Markovian quantum simulation of Fenna-Matthew-Olson (FMO) complex using one-parameter semigroup of generators

arxiv:90.03085v [quant-ph] Jan 09 Markovian quantum simulation of Fenna-Matthew-Olson (FMO) complex using one-parameter semigroup of generators M.Mahdian and H.Davoodi Yeganeh Faculty of Physics, Theoretical and astrophysics department, University of Tabriz, 5665-63 Tabriz, Iran Abstract The application of the theory of open quantum systems in biological processes such as photosynthetic complexes, has been recently received renewed attention. In this paper, a quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of FMO complex exist in photosynthesis using a universal set of one-parameter semigroup of generators. We investigate constituent of each generator that obtain from spectral decomposition of Gorini-Kossakowski-Sudarshan (GKS) matrix by using of linear combination and unitary conjugation. Also we design quantum circuit for implement of these generators. Keyword FMO complex, Linear combination, Unitary conjugation, GKS matrix mahdian@tabrizu.ac.ir h.yeganeh@tabrizu.ac.ir

Introduction In the recent years the study of biological systems in which quantum dynamics is visible and theory of open quantum system is applicable to describe dynamics of these systems has attracted much attention []. One of these biological systems in planets and a group of prokaryotes like green sulfur bacteria utilize photosynthesis as a process to produce energetic chemical compounds by free solar energy. Harvesting of light energy and its conversion to cellular energy currency is mainly done in photosystem complexes present in all photosynthetic organisms []. Photosystem is mainly constructed of two linked sections, antenna unit includes several proteins referred as Light Harvesting Center (LHC) which absorbs energy of light and conducting it to reaction center. Both of LHC and reaction center contain pigment molecules, and ability of pigment molecules in absorbing various light wavelengths, increases accessible spectrum for photosynthesis process. After absorbing a photon, the FMO antennae complex transfers it to the reaction center and behave like quantum wire between antenna and reaction center[3]. The FMO complex structure is relatively simple, consisting of three monomers. Each monomer environments including seven bacterial protein or molecule called bacteriochlorophyll a (BChl a). It is important to know that key to photosynthesis is the interaction of light with the electronic degrees of freedom of the pigment molecules which is quantum mechanical in nature and also longlived quantum coherence among the electronic excited states of the multiple pigments in FMO complex has been shown by D electronic spectroscopy. Pigments go to excited state receiving light energy, so the state of pigments could be evaluated in both excited and ground states of molecular orbital energies due to thermal changes and also behave like two-level system. Several researcher has studied the electronic excitation transfer by diverse methods such as Forster theory in weak molecular interaction limit or by Redfield master equations derived from Markov approximation in weak coupling regime between molecules and environment [3] [4] [5] [6]. Effective dynamics FMO complex is modeled by a Hamiltonian which describes the coherent exchange of excitations between sites and local Lindblad terms that take into account the dissipation and dephasing caused by the surrounding environment [7]. Classical computers fail to efficiently simulate quantum systems with complex many-body interactions due to the exponential growth of variables for characterizing these systems so quantum simulation was proposed to solve such an exponential explosion problem using a controllable quantum system

which was originally conjectured by Feynman[8]. The dynamics evolution of closed systems is described by unitary transformation, it can be simulated directly with quantum simulator but in the real world all quantum systems are invariably in contact with an environment and are an open quantum system, usually dynamic evolution of these systems due to decoherence and dissipation is non-unitarian. Generally, dynamics of open quantum system is very complex and often used to describe the dynamics of proximity like the born and Markov approximations are used [9]. A lot of analytical and numerical methods have been employed to simulate the dynamics of open quantum systems like composition framework for the combination and transformation of semigroup generators, simulation of Markovian quantum dynamics by logic network and in particular simulation of arbitrary quantum channels [0] [] [] [3] [4] [5]. However, in these methods exists no universal set of nonunitary processes through which all such processes can be simulated via sequential simulations from the universal set, but they are applicable in many of problem. Rayn et al. introduce efficient method to simulate Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be through sequential simulations of processes from the universal set. They use of linear combination and unitary conjugation to simulate Markovian open quantum systems [6]. The simulation dynamics of light-harvesting complexes is highly regarded and a large number of various experimental and analytically studies has been conducted on them. Finding spectral density by molecular dynamics and numerical method have been studied in [7] [8] and the system dynamics simulation have been investigated based on two-dimensional electronic spectroscopy [9], superconducting qubits [0]. In this paper we use of linear combination and unitary conjugation to simulate Markovian non-unitary processes in photosynthetic FMO complex. We design quantum circuit to implement unitary operation that apply in simulation. The remainder of the paper is organized as follows. In Sec. we give a brief description of effective dynamics FMO complex. We descript method to universal simulation of Markovian open quantum systems. in Sec.3. In Section 4 calculations to simulate the non-unitarian processes in photosynthetic FMO complex will be study. We finally express our results in Sec.5. 3

