Digital Quantum Simulation of Hadronization in Yang-Mills Theory
De-Sheng Li, Chun-Wang Wu, Ming Zhong, Wei Wu, Ping-Xing Chen
DDigital Quantum Simulation of Yang-Mills Theory and Hadronization
De-Sheng Li, ∗ Chun-Wang Wu, † Ming Zhong, ‡ Wei Wu, § and Ping-Xing Chen ¶ Interdisciplinary Center for Quantum Information,National University of Defense Technology, Changsha 410073, P.R.China Department of Physics, National University of Defense Technology, Changsha 410073, P.R.China
A quantum algorithm with polynomial complexity to non-perturbatively calculate the Dysonseries of SU ( N ) Yang-Mills theory is formulated in terms of quantum circuits. Applying it to thequantum simulation of quantum chromodynamics (QCD), the quark and gluon’s wave functionsevolved from the initial states by the interactions can be observed and the information from wavefunctions can be extracted at any discrete time. This may help us understand the natures of thehadronization which has been an outstanding question of significant implication on high energyphenomenological studies. Yang-Mills theory plays a fundamental role in theconstructions of the Standard Model (SM) of particlephysics. One important property of the theory is asymp-totic freedom demonstrating that the interaction be-comes weaker as the energy scale evolves higher. Thismakes the theory rigidly predictable by perturbationwhen the energy of the physics processes are high enough.While in the opposite infrared limit, the interaction isso large that the perturbation theory breaks down andsome interesting phenomena are out of understandingwithout obscurity. One significant example of the non-perturbative effects is the colour confinement in quantumchromodynamics (QCD) which is a prototype of Yang-Mills theory with SU (3) colour symmetry. Due to theconfinement, the colour charged quarks and gluons cannot be isolated and observed directly at low energy. Theymust clump together to form observable hadrons.The nature of hadronization is far away from wellexplored and remains as a big mystery in high energyphysics. On the other hand, the knowledge of hadroniza-tion is indispensable in investigating the new physics be-yond the SM, since hadrons appear inevitably in everyhigh energy physics experiment. Plenty of endeavourshave been made since 1960s and various phenomenologi-cal models can give us some information on hadronizationso that it can be taken into account in the experimentaldata analysis. More precise and comprehensive knowl-edge of the hadronization effects is required as more pre-cise experiments devoting to study the new physics.The most promising attempt to account for the ef-fects in the near future is believed to employ the lat-tice theory [1], a method to quantize a field theory ona discretized lattice and permit numerical calculationsin strong-coupling limit. Though it has achieved muchprogress, the capability of the lattice computations reliesheavily on the performance of computers available, whichin turn hinders it’s applications since the development ofthe classical computer meets the bottleneck increasingly.The breakthrough of the bottleneck is the quantumcomputer introduced notionally by Feyman three decadesago [2], and has speeded up development recently [3–8].The quantum computer is structurally powerful to sim- plify the calculation and adaptable to the complex sys-tem studies in essence [9–11]. The idea of using it toexplore the physics system of quantum field theory hasset out to attract increasing interest. A two trapped ionsscenario on quantum simulation of quantum field theorywith fermion-boson-antifermion interaction in two space-time dimensions was proposed [12]. The quantum al-gorithm to calculate the scattering amplitude of scalarquartic self-interaction in four and fewer