Evolutionary Design in Biological Quantum Computing
11 Evolutionary Design in Biological Quantum Computing
Gabor Vattay , ∗ , Stuart Kauffman ∗ E-mail: [email protected]
Abstract
The unique capability of quantum mechanics to evolve alternative possibilities in parallel is appealingand over the years a number of quantum algorithms have been developed offering great computationalbenefits. Systems coupled to the environment lose quantum coherence quickly and realization of schemesbased on unitarity might be impossible. Recent discovery of room temperature quantum coherence in lightharvesting complexes [1–4] opens up new possibilities to borrow concepts from biology to use quantumeffects for computational purposes. While it has been conjectured that light harvesting complexes suchas the Fenna-Matthews-Olson (FMO) complex in the green sulfur bacteria performs an efficient quantumsearch similar to the quantum Grover’s algorithm [1, 6, 7] the analogy has yet to be established.In this work we show that quantum dissipation plays an essential role in the quantum search performedin the FMO complex and it is fundamentally different from known algorithms. In the FMO complexnot just the optimal level of phase breaking is present to avoid both quantum localization and Zenotrapping [5, 8] but it can harness quantum dissipation as well to speed the process even further up. Withdetailed quantum calculations taking into account both phase breaking and quantum dissipation weshow that the design of the FMO complex has been evolutionarily optimized and works faster than purequantum or classical-stochastic algorithms. Inspired by the findings we introduce a new computationalconcept based on decoherent quantum evolution. While it is inspired by light harvesting systems, thenew computational devices can also be realized on different material basis opening new magnitude scalesfor miniaturization and speed.
Introduction
In the last five years it became apparent that some biological systems can benefit from quantum effectseven at room temperature. It has been shown experimentally that quantum coherence can stay alivefor an anomalously long time in light harvesting complexes [1–4]. In these systems excitons initiated bythe incoming photons should travel really fast throughout a chain of chromophores in order to reach thereaction center where they are converted to chemical energy. Excitons decay within 1 nanosecond anddissipate energy back to the environment if they cannot find the photosynthetic reaction center via randomhopping within that characteristic time. With classical diffusion via thermal hopping that time is easilyconsumed, thus evolution should have found more optimal ways to reach that goal. Quantum mechanicsis very helpful in this respect as it allows the system to explore many alternative paths in parallel and candiscover the optimal one faster than a classical random search would do. However, quantum mechanicshas adverse effects too. Anderson localization can prevent excitons to travel large distances. Couplingthe system to the environment breaks phase coherence and can destroy this negative effect of quantumlocalization. Too much phase breaking however slows down the propagation again via the Zeno effect.At the right amount of phase breaking environmental decoherence and quantum evolution collaborate toachieve optimal performance and efficiency. The Environment Assisted Quantum Transport (ENAQT)theory [5, 6] accounts for the interplay of these two effects and can explain the existence of a transportefficiency optimum at room temperature relative to both pure quantum or pure classical transport. a r X i v : . [ c ond - m a t . d i s - nn ] N ov ENAQT explains the quick quantum exploration of the search space at optimal phase breaking. Oncethe exciton can reach nearly ergodically the chromophore sites random trapping delivers of the excitonto the reaction center.
