H. Weingarten, Phys. Rev.
D22 , 2465(1980).[46] D. H. Weingarten and D. N. Petcher, Phys. Lett. ,333 (1981).[47] D. Weingarten, Phys. Lett. , 57 (1982), [,631(1981)].[48] E. H. Fradkin and S. H. Shenker, Phys. Rev. D , 3682(1979).[49] P. Lisboa and C. Michael, Phys. Lett. , 303 (1982).[50] J. C. Halimeh and P. Hauke, (2019), arXiv:2001.00024[cond-mat.quant-gas].[51] H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS),(2020), arXiv:2005.12688 [quant-ph].[52] R. C. Edgar, Nucl. Phys. B200 , 345 (1982).[53] M. Fukugita, T. Kaneko, and M. Kobayashi, Nucl. Phys.