dimensionalspacetime was developed and found to have exponen-tial speedup [13, 14]. A quantum simulation of fermionantifermion scattering mediated by gauge boson was ex-perimentally implemented in a trapped ion very recently[15]. One impressive common feature of these studies isthat quantum simulation works in both perturbative andnon-perturbative regimes.Given it’s significance in high energy physics, it is de-served to make some efforts in quantum simulation ofYang-Mills theory. We will try to fill this gap in thepaper, presenting a scheme to describe the SU ( N ) Yang-Mills theory by digital quantum simulator. The quanti-zation to Yang-Mills gauge theory is rather different andcomplicated as compared to the pure scalar and abeliangauge theory, in that ghost fields are introduced into theLagrangian accompanying by the gauge fixing. We for-mulate a canonical quantization based algorithm to cal-culate the Dyson series of the theory non-perturbativelyand thus can be made use to study the system in bothperturbative and non-perturbative regimes. Using qubitsthe number of which increases polynomially with the lat-tices to simulate the interactions, the algorithm has ex-ponential speedup. Moreover, the dynamical informationfrom the evolving wave functions of particles can be ex-tracted at any discrete time. This may help us dig intothe understanding of hadronization and low energy QCD.We begin with a short introduction to the formalismof SU ( N ) Yang-Mills theory and it’s canonical quan-tization. With these indispensable knowledge, we willthen present an elaboration on the quantum simulationof the theory and discuss it’s possible application on thehadronization of QCD.For definiteness and practicality, we study a one- a r X i v : . [ qu a n t - ph ] O c t flavor non-abelian gauge theory with SU ( N ) symmetry.The extension to arbitrary N f duplicates of fermions isstraightforward. The Lagrangian density is [16] L = −
14 ( F aµν ) + ¯ ψ i ( iδ ij γ µ ∂ µ + gγ µ A aµ t aij − mδ ij ) ψ j −
12 ( ∂ µ A aµ ) + ( ∂ µ ¯ c a ) ∂ µ c a + gf abc ( ∂ µ ¯ c a ) A bµ c c , (1)where ψ i and A aµ are fermion and gauge fields with i = 1 , · · · , N and a = 1 , · · · , N −
1. The Dirac ma-trices γ µ are defined so that { γ µ , γ ν } = 2 η µν I × withthe convention η µν = diag (1 , − , − , −
1) for the space-time metric. The SU ( N ) generators t a are representedby the traceless and Hermitian matrices. The gauge fieldstrength reads F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν , (2)where f abc are the structure constants of the group. Thefirst two terms in Eq. (1) compose the Yang-Mills theory.To make a proper quantization, one must introduce thegauge fixing and the resulting Faddeev-Popov ghosts inthe other terms. The longitudinal gauge fields growingout of the gauge fixing and the ghost fields c a (¯ c a ) do notcorrespond to physical particles at all.We canonically quantize the theory in interaction pic-ture where the fields behave freely so that the field op-erators are expanded in terms of the orthonormal andcomplete solutions of the free equations of motion withthe annihilation and creation operators as the coefficients A aµ ( x ) = (cid:90) d p (2 π ) (cid:112) ω (cid:126)p (cid:88) l ( a a(cid:126)p,l (cid:15) lµ e − ip · x + a a † (cid:126)p,l (cid:15) l ∗ µ e ip · x ) ,ψ i ( x ) = (cid:90) d p (2 π ) (cid:112) ω (cid:126)p (cid:88) s ( b s(cid:126)p,i u sp e − ip · x + c s † (cid:126)p,i v sp e ip · x ) ,c a ( x ) = (cid:90) d p (2 π ) (cid:112) ω (cid:126)p ( d a(cid:126)p e − ip · x + d a † (cid:126)p e ip · x ) , ¯ c a ( x ) = (cid:90) d p (2 π ) (cid:112) ω (cid:126)p ( e a(cid:126)p e − ip · x − e a † (cid:126)p e ip · x ) . (3)The ω (cid:126)p = p = (cid:112) | (cid:126)p | + m denotes the energy of theparticles created or