Results and Discussion
While ENAQT assures the fast spreading of probability over the light harvesting complex, it does notguide the exciton to the reaction center. The reason for this is that quantum mechanics and phasebreaking leads to a uniform probability distribution over the state space. The reduced density matrix ofa system with Hamiltonian H is described by the Lindbad equation [11] ∂ t (cid:37) + i (cid:126) [ H, (cid:37) ] = 12 (cid:88) j (cid:2) V j (cid:37), V + j (cid:3) + (cid:2) V j , (cid:37)V + j (cid:3) , (1)where the operators V j describe the coupling of the system and the environment. In light harvestingsystems the Hamiltonian H nm is a discrete, where the chromophore sites are indexed by n = 1 , ..., N . Incase the chromophores are coupled to the environment independently the generators are simply diagonal V j = √ γ φ · | j (cid:105)(cid:104) j | , where γ φ is the rate of phase breaking. The Lindblad equation keeps the densitymatrix normalized during the evolution Tr { (cid:37) } = 1 and its diagonal elements (cid:37) nn stay positive and givethe probability of finding the exciton on site n . At the optimal level of phase breaking the system relaxesquickly to the uniform probability distribution (cid:37) nn = 1 /N . Trapping to the reaction center is describedby the imaginary Hamiltonian − i (cid:126) κ | r (cid:105)(cid:104) r | , where r is the site of the reaction center and κ is the trappingrate. Assuming rapid relaxation to the uniform distribution the bulk of the time an exciton needs to gettrapped by the reaction center is determined by the fraction of time it spends on the chromophore ofthe reaction center. The reaction center is able to catch an exciton siting on it in average time 1 /κ andthe exciton spends (cid:37) rr fraction of its time on the chromophore. The average transport time is then theproduct (cid:104) τ (cid:105) ≈ / ( (cid:37) rr κ ) = N/κ . One of the best studied light harvesting systems is the FMO complex [10]which consists of N = 7 chromophores. We use this example thrughout this paper. It has been shown [5]that ENAQT is optimal in this system at phase breaking rates of γ φ = 300cm − corresponding to roomtemperature. At trapping rate 1ps − the exciton needs about 7 ps to reach the reaction center, which isconsistent with this estimate.Since at optimal phase breaking the transport time depends only on the number of sites and on thetrapping rate the concrete form of the FMO Hamiltonian plays no role as long as the relaxation to theuniform distribution is sufficiently fast. Accordingly, Hamiltonians with extended wave functions shouldbe slightly more efficient than localized systems since the exciton is not trapped and the relaxation tothe uniform distribution is somewhat faster. We demonstrate this in case of the FMO complex wherethe Hamiltonian H nm has been obtained from spectroscopy [12]. The diagonal part of the Hamiltonianconsists of the site energies of the chromophores. The off diagonal hopping terms describe the transitionbetween sites. We can modify the localization properties of this Hamiltonian by rescaling the diagonalelements relative to the off diagonal elements H (cid:48) nm = H nm + ( λ − δ nm H nn , where λ is the tuningparameter. For λ = 1 we recover the original Hamiltonian. For λ > λ → ∞ , while for 0 ≤ λ < λ . In Fig. 2 we show the transport efficiency and transport time as a function of λ atoptimal phase breaking calculated with the parameters of Ref. 5. The details are in the supplementarymaterial. Both of them change monotonically with λ and the transport is slightly more efficient andfaster for the delocalized case as we expected. The real FMO complex at λ = 1 is not optimal in anysense. As we show next, evolution optimized the transport process further by guiding the excitons to thereaction center which sits at the lowest energy site and the design of the FMO complex is in fact highlyoptimal. To show this we have to go beyond the Lindblad equation in order to account for the relaxationto thermal equilibrium.One way to study the relaxation to the correct thermal equilibrium is to use the Redfield equationsdescribing the interaction of the system and the environmental bath. The Redfield equation can be castin a form similar to the Lindblad equation [13, 14] ∂ t (cid:37) + i (cid:126) [ H, (cid:37) ] = (cid:88) j (cid:2) V + j (cid:37), V j (cid:3) + (cid:2) V