FMO complex FMO complex is generally constituted of multiple chromophores which transform photons into exactions and transport to a reaction center. As already mentioned that efficient dynamics FMO complex express by combining Hamiltonian which describe the coherent exchange of excitations between sites, and local Lindblad terms that take into account the dephasing and dissipation caused by the external environment [7] and only here we consider non-unitarian processes. For expressing the dynamics of the non-unitarian part assumed that the system is susceptible simultaneously to two distinct types of noise processes, a dissipative process and dephasing process. Dissipative processes pass on excitation energy with rate Γ j to the environment and dephasing process destroys the phase coherence with rate γ j of site j th and the both processes by using the Markovian master equation with local dephasing and dissipation terms. We approached the Markovian master equation for FMO complex, dissipative and dephasing processes are captured, respectively, by the Lindblad super-operators as: 7 L diss (ρ) = Γ j ( σ j + σj ρ ρσ j + σj + σj ρσ j + ), () j= 7 L deph (ρ) = γ j ( σ j + σj ρ ρσ j + σj + σ j + σj ρσ j + σj ). () j= Where σ + j = j 0 and σ j = 0 j are raising and lowering operators for site j respectively and j = g... e j... g 7 denote one excitation in the site j. That mean s j are basis of single excitation space. Finally, the total transfer of excitation is measured by the population in the sink. 3 Universal simulation of Markovian open quantum systems Assume that we have quantum system with Hilbert space H S = C d and a state of this system can be describe by density matrix ρ M d (C). Density matrix evolves according to a quantum Markovian master equation d ρ(t) = Lρ(t), (3) dt 4

where L is the generator of one parameter semigroup of quantum channels {T (t)} [9]. At time t > t 0 state of quantum system obtained from ρ(t) = T (t t 0 )ρ(t 0 ). In this case we almost can write Lρ(t) follow as L(ρ) = i[ρ, H] + d l,k A l,k (F l ρf k {F k F l, ρ}), (4) where H is Hermitian operator(h M d (C)), A l,k s are element s matrix A M d (C) and {F i } is basis for the space of traceless matrix s in M d (C). Eq.(4) is is known as the Gorini, Kossakowski, Sudarshan and Lindblad (GKSL) form of the quantum Markov master equation and A is the GKS matrix in this trem. By diagonalisation of the GKS matrix A we obtain Lindblad master equation n L(ρ) = i[ρ, H] + γ k (L k ρl k k= {L k L k, ρ}), (5) where n is the number of non-zero eigenvalues of A. Theorem For simulations of Markovian semigroups, using linear combination and conjugation by unitaries, an arbitrary Markovian semigroup generated by L with H S = C d, it is necessary and sufficient to be able to simulate all Markovian semigroups whose generator is specified by a GKS matrix from the d 3 parameter family [6]. where A(θ, α R, α I ) = a(θ, α R, α I )a(θ, α R, α I ), a(θ, α R, α I ) = cos(θ)ã R ( α R ) + i sin(θ)ã I ( α I ), for θ [0, π/4] and ã R ( α R ) ã I ( α I ) R d. and a R = cos(α R ) (6) a R = sin(α R ) cos(α R ) (7)... a R d = sin(α R )... sin(αd 3) R cos(αd ) R (8) a R d = sin(α R )... sin(αd 3) R sin(αd ) R (9) a I = cos(α) I (0) 5

a R = sin(α I ) cos(α I ) ()... a I d d = sin(α I )... sin(α I d d ) () a I d d = sin(α I )... sin(α I d d ) sin(α I d d ). (3) Linear combination mean s if L a and L b be be the generators of Markovian semigroups {T (a) } and {T (b) }, we can write L a+b = L a + L b is generator of a Markovian semigroup {T (a+b) (t)}[]. By using Lie-Trotter theorem []we can write {T (a+b) (t)} = lim [T (a) (t/n)t (b) (t/n)] n, (4) n inf note that we can generalize linear combination to any number of generators. Unitary conjugation, mean s via unitary operator U SU(d) we can implement new Markovian semigroup {T U (t)} {U T (t)u}, (5) where U(ρ)=UρU. See reference [] [6] for details and proof of Theorem. 4 Simulation of non-unitary processes FMO complex We begin by transforming Lindblad master equation into the GKS form. After obtaining the GKS matrix A, we decompose A into the linear combination of rank generators through the spectral decomposition. Then each constituent generator a k a k decomposed into the unitary conjugation of a semigroup from the universal set. 4. Dissipative processe For dissipative processe we have 7 L diss (ρ) = Γ j ( σ j + σj ρ ρσ j + σj + σj ρσ + j), (6) j= 6