annihilated by the coefficient oper-ators. The (cid:15) lµ are polarization vectors of the masslessgauge fields. The u sp and v sp are the Dirac spinors for theparticles and anti-particles respectively.Giving the equal time commutation to gauge field andanti-commutation relations to the other fields with theircanonical momentums, we then have the relations interms of the annihilation and creation operators[ a a(cid:126)p,l , a b † (cid:126)p (cid:48) ,m ] = (2 π ) δ (3) ( (cid:126)p − (cid:126)p (cid:48) ) δ lm δ ab , { b r(cid:126)p,i , b s † (cid:126)p (cid:48) ,j } = { c r(cid:126)p,i , c s † (cid:126)p (cid:48) ,j } = (2 π ) δ (3) ( (cid:126)p − (cid:126)p (cid:48) ) δ rs δ ij , { d a(cid:126)p , e b † (cid:126)p (cid:48) } = { e b(cid:126)p , d a † (cid:126)p (cid:48) } = − (2 π ) δ (3) ( (cid:126)p − (cid:126)p (cid:48) ) δ ab , (4) with all the other commutators or anticommutators van-ished. Note that we have the anti-commutation relationfor the annihilation operator and the creation operatorof ghost and anti-ghost field in the last equation.We now switch to the quantum simulation. The maintasks involve giving maps between qubits and particlestates, digitizing the interactions by matrices and evolv-ing the qubits by the operations of the matrices via quan-tum circuits. Note that we study within the interactionpicture rather than the Schr¨ o dinger picture in [13, 14].The particles are described by the Fock states labelingwith the momentum, spin and color charge as the degreesof freedom. To have a digital quantum simulation, thecontinuous three dimensional momentum space needs tobe discretized to the lattice space [13, 14]Γ = a Z P , (5)where the Z P denotes a ˆ P × ˆ P × ˆ P lattice in three dimen-sional momentum space. The ˆ P is equal to Int ( P/a ) with P the range of momentum component and a the latticespacing. The number of the lattice cites is V = ˆ P .The multi-particle state of quantum field theory is Fockspace valued. The bases to expand the Fock space of theYang-Mills theory are F = {⊗ (cid:126)p ∈ Γ | n al , m si , ¯ m rj , l b , ¯ l c (cid:105) (cid:126)p } , (6)where n al , m si , ¯ m ri , l a and ¯ l a are occupation numbers ofgauge boson, fermion, anti-fermion, ghost and anti-ghostat lattice site (cid:126)p ∈ Γ respectively. They are all integerswith 0 (cid:54) n al (cid:54) N and 0 (cid:54) m si , ¯ m ri , l a , ¯ l a (cid:54)
1. We haveintroduced a truncation N to the occupation number ofboson in each lattice, which is practically reasonable sincethe number of bosons available in an experiment is alwayslimited. For an illustration, the vacuum of the free theoryis given by | V ac (cid:105) = ⊗ (cid:126)p ∈ Γ | al , si , rj , b , c (cid:105) (cid:126)p . (7)The states with different number of free particles can berelated to each other by the creation and annihilationoperators¯ a a † (cid:126)p,l | n al (cid:105) (cid:126)p = √ n + 1 | ( n + 1) al (cid:105) (cid:126)p , ¯ a a † (cid:126)p,l |N al (cid:105) (cid:126)p = 0 , (8)¯ a a(cid:126)p,l | n al (cid:105) (cid:126)p = √ n | ( n − al (cid:105) (cid:126)p , ¯ a a(cid:126)p,l | al (cid:105) (cid:126)p = 0 , (9) b s † (cid:126)p,i | si (cid:105) (cid:126)p = | si (cid:105) (cid:126)p , b s † (cid:126)p,i | si (cid:105) (cid:126)p = 0 , (10) b s(cid:126)p,i | si (cid:105) (cid:126)p = | si (cid:105) (cid:126)p , b s(cid:126)p,i | si (cid:105) (cid:126)p = 0 , (11)where ¯ a a † (cid:126)p,l and ¯ a a(cid:126)p,l are the truncated creation and anni-hilation operators of bosons, and b s † (cid:126)p,i and b s(cid:126)p,i are that forthe fermions.According to [17], the one-to-one map for gauge bosonstates to qubits can be obtained | al (cid:105) (cid:126)p ↔ | ↑ ↓ ↓ · · · ↓ N (cid:105) (cid:126)p,a,l , | al (cid:105) (cid:126)p ↔ | ↓ ↑ ↓ · · · ↓ N (cid:105) (cid:126)p,a,l , ... | N al (cid:105) (cid:126)p ↔ | ↓ ↓ ↓ · · · ↑ N (cid:105) (cid:126)p,a,l , (12)and that between fermion states and qubits are | si (cid:105) (cid:126)p ↔ | ↑(cid:105) (cid:126)p,i,s , | si (cid:105) (cid:126)p ↔ | ↓(cid:105) (cid:126)p,i,s . (13)Summing up all qubits which expand the Fock space forone-flavor SU ( N ) Yang-Mills theory, the total number ofthe working qubits is [2( N − N + 1) + 4 N ] V . Thespace complexity of working qubits is polynomial.The Jordan-Wigner mapping gives the matrix repre-sentations to the creation and annihilation operators¯ a a † (cid:126)p,l = N (cid:88) h =0 √ h + 1 σ h,(cid:126)p,a,l − σ h +1 ,(cid:126)p,a,l + , ¯ a a(cid:126)p,l = N (cid:88) h =0 √ h + 1 σ h,(cid:126)p,a,l + σ h +1 ,(cid:126)p,a,l − ,b s † (cid:126)p,i = ( (cid:89) α − σ αz ) σ (cid:126)p,i,s + ,b s(cid:126)p,i = ( (cid:89) α − σ αz ) σ (cid:126)p,i,s − , (14)with σ ± = ( σ x ± iσ y ) and α ∈ { ( (cid:126)p (cid:48) , i (cid:48) , s (cid:48) ) |K ( (cid:126)p (cid:48) , i (cid:48) , s (cid:48) ) < K ( (cid:126)p, i, s ) } , where K ( (cid:126)p, i, s ) ∈ N is the primary key of( (cid:126)p, i, s ). The mapping digitally expresses the action ofthe creation and annihilation operators to the particlestates by the operation of the Pauli matrices σ x,y,z onthe qubits. In the interaction picture, the Hamiltonian iscomposed of creation and annihilation operators, whichmakes the simulation of the evolution of the particlestates in a straightforward manner.The evolution of the particle state in the interactionpicture is | Ψ( t ) (cid:105) = U ( t, t ) | Ψ( t ) (cid:105) , (15)where | Ψ( t ) (cid:105) is the initial state. The U ( t, t ) is the evo-lution operator (Dyson series) U ( t, t ) = I, U ( t, t ) = U ( t, t (cid:48) ) U ( t (cid:48) , t ) . (16)Dividing the time interval into slices of duration ∆ t =( t − t ) /n , we have U ( t, t ) = U ( t, t + ( n − t ) · · · U ( t + ∆ t, t ) . When ∆ t is small enough, we have U ( t + ∆ t, t ) = e − iH I ( t )∆ t (17) with H I the interaction part of the Hamiltonian H I = (cid:90) d x H I . (18)The Hamiltonian density of Yang-Mills theory H I is asum of four types of Hermitian interactions: fermion-boson H F I , four-boson H G I , three-boson H G I andghost-boson H F P I H I = H F I + H G I + H G I + H F P I = − gA aµ ¯ ψ i γ µ t aij ψ j + 14 g ( f eab A aµ A bν )( f ecd A µc A νd )+ gf abc ( ∂ µ A aν ) A µb A νc − gf abc ( ∂ µ ¯ c a ) A bµ c c , Using the Trotter formula [18] e i ( A + B )∆ t = e iA ∆ t e iB ∆ t + O (∆ t ) (19)where A and B are Hermitian operators and taking ∆ t asa small enough quantity so that the terms proportionalto ∆ t can be ignored, we have e − iH I ( t )∆ t = e − iH FI ∆ t e − iH G I ∆ t e − iH G I ∆ t e − iH FPI ∆ t . The task following is to substitute (3) and (14) into theabove equation. With repeated use of the Trotter for-mula, it can be written as continued product of the ex-ponential of the Pauli matrices. The operation of eachfactor of the product on qubits can be implemented bya quantum circuit. For illustration, we take e − iH FI ∆ t asan example to show the procedure.With (3), the Hamiltonian H F I can be written as H F I = − g (cid:90) d x ( A a + µ ¯ ψ + i γ µ t aij ψ + j + A a + µ ¯ ψ + i γ µ t aij ψ − j + A a + µ ¯ ψ − i γ µ t aij ψ + j + A a + µ ¯ ψ − i γ µ t aij ψ − j + H.c. ) , where the superscripts + and − on the fields denote thepositive and negative frequency components. We pickout a typical structure in H F I H = − g (cid:90) d x ( A a + µ ¯ ψ − i γ µ t aij ψ + j + H.c. ) , and give a detailed analysis.Discretizing the momentum space and applying thegauge boson occupation number truncation, we can havethe equation after expressed with the creation and anni-hilation operators H = − g (cid:88) (cid:126)p ,(cid:126)p ,(cid:126)p ∈ Γ a a a (2 