j , (cid:37)V − j (cid:3) , (2)where the operators can be written in energy representation as (cid:2) V + j (cid:3) ab = [ V j ] ab / (1 + e β ( E a − E b ) ) and (cid:2) V − j (cid:3) ab = [ V j ] ab / (1 + e − β ( E a − E b ) ). The operators coupling the bath and the environment are physicalobservables hence self-adjoint V j = V + j . The equilibrium solution of this equation is the Boltzmann distri-bution (cid:37) = exp( − βH ) /Z , where Z = Tr { exp( − βH ) } is the partition function. The uniform distributionis recovered for infinite temperature β = 0.For high temperatures we can expand this equation for small β . The first two terms in the expansionare basis independent ∂ t (cid:37) + i (cid:126) [ H, (cid:37) ] = 12 (cid:88) j [ V j , [ V j , (cid:37) ]] + β V j , { [ H, V j ] , (cid:37) } ] , (3)while the third term in the expansion is zero in general as we show in the supplementary material. Thefirst term is the Lindblad equation for self-adjoint operators V j . The second term describes quantumdissipation, which is missing from the Lindblad equation. Caldeira and Leggett (CL) showed that thereduced density matrix of open quantum systems coupled to a high temperature bath experience bothphase breaking and quantum dissipation and satisfy the equation ∂ t (cid:37) + i (cid:126) [ H, (cid:37) ] = 12 γ φ [ x, [ x, (cid:37) ]] − i (cid:126) β m γ φ [ x, { p, (cid:37) } ] . (4)Our new equation (3) gives back the CL equation as a special case for the Hamiltonian H ( x, p ) = m p + U ( x ) with coupling V = √ γ φ x and it is valid for a much larger class of Hamiltonians andoperators V . In particular for discrete Hamiltonians H nm describing the exciton dynamics in lightharvesting complexes and for environmental couplings V j = √ γ φ · | j (cid:105)(cid:104) j | it takes the form ∂ t (cid:37) nm + i [ H, (cid:37) ] nm = − γ φ (1 − δ nm ) (cid:37) nm − (1 − δ nm ) γ φ β { H, (cid:37) } nm (5) − γ φ β H nm (cid:37) mm + (cid:37) nn H nm − H nn (cid:37) nm − (cid:37) nm H mm ) . The most important feature of this equation is that the quantum dissipative term cannot be chosenarbitrarily in models of exciton dynamics. The Hamiltonian and the generators V j determine both phasebreaking and dissipation uniquely. Also the order of magnitude the dissipative term relative to thephase breaking term is determined by the ratio of the size of the typical Hamilton matrix element andthe temperature. In light harvesting systems these are comparable and quantum dissipation cannot beneglected.Quantum dissipation speeds up the transport process in light harvesting complexes. If the site energiesat the reaction center are lower than in the other parts of the complex the equilibrium density is higherand the exciton spends longer time on the chromophore related to the reaction center and is trappedwith higher probability. The average time is again (cid:104) τ (cid:105) = 1 / ( κ(cid:37) rr ) but now the probability is given bythe Boltzmann factor (cid:37) rr = (cid:104) r | e − βH | r (cid:105) /Z . In case of the FMO complex this probability is about 40%and the transport time would drop to a mere 2 . . (cid:37) rr ≈ ψ ( k ) n as (cid:37) nn = (cid:88) k | ψ ( k ) n | e − βE k Z . (6)If the system is completely delocalized the wave functions are extended | ψ ( k ) n | ≈ /N and the diagonalelements of the density matrix become uniform (cid:37) nn ≈ /N independently of the energy levels E k ofthe system. In this case the relaxation to the equilibrium is fast since the extended wave functionsoverlap strongly with the exciton starting on one of the chromophores, but the exciton spends time oneach chromophore nearly equally. If the system is strongly localized the wave functions are concentratedon single sites | ψ ( k ) n | ≈ δ nk and (cid:37) nn ≈ e − βE n /Z , where the energy levels are close to the site energies E n ≈ H nn . To have localization the site energies should be much larger than the hopping terms in theHamiltonian. In equilibrium the exciton would spend long time in the neighborhood of the chromophorewith the lowest site energy, but the relaxation time to this equilibrium is very large. The overlap of thewave function localized on the lowest energy site with the initial site of the exciton is very small and theexciton stays localized near to its entry site for a very long time. In Fig. 2 we show both the transportefficiency and transport time for the FMO complex at the optimal phase breaking for different λ -s tuningthe localization length of the system. For λ > λ < λ ≈