for obtain GKS form Eq.(6) we use the fact of {F k } is basis for the space of traceless matrix s in M 8 (C). Other hand {if k } is a basis for su(8). Basis of su(8) is {F i } 7 i= d l, d l = l l(l + ) [ j j l l + l + ], (7) j= {F i } 35 i=8 {σ j,k x } 7 j= j<k 8, σ j,k x = ( j k + k j ), (8) {F i } 63 i=36 {σ j,k y } 7 j= j<k 8, σ j,k y = ( i j k + i k j ). (9) Now we can write σ + j and σ j as below σ + j = (F j+7 if j+35 ), σ j = (F j+7 + if j+35 ). (0) So by putting up in Eq.(6) GKS matrix can be obtained. element of GKS matrix A are Nonvanishing Now we decompose A a 8,8 = Γ a 36,36 = Γ a 8,36 = iγ a 36,8 = iγ a 9,9 = Γ a 37,37 = Γ a 9,37 = iγ a 37,9 = iγ a 0,0 = Γ 3 a 38,38 = Γ 3 a 0,38 = iγ 3 a 38,0 = iγ 3 a, = Γ 4 a 39,39 = Γ 4 a,39 = iγ 4 a 39, = iγ 4 a, = Γ 5 a 40,40 = Γ 5 a,40 = iγ 5 a 40, = iγ 5 a 3,3 = Γ 6 a 4,4 = Γ 6 a 3,46 = iγ 6 a 46,3 = iγ 5 a 4,4 = Γ 7 a 4,4 = Γ 7 a 4,4 = iγ 7 a 4,4 = iγ 7 7 A = λ k a k a k, () k= by λ k = Γ k and a k have nonvanishing elements a k+7 = i and a k+35 =. Now each generator a k a k of the linear combination should decomposed into the unitary conjugation of a semigroup from the universal set. In general form we can write e iψ k a k = cos(θ k )â R k + i sin(θ k )â I k, () 7

â R k and â I k is real and imaginary part of a k respectively and ψ is phase transformation. If â R k.â I k = 0 and â R k = â I k = phase transformation ψ = 0 and θ k [0, π/4]. In our problem ψ k = 0, θ k = π/4, α k R = 0 and α k I = (a,..., a 35 ) T by a i = π/, i =,,...34 and a 35 = 3π/ for k =,,...7. So we can implement a k a k = G U (k)[a(k) (θ k, α k R, α k)]g I T U, (3) (k) where A (k) (θ k, α k R, α k) I an element of the universal set of semigroup generators and G U (k) = Int(U (k) ) by considering U (k) = U () k = U n U lm that U n is single unitary operation R y ( π) and U lm is two qubit gate. Furthermore if L k is generator of a Markovian semigroup we can simulate any channel T k (t) = exp(tl ak a from the semigroup generated by a k) k a k, T k (t)(ρ) = U (k) (T A (k)(t)[u (k) ρu (k) ])U (k), (4) where T A (k)(t) = exp(tl A k)). ( We drive this equation s for semigroup generated by a a directly. For a we obtain â R = 36, â I = 8. The next step is finding ã R and ã I, for this work we use of map f : su(8) R 63 that define as f(if ) = j. If define ÂR f (â R ), we have by using of matrix U () matrix ÂR comes diagonal. U () =  R = if 36, (5) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, (6) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 For imaginary part à R d, U ()  R U () = if. (7)  I = if 8, (8) 8

and A I U () Â I U () = if 36 ÃI, (9) we need not to finding U () because A I is desired form. So f(ãr d,) =, f(ãi ) = 36. (30) If we define ã R f(ãr d,) and ã I f(ãi ) we haven t second unitary transformation because ã R and ã I have desired form. So by defining U () = U () and G U () = Int(U () ), we can implement a a = G U ()[A () (θ, α R, α I )]G T U (), (3) where A () (θ, α R, α I ) an element of the universal set of semigroup generators, by θ = π/4, α R = 0 and α R = 0 and α I = (a,..., a 35 ) T by a i = π/, i =,,...34 and a 35 = 3π/. Furthermore if L is generator of a Markovian semigroup we can simulate any channel T (t) = exp(tl a a ) from the semigroup generated by a a, T (t)(ρ) = U () (T A ()(t)[u () ρu () ])U (), (3) where T A ()(t) = exp(tl A )). ( Now we design quantum circuit for implement of U (i). At first we obtain quantum circuit for implement U () and for other U (i) similarly circuit can be designed. By finding action of unitary operation U () on the q, q, q 3 where three qubits space bases we obtain below conditions.. If second qubit was in state state of qubit s first and third gets be NOT. q q q 3 R y ( π ) X Figure : Quantum circuit for implement of U ()dissipative 9