π ) (cid:112) ω (cid:126)p ω (cid:126)p ω (cid:126)p δ (3) ( (cid:126)p − (cid:126)p + (cid:126)p ) (cid:88) l,r,s [¯ a a(cid:126)p ,l b r † (cid:126)p ,i b s(cid:126)p ,j ¯ u rp ( (cid:15) l · γ ) t aij u sp e − iω t + H.c. ] , (20)with ω = ω (cid:126)p − ω (cid:126)p + ω (cid:126)p and δ (3) being the three-dimensional discrete delta function. The a i denotes thespacing of the momentum (cid:126)p i lattice. The repeated indices ! H! H!H! H!R! R † !H! H! |0 (cid:7) ! W o r k i n g ! Q u b i t s ! A u x i l i a r y ! Q u b i t ! e (cid:1)(cid:2)(cid:4)(cid:6)(cid:5) ! |0 (cid:7) ! FIG. 1. Quantum circuits which perform unitary transforma-tion e − i ∆ tσ x σ x σ y ( (cid:81) α σ αz ) σ rx on working qubits. The symbol H means Hadamard operation. The Z is the σ z matrix and the R is a transformation: Rσ y R † = σ z . This unitary transfor-mation is one part of transformation in e − iH ξ ∆ t . There arelots of CNOT operations and single qubit operations beingused in this quantum circuits. But only one auxiliary qubitis needed. in the square bracket is understood as the summationconvention.Applying the Jordan-Wigner mappings (14) to theterms in the bracket, it is found that the H is sum ofterms which are products of Pauli matrices H ξ = λ ξ ⊗ Mr =1 σ rc ( r ) , H = (cid:88) ξ H ξ , (21)where λ ξ ∈ R is the effective coupling strength of H ξ .The Pauli matrix σ rc ( r ) acts on the r th qubit, with c ( r ) ∈{ , , , } and σ c ( r ) = ( I, σ x , σ y , σ z ). With repeated useof the Trotter formula, it is easy to have e − iH ∆ t = (cid:89) ξ e − iH ξ ∆ t , (22)and the similar expression for e − iH FI ∆ t as well.As an unitary transformation, the action of the matrix e − iH ∆ t to the qubits can be effectively implemented by aquantum circuits [18]. This is an advantage of the quan-tum simulation over the classical one which performs thecalculation by perturbation in expanding the exponentialinto power series when the coupling is in the weak limitor by the lattice gauge theory to numerically evaluate thefunctional integral in the strong coupling limit.An example of quantum circuit to simulate the e − i ∆ tσ x σ x σ y ( (cid:81) α σ αz ) σ rx operation on working qubits isgiven in FIG. 1 where the symbol H denotes Hadamardoperation and R can be chosen as R = 1 √ (cid:18) − i − i (cid:19) , (23)and we have Rσ y R † = σ z . The unitary transformationis one part of transformation in e − iH ξ ∆ t . With the same treament, the operations of e − iH G I ∆ t , e − iH G I ∆ t and e − iH FBI ∆ t can be implemented by quan-tum circuits, which add up to the quantum simulationof the evolution (15). These quantum circuits are con-structed by a vast quantity of CNOT operations, sin-gle qubit operations and one auxiliary qubit. Sum-ming up the CNOT operations used in quantum circuits,the time complexity to simulate e − iH FI ∆ t , e − iH G I ∆ t , e − iH G I ∆ t and e − iH FPI ∆ t is lower than O ( N V ( N + 1)), O ( N V ( N + 1) ), O ( N V ( N + 1) ) and O ( N V ( N +1)) respectively. This leads to the time complexity of thesimulation U ( t, t ) to be O ( N V ( N + 1) n ).A good venue to make use of the algorithm is QCDwhere simulation can be performed to study the hadronphysics in the future when the quantum computer be-comes full-developed. One interesting application mightbe the visualization of the hadronization, by giving dy-namic information on how the unobservable free quarksform into a hadron. For demonstration purpose, wesketch out an example of meson with one flavor. Theinitial state to evolve is the three-colour superposition ofa pair of free quark and anti-quark with equal amplitude | Ψ( t ) (cid:105) = 1 √ (cid:88) i =1 | q s (cid:126)p ,i ¯ q s (cid:126)p ,i (cid:105) . (24)To prepare the color singlet initial state in experiment,one can first