1, where the states are neither toolocalized nor too much extended and realize the tread-off. Note, that the Hamiltonian is reconstructedfrom experiments and it carries some level of error. In Fig. 1 we can see that the localization length ofthe real FMO is just half way between the fully localized case, where the wave functions are concentratedon a single site and the maximally delocalized case, where the states are spread the most.We think that this picture is quite general. If we consider larger transport systems the optimumwould again lie somewhere midway between the extended and localized cases. Since the localization-delocalization transition is getting sharper with increasing system size these systems can only be foundat parameters near the metal-insulator threshold. To demonstrate this in Fig. 3 we show the the transportefficiency for the golden mean Harper model which is one of the simplest models on which the metal-insulator transition can be studied [15]. In this model we can see qualitatively the same behavior and anoptimal transport near the localization delocalization (or metal-insulator) transition. It is important tonote that even in this large system the transport time at the optimal phase breaking is still determinedby the shape of the equilibrium distribution and the relaxation time is negligible. It seems advantageousfor biologically relevant quantum transport to tune the system into the critical point of the localization-delocalization transition.How could we use this mechanism to build new types of computers? In the light harvesting casethe task of the system is to transport the exciton the fastest possible way to the reaction center whoseposition is known. In a computational task we usually would like to find the minimum of some complexfunction f n . For the simplicity let this function have only discrete values from 0 to K . If we are able tomap the values of this function to the electrostatic site energies of the chromophores H nn = (cid:15) f n and wedeploy reaction centers near to them trapping the excitons with some rate κ and can access the current ateach reaction center it will be proportional with the probability to find the exciton on the chromophore j n ∼ κ(cid:37) nn . Since the excitons will explore the Boltzmann distribution the currents will reflect that j n = κ (cid:104) n | e − βH | n (cid:105) /Z . There are three conditions which should be valid simultaneously: 1, The systemshould operate at the optimal phase breaking which then should be in the order of magnitude of the energysteps γ φ ∼ O ( (cid:15) ). 2, In the worst case scenario the minimum current is elevated with a factor e β(cid:15) relativeto the second smallest minimum. To be able to detect this the energies should be of the order of thethermal energy (cid:15) ∼ O ( k B T ). 3, The hopping terms H nm between the chromophores should be optimalto keep the system at the border of the localization-delocalization transition. The first two conditionscan be easily met since the phase breaking is usually of the same order as the thermal energy γ φ ∼ k B T .The third condition can be realized by placing the chromophores interacting via the dipole interaction toan optimal distance from each other randomly so that the quasy random H nm matrix elements keep thesystem at the localization-delocalization threshold. Conversely, given a random arrangement of H nm -sthe parameter (cid:15) can be tuned so that the system gets to the localization-delocalization threshold.This quantum-classical optimization method discovered by evolution seems to be superior to theoptimization methods developed so far. Classical stochastic optimization techniques can be trapped inlocal minima for long times and careful annealing techniques are required to reach the correct minimum.Even then sites are discovered in a classical sequential manner and it takes the process long times to findthe minimum. Quantum mechanics is more advantageous as it is able to explore the sites in parallel, butthe discovery process is hampered by Anderson localization especially near local minima. An optimalamount of phase breaking can destroy the interferences causing this and can ensure the ergodic explorationof the states while quantum dissipation