. If first and second qubit was in state 0, state of third qubit will be rotation. 3. If first qubit were in state and second qubit in state 0, state of third qubit will be flip. Given the above conditions we need two CNOT gate, a single qubit gate(r y ( π/)) and a X gate. quantum circuit for implement of U () shown in Fig. 4. Dephasing processe Similary in previous section we obtain GKS matrix for this process and decompose it. So we have 7 A = λ k a k a k, (33) k= with λ k = 4γ k for k =, 3,.., 7 and λ = 4 γ. a have nonvanishing elements a 0 = i(+ ) 4+, a 38 = 4+. Nonvanishing element of a are a 6 = 5 ( i), a 47 = 5 and for a 3 : a = 5 ( i), a 40 = 5. As the same way nonvanishing elements of a 4, a 5, a 6, a 7 are (a 0 = i, a 47 = ),(a 9 = i, a 46 = ),(a 6 = i, a 45 = ),(a = i, a 39 = )respectively. ψ k = π/ for k =,, 3 and equal with zero for k = 4,..., 7. θ = arccos( + 4+ ), θ = θ 3 = arccos( 5 ) and θ k = π/4 for k = 4,..., 7. Furthermore we can obtain α R,7 = 0 and α I,7 = (a,..., a 35 ) T by a i = π/, i =,,...34 and a 35 = 3π/ for a,3 can be written α R,3 = 0, α I,3 = π, and for a 4,5,6 we obtain that α R 4,5,6 = α I 4,5,6 = π. Now we consider semigroup generated by a a and decompose it into the unitary conjugation of a semigroup from the universal set. we begin by â R = 0, â I = 38. So f (â R ) = f ( 0 ) = if 0 = ÂR, 0

now by using of U () we can diagnose the matrix ÂR. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 U () 0 0 0 0 0 0 0 =, (34) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Then we ll have for an imaginary part and U () Â R U () = if = ÃR d,, f (â I ) = f ( 38 ) = if 38 = ÂR, U () Â I U () = if 36 = ÃI, because ÂI is desired form, no need to find a matrix U (). If we define ã R f(ãr d,) = and ã I f(ã I ) = 36. We need not to second unitary transformation because ã R and ã I are have the desired form. By consider U () = U () and G U () = Int(U () ) similar to the previous one a a = G U ()[A () (θ, α R, α I )]G T U (), (35) where A () (θ, α R, α I ) an element of the universal set of semigroup generators, by θ = arccos( + 4+ ), αr = 0 and α I = (a,..., a 35 ) T by a i = π/, i =,,...34 and a 35 = 3π/. Furthermore if L is generator of a Markovian semigroup we can simulate any channel T (t) = exp(tl a a ) from the semigroup generated by a a, T (t)(ρ) = U () (T A ()(t)[u () ρu () ])U (), (36) where T A ()(t) = exp(tl A ()). Note in any case must be calculated U () i and U () i for i =,..., 6 except in case a,7 that their calculation is straightforward. Now we design quantum circuit to implement of U (). We drive below conditions by finding action of U () on three qubits space bases.

. If first and second qubit was in state 0 and state of qubit third will be rotation. If first qubit was in state and second qubit in state 0, state of qubit third will be flip. 3. If second qubit was in state state of qubit s first and third gets be NOT. 4. For state 00 does not exist output. Furthermore via two CNOT gate and single qubit gate R y ( π ) and a one gate X, unitary operation U () can be implement. Quantum circuit of U () shown in Fig.. For other U () i quantum circuit can be design analogous U (). q q q 3 R y ( π ) X Figure : Quantum circuit designed by using of single qubit gate s and CNOT gate for implement of U ()dephasing. 5 Conclusion In this paper we have investigated universal simulation of Markovian dynamic s of FMO complex. At first we have transformed Lindblad master equation into the GKS form for non-unitarian processes in FMO complex. Next decomposed GKS matrix into the linear combination of rank generators through the spectral decomposition. Then each constituent generator a k a k decomposed into the unitary conjugation of a semigroup from the universal set. Finally, quantum circuit had designed for implement unitary matrix s that applied for simulation of the structure generator s.

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