produce, according to Eq. (13), two qubitsto represent the two-quark state | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) . (25)Then with a specific unitary transformation T acting onthe qubits, one can have the qubits representation of theinitial state | Ψ( t ) (cid:105) = T | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) . (26)The operation of the transformation T can be imple-mented by quantum circuits. The complexity to prepareinitial state is O (1). At last, one can turn on the quantumcircuits and adjust the value of the coupling constant g in the evolution operator U ( t, t ) to accomplish the sim-ulation of the evolution from the initial state to a meson U ( t, t ) | Ψ( t ) (cid:105) → | M eson (cid:105) . (27)In the simulation, one can extract information on thebehavior of the particles at any k-slice discrete time t = t + k ∆ t from the evolving wave function | Ψ( t ) (cid:105) , with k =0 , · · · n . A typical and straightforward measurement isthe possibility P ( n q , (cid:126)p, t ) of each kind of particles withoccupation number n q at momentum (cid:126)p , by counting thenumber of the corresponding qubit collapsing into spin | ↑(cid:105) or | ↓(cid:105) state.In summary, we have presented a quantum algorithmto non-perturbatively simulate the Yang-Mills theory.Both the space complexity and time complexity of this al-gorithm is polynomial. This efficient algorithm may pavethe way to study the physics in and beyond the StandardModel of particle physics with quantum computers in thefuture.This work is supported by National Basic ResearchProgram of China under Grant No. 2016YFA0301903,and the National Natural Science Foundation of Chinaunder Grants Nos. 11174370, 11304387, 61632021,11305262, 61205108, 11475258 and 11574398. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] (corresponding author) § email: [email protected] ¶ email: [email protected] (corresponding author)[1] K. G. Wilson, Physical Review D , 2445 (1974).[2] R. P. Feynman, International Journal of TheoreticalPhysics , 467 (1982).[3] S. Lloyd, Science , 1073 (1996).[4] T. Byrnes and Y. Yamamoto, Physical Review A ,022328 (2006).[5] B. P. Lanyon, C. Hempel, D. Nigg, M. M¨uller, R. Ger-ritsma, F. Z¨ahringer, P. Schindler, J. Barreiro, M. Ram-bach, G. Kirchmair, et al., Science , 57 (2011). [6] I. Georgescu, S. Ashhab, and F. Nori, Reviews of ModernPhysics , 153 (2014).[7] L. Lamata, A. Mezzacapo, J. Casanova, and E. Solano,EPJ Quantum Technology , 9 (2014).[8] G. S. Paraoanu, Journal of Low Temperature Physics , 633 (2014).[9] J. I. Cirac and P. Zoller, Nature Physics , 264 (2012).[10] R. Blatt and C. F. Roos, Nature Physics , 277 (2012).[11] I. Arrazola, J. S. Pedernales, L. Lamata, and E. Solano,Scientific Reports , 30534 (2016).[12] J. Casanova, L. Lamata, I. Egusquiza, R. Gerritsma,C. Roos, J. Garc´ıa-Ripoll, and E. Solano, Physical Re-view Letters , 260501 (2011).[13] S. P. Jordan, K. S. M. Lee, and J. Preskill, Science ,1130 (2012).[14] S. P. Jordan, K. S. M. Lee, and J. Preskill, QuantumInformation and Computation , 1014 (2014).[15] X. Zhang, K. Zhang, Y. Shen, S. Zhang, J. Zhang,M. Yung, J. Casanova, J. S. Pedernales, L. Lamata,E. Solano, and K. Kim, Nature Communications , 195(2018).[16] T. Kugo and I. Ojima, Physics Letters B , 459 (1978).[17] R. Somma, G. Ortiz, E. Knill, and J. Gubernatis, Inter-national Journal of Quantum Information , 189 (2003).