takes all the advantages of the classical stochastic optimizationand establishes the Boltzmann distribution which elevates the proper minimum. The physical speed ofthe process is determined by the inverse trapping rate 1 /κ which is in the order of picoseconds.Current computers operate with about 4 GHz processors, where the cycle time of logical operations is250 picoseconds. Computers based on artificial light harvesting complexes could have units with 100-1000times larger efficiency at room temperature. But, it is also possible to realize such systems on excitons oforganic molecules or on Hamiltonians arising in nuclear matter, which would provide a virtually endlesssource of improvement both in time and miniaturization below the atomic scale. Since the realizationof this mechanism seems now relatively easy, it is an important question if it has been realized in lightharvesting systems or is also present in other biological transport or optimization processes. Especiallyin the human brain [16]. Materials and Methods
The Redfield equation can be cast into a form similar to the Lindblad equation (see W. T. Pollard andR. A. Friesner, J. Chem. Phys. 100, 5054 (1997)). In energy representation: ∂ t (cid:37) ab + i (cid:126) [ H, (cid:37) ] ab = (cid:88) j (cid:2) V + j (cid:37), V j (cid:3) ab + (cid:2) V j , (cid:37)V − j (cid:3) ab , (7)where the operators can be written in energy representation as (cid:2) V + j (cid:3) ab = [ V j ] ab / (1 + e β ( E a − E b ) ) and (cid:2) V − j (cid:3) ab = [ V j ] ab / (1 + e − β ( E a − E b ) ). The operators coupling the bath and the environment are physicalobservables hence self-adjoint V j = V + j . The equations can be written also in the form of ∂ t (cid:37) ab + i (cid:126) [ H, (cid:37) ] ab = (cid:88) cd R abcd (cid:37) cd , (8)where R abcd = V ac V db e β ( E a − E c ) − (cid:88) i V ai V ic δ db e β ( E i − E c ) + V ac V db e − β ( E d − E b ) − (cid:88) i V ai V ic δ bd e − β ( E a − E i ) . (9)The energy representation of the coupling operator is V jab = √ γ φ ψ aj ∗ ψ bj , where ψ aj is the energy eigenstatecorresponding to E a and site index j . We can then transform back the equations into site representationand can carry out the summations for j yielding ∂ t (cid:37) nm + i (cid:126) [ H, (cid:37) ] nm = γ φ (cid:88) kl K nmkl (cid:37) kl , (10)where K nmkl = A + ( n, k, m ) δ ml + δ nk A − ( m, l, n ) − A + ( n, k, n ) δ ml − δ nk A − ( m, l, m ) , (11)and A ± ( n, m, j ) = (cid:88) ab ψ ∗ an ψ aj ψ ∗ bj ψ bm e ± β ( E a − E b ) . (12) We can expand the operators in the Redfield equations in energy representation up to the second powerof β as [ V j ] ab e β ( E a − Eb ) = 12 [ V j ] ab − β E a [ V j ] ab − [ V j ] ab E b ) . (13)Note that the β term is identiaclally zero. The second term is independent of the representation andcan be written as V ± j = 12 V j − ± β H, V j ] , (14)where we use the commutator [ H, V j ] , = HV j − V j H . Substituting this into the Redfield equation (7)yields ∂ t (cid:37) + i (cid:126) [ H, (cid:37) ] = 12 (cid:88) j [ V j , [ V j , (cid:37) ]] + β V j , { [ H, V j ] , (cid:37) } ] . (15) The Harper model is defined by the one dimensional chain with site energies H nn = 2 λJ cos(2 πGn ) andhopping terms H n,n +1 = J , where G = ( √ − / λ is the tuning parameter.If λ = 1 the system is at the critical point of the localization-delocalization transition. For λ > λ < K and the phase breaking at 300 cm − (in spectroscopic wavenumber units) similar tothe FMO complex the parameter J defines the energy scale of the model. For values J ≈ − cm − the phase breaking seems to be optimal in a chain of length N = 30 and the best efficiency and thesmallest transport time is attained at the critical point of the metal-insulator transition at λ = 1. Transport time and efficiency calculations coincide with those presented in Patrick Rebentrost, MasoudMohseni, Ivan Kassal, Seth Lloyd and Aln Aspuru-Guzik,
Environment-assisted quantum transport , NewJournal of Physics 11 (2009) 033003. The same trapping rate κ = 1 ps − and exciton decay rate Γ = 1 ns − is used throughout this paper. The transport efficiency and transport time is calculated with the numericalinversion of the superoperator discussed in M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru-Guzik, Environment-assisted quantum walks in photosynthetic energy transfer , J. Chem. Phys. 129 174106(2008).
AcknowledgmentsReferences
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J. Chem. Phys. 129 174106 (2008)7. Grover L.KA fast quantum mechanical algorithm for database search,Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 2128. S. LloydQuantum coherence in biological systemsJournal of Physics: Conference Series 302, 012037 (2011)9. A. O. Caldeira and A. J. LeggettPath integral approach to quantum Brownian motionPhysica A , 587-616 (1983)10. Olson J.M.The FMO proteinPhotosynthesis Research 80: 181187, 2004.11. Lindblad G.On the generators of quantum dynamical semigroupsCommunications in Mathematical Physics (1976), 119–130 (1976)12. Minhaeng Cho, Harsha M. Vaswani, Tobias Brixner, Jens Stenger, and Graham R. FlemingExciton Analysis in 2D Electronic SpectroscopyPhys. Chem. B, 2005, 109 (21), pp 10542-10556 (2005)13. W. Thomas Pollard and Richard A. FriesnerSolution of the Redfield equation for the dissipative quantum dynamics of multilevel systemsJ. Chem. Phys. 100, 5054 (1994)14. D. Kohen, C. C. Marston, and D. J. TannorPhase space approach to theories of quantum dissipationJ. Chem. Phys. 107, 5236 (1997)15. Evangelou, S. N. and Pichard, J.-L.Critical Quantum Chaos and the One-Dimensional Harper ModelPhys. Rev. Lett. , 8, 1643–1646 (2000)16. Kauffman S.A.Answering Descartes: Beyond TuringIn S. Barry Cooper and Andrew Hodges (editors) ”The Once and Future Turing: Computing theWorld”, Cambridge University Press (2012)reprinted in Proceedings of the Eleventh European Conference on the Synthesis and Simulation ofLiving Systems Edited by Tom Lenaerts, Mario Giacobini, Hugues Bersini, Paul Bourgine, MarcoDorigo and Ren Doursat, MIT Press (2011) \http://mitpress.mit.edu/books/chapters/0262297140chap4.pdf Figure Legends L o ca li za ti on l e ng t h ( nu m b e r o f s it e s ) Figure 1.
Average localization length of the FMO complex as a function of the tuning parameter λ .The value λ = 1 corresponds to the real FMO complex. For λ > <
1) the diagonal elements of theHamiltonian matrix are magnified (shrinked) causing more (less) localization. The localization length isthe reciprocal of the inverse participation ratio ξ = 1 / IPR calculated as an average for all the N = 7eigenfunctions ( k ) and sites n of the FMO complex IPR = ( (cid:80) Nn,k =1 | ψ ( k ) n | ) /N . The localization lengthshows how many sites are involved in a given energy eigenfunction in average. Conversely it also showshow many energy eigenstates overlap in a given site. The value ξ = 1 means that the energyeigenfunctions are localized on a single state while ξ = 4 seems to be the largest level of attainabledelocalization. The FMO complex is half way between the fully localized and delocalized cases.0 (cid:104) E ff i c i e n c y Efficiency T=277 KEfficiency T= (cid:39) T r a n s po r t ti m e [ p i c o s ec ond ] Transport time T=277 KTransport time T= (cid:39)
Figure 2.
Transport efficiency and transport time in the FMO complex as a function of the tuningparameter λ at optimal phase breaking γ φ = 300 cm − . Solid curves show transport efficiency forambient temperatures T = 277 (black) and for the case when quantum dissipation is not present T = ∞ (red). The presence of quantum dissipation increases the transport efficiency for all parameters λ .Without quantum dissipation the delocalized systems λ < λ > .
8% in the optimal point near λ ≈ λ = 1.) Dashed lines show the transport time. Thereis a dramatic speedup of transport due to quantum dissipation. The transport time drops from about 7picoseconds to 3.5 picoseconds for the FMO complex. The transport time without quantum dissipation(blue) changes monotonically with the localization and fastest for the most delocalized case. Withquantum dissipation (green) the transport time is about minimal for the real FMO complex which is inbetween the localized and delocalized cases.1 (cid:104) E ff i c i e n c y J=2000J=1000J=5000 0.5 1 1.5 202004006008001000 T r a n s po r t ti m e [ p i c o s ec ond ] Figure 3.
Transport efficiency and transport time for the golden mean Harper model as a function ofthe tuning parameter. The Harper model is defined by the one dimensional chain with site energies H nn = 2 λJ cos(2 πGn ) and hopping terms H n,n +1 = J , where G = ( √ − / λ is the tuning parameter. If λ = 1 the system is at the critical point of the localization-delocalizationtransition [15]. For λ > λ < K and the phase breaking at 300 cm − (inspectroscopic wavenumber units) similar to the FMO complex the parameter J defines the energy scaleof the model. For values J ≈ − cm − the phase breaking seems to be optimal in a chain oflength N = 30 and the best efficiency and the smallest transport time is attained at the critical point ofthe metal-insulator transition at λ = 1. Solid lines show transport efficiency for J = 500 , λ = 1. The transport time of 20 picoseconds is in accordance with the trapping rate1 /κ = 1 ps and the Boltzmann factor giving probability (cid:37) rr = 1 /