[18] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Pressand Tsinghua University Press, 2015) pp. 204–211. Suplementary materialThe expression of H . In the case K ( (cid:126)p , i, r ) (cid:54) K ( (cid:126)p , j, s ), the explicit expression of the factor I in H is presentedin terms of the Pauli matrices I = (cid:88) l,r,s [¯ a a(cid:126)p ,l b r † (cid:126)p ,i b s(cid:126)p ,j ¯ u r ( p )( (cid:15) l · γ ) t aij u s ( p ) e − iω t + H.c. ]= 18 (cid:88) l,r,s N (cid:88) h =0 (cid:2)(cid:0) σ h,(cid:126)p ,a,lx σ h +1 ,(cid:126)p ,a,lx + σ h,(cid:126)p ,a,ly σ h +1 ,(cid:126)p ,a,ly (cid:1) (28) (cid:32) ( − σ (cid:126)p ,i,ry )( (cid:89) α − σ αz ) σ (cid:126)p ,j,sx W a,r,s ,ij,l + σ (cid:126)p ,i,ry ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sy W a,r,s ,ij,l + σ (cid:126)p ,i,rx ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sx W a,r,s ,ij,l + σ (cid:126)p ,i,rx ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sy W a,r,s ,ij,l (cid:33) + (cid:0) σ h,(cid:126)p ,a,ly σ h +1 ,(cid:126)p ,a,lx − σ h,(cid:126)p ,a,lx σ h +1 ,(cid:126)p ,a,ly (cid:1)(cid:32) ( − σ (cid:126)p ,i,ry )( (cid:89) α − σ αz ) σ (cid:126)p ,j,sx W a,r,s ,ij,l − σ (cid:126)p ,i,ry ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sy W a,r,s ,ij,l − σ (cid:126)p ,i,rx ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sx W a,r,s ,ij,l + σ (cid:126)p ,i,rx ( (cid:89) α − σ αz ) σ (cid:126)p ,j,sy W a,r,s ,ij,l (cid:33)(cid:35) , with α ∈ { ( (cid:126)p (cid:48) , i (cid:48) , s (cid:48) ) |K ( (cid:126)p , i, r ) < K ( (cid:126)p (cid:48) , i (cid:48) , s (cid:48) ) < K ( (cid:126)p , j, s ) } . The W a,r,s ,ij,l , W a,r,s ,ij,l ∈ R are defined by¯ u r ( p )( (cid:15) l · γ ) t aij u s ( p ) e − iω t = W a,r,s ,ij,l + iW a,r,s ,ij,l . (29)We have checked that the I is Hermitian and thus e − iH ∆ t can be simulated by quantum circuits. The CNOToperations used to simulate it is less than 512(2 + V N )( N − N ( N + 1). To simulate e − iH FI ∆ t , we need less than64 V (cid:2) (1 + N ) ( V −
3) + 64 V (cid:3) N ( N − N + 1) CNOT operations. The expression for four-gluon interaction.
The decomposition of four gluon interaction is rather complicated.We use a term H in H G I as an example H = g (cid:90) d xf eab f ecd A a + µ A b + ν A µc − A νd − . (30)After performing the discretization and gluon occupation number truncation, one can have H = g (cid:88) (cid:126)p ∈ Γ (cid:88) (cid:126)p ∈ Γ (cid:88) (cid:126)p ∈ Γ (cid:88) (cid:126)p ∈ Γ a a a a π ) √ ω (cid:126)p ω (cid:126)p ω (cid:126)p ω (cid:126)p δ (3) ( (cid:126)p + (cid:126)p − (cid:126)p − (cid:126)p ) f eab f ecd (31) (cid:88) l ,l ,l ,l ¯ a a(cid:126)p ,l ¯ a b(cid:126)p ,l ¯ a c † (cid:126)p ,l ¯ a d † (cid:126)p ,l (cid:15) µ,l (cid:15) ν,l (cid:15) µ ∗ l (cid:15) ν ∗ l e − iω t , with ω = ω (cid:126)p + ω (cid:126)p − ω (cid:126)p − ω (cid:126)p . We calculate the factor I in H I = 12 (cid:88) l ,l ,l ,l (cid:16) ¯ a a(cid:126)p ,l ¯ a b(cid:126)p ,l ¯ a c † (cid:126)p ,l ¯ a d † (cid:126)p ,l (cid:15) µ,l (cid:15) ν,l (cid:15) µ ∗ l (cid:15) ν ∗ l e − iω t + ¯ a d(cid:126)p ,l ¯ a c(cid:126)p ,l ¯ a b † (cid:126)p ,l ¯ a a † (cid:126)p ,l (cid:15) ∗ µ,l (cid:15) ∗ ν,l (cid:15) µl (cid:15) νl e iω t (cid:17) , (32)and give its expression in two cases categorized by the quantum numbers of the particles.For the case ( (cid:126)p , a, l ) = ( (cid:126)p , c, l ) (cid:54) = ( (cid:126)p , b, l ) = ( (cid:126)p , d, l ) or ( (cid:126)p , a, l ) = ( (cid:126)p , d, l ) (cid:54) = ( (cid:126)p , b, l ) = ( (cid:126)p , c, l ) , (33)we have I = ¯ a a(cid:126)p ,l ¯ a c † (cid:126)p ,l ¯ a b(cid:126)p ,l ¯ a d † (cid:126)p ,l W ,l ,l ,l ,l , (34)with ¯ a a(cid:126)p ,l ¯ a c † (cid:126)p ,l = N (cid:88) h ,h =0 (cid:112) ( h + 1)( h + 1)116 (cid:2) ( σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l x + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l y )( σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l x + σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l y )+( σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l y − σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l x )( σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l y − σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l x ) (cid:3)(cid:12)(cid:12) h (cid:54) = h + N (cid:88) h h + 14 ( I h ,(cid:126)p ,a,l + σ h ,(cid:126)p ,a,l z )( I h +1 ,(cid:126)p ,a,l − σ h +1 ,(cid:126)p ,a,l z ) , in which the repeated indices are not summed.When the quantum numbers of the bosons ( (cid:126)p , a, l ), ( (cid:126)p , b, l ), ( (cid:126)p , c, l ) and ( (cid:126)p , d, l ) do not equal to each other,we have another expression for I I = 1256 (cid:88) l ,l ,l ,l N (cid:88) h ,h ,h ,h =0 (cid:112) ( h + 1)( h + 1)( h + 1)( h + 1)[(Ω Ω − Ω Ω ) W ,l ,l ,l ,l − (Ω Ω + Ω Ω ) W ,l ,l ,l ,l ] , (35)where the Ω , Ω , Ω , Ω are Ω = ( σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l x + σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l y + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l x + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l y − σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l y + σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l x + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l y − σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l x ) , Ω = ( − σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l x − σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l y + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l x + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l y − σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l y − σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l x σ h +1 ,(cid:126)p ,b,l y + σ h ,(cid:126)p ,a,l x σ h +1 ,(cid:126)p ,a,l x σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l x + σ h ,(cid:126)p ,a,l y σ h +1 ,(cid:126)p ,a,l y σ h ,(cid:126)p ,b,l y σ h +1 ,(cid:126)p ,b,l x ) , Ω = ( σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l x + σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l y + σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l x + σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l y − σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l x + σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l y + σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l x − σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l y ) , Ω = ( − σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l x − σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l y + σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l x + σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l y − σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l x − σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l y σ h +1 ,(cid:126)p ,d,l x + σ h ,(cid:126)p ,c,l x σ h +1 ,(cid:126)p ,c,l x σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l y + σ h ,(cid:126)p ,c,l y σ h +1 ,(cid:126)p ,c,l y σ h ,(cid:126)p ,d,l x σ h +1 ,(cid:126)p ,d,l y ) . Note that the repeated indices a, b, c, d , (cid:126)p , (cid:126)p , (cid:126)p , (cid:126)p , l , l , l , l are not summed and the coefficients W ,l ,l ,l ,l , W ,l ,l ,l ,l ∈ R in Eqs. (34, 35) are defined by (cid:15) µ,l (cid:15) ν,l (cid:15) µ ∗ l (cid:15) ν ∗ l e − iω t = W ,l ,l ,l ,l + iW ,l ,l ,l ,l . (36)We need less than 65536( N + 1) CNOT operations to simulate e − iI ∆ t and 655360( N − V ( N + 1) CNOToperations to simulate e − iH G I ∆ t . The unitary transformation T . The unitary transformation T is T | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) = 1 √ (cid:88) i =1 | q s (cid:126)p ,i ¯ q s (cid:126)p ,i (cid:105) . (37)We label the states as | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , | (cid:105) = | q s (cid:126)p , ¯ q s (cid:126)p , (cid:105) , (38)then T can be represented by the states as a 9 × T = 1 √ (cid:0) − i √ (cid:1) (cid:0) − i + √ (cid:1) √ √ √ − (cid:0) i √ (cid:1) − (cid:0) i + √ (cid:1) √ √ √ i ..