Quantum Algorithm for Molecular Properties and Geometry Optimization

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1 Quantum Algoritm for Molecular Properties an Geometry Optimization Te Harvar community as mae tis article openly available. Please sare ow tis access benefits you. Your story matters. Citation Publise Version Accesse Citable Link Terms of Use Kassal, Ivan, an Alán Aspuru-Guzik. 29. Quantum algoritm for molecular properties an geometry optimization. Journal of Cemical Pysics 131(22): oi:1.163/ August 17, 218 7:2:21 PM EDT ttp://nrs.arvar.eu/urn-3:hul.instrepos: Tis article was ownloae from Harvar University's DASH repository, an is mae available uner te terms an conitions applicable to Open Access Policy Articles, as set fort at ttp://nrs.arvar.eu/urn-3:hul.instrepos:as.current.terms-ofuse#oap (Article begins on next page)

2 Quantum Algoritm for Molecular Properties an Geometry Optimization arxiv: v1 [quant-p] 13 Aug 29 Ivan Kassal an Alán Aspuru-Guzik Department of Cemistry an Cemical Biology, Harvar University, Cambrige, MA 2138 (Date: August 13, 29) It is known tat quantum computers, if available, woul allow an exponential ecrease in te computational cost of quantum simulations. We exten tis result to sow tat te computation of molecular properties (energy erivatives) coul also be spe up using quantum computers. We provie a quantum algoritm for te numerical evaluation of molecular properties, wose time cost is a constant multiple of te time neee to compute te molecular energy, regarless of te size of te system. Molecular properties compute wit te propose approac coul also be use for te optimization of molecular geometries or oter properties. For tat purpose, we iscuss te benets of quantum tecniques for ewton's meto an Houseoler metos. Finally, global minima for te propose optimizations can be foun using te quantum basin opper algoritm, wic oers an aitional quaratic reuction in cost over classical multi-start tecniques. Applying ab initio metos of quantum cemistry to particular problems often requires computing erivatives of te molecular energy. For instance, obtaining a molecule's electric properties relies on te ability to compute erivatives wit respect to external electromagnetic els. Likewise, computing te graient of te molecular energy wit respect to te nuclear coorinates is te most commonly use meto for te proper caracterization of potential energy surfaces an for optimizing te geometry of all but te smallest molecules. Te computation of tese kins of erivatives, known as molecular properties, is nowaays a routine matter wen it comes to low-orer erivatives or small systems (or bot). Tis is largely ue to avances in analytical graient tecniques, wic allow for explicit property evaluation witout resorting to numerical ierentiation [1, 2, 3, 4, 5, 6, 7]. everteless, te computation of iger-orer erivatives is often proibitively expensive, even toug suc erivatives are often neee. For example, tir- an fourt-orer anarmonic constants are sometimes require to accurately compute a vibrational absorption spectrum [3] or eciently etermine te location of transition states on complex potential energy surfaces [6]. Oter properties of interest, suc as yperpolarizabilities, Raman intensities, or vibrational circular icroism, are also cubic or quartic erivatives. In tis report, we sow tat quantum computers, once available, will be able to bypass some of te ig cost of computing tese properties. In particular, we sow tat any molecular property can be evaluate on a quantum computer using resources tat, up to a small constant, are equal to tose require to compute te molecular energy once. We ave previously caracterize te avantage of quantum computers at bot computing molecular energies [8, 9] an simulating cemical reaction ynamics [1], an te present work extens our program to molecular properties. Tis paper begins wit a brief overview of classical Electronic aress: kassal@fas.arvar.eu Electronic aress: aspuru@cemistry.arvar.eu tecniques for te evaluation of molecular properties, bot numerical an analytical. We ten introuce te quantum algoritm for molecular properties, an iscuss its avantages an isavantages wit respect to classical tecniques. We conclue wit geometry optimization as a particular example, an we sow tat it can benet from an aitional quaratic spee-up troug Grover's searc [11]. I. THE CLASSICAL METHODS Given an external perturbation µ, te total molecular electronic energy can be expane in a Taylor series E(µ) = E () + µ E (1) µ E (2) µ +... (1) were te coecients E (n) are calle te molecular properties an escribe te response of te system to te applie perturbation [7]. We consier time-inepenent properties, wic can be obtaine by ierentiating te energy at µ =, E (n) = n E µ n. (2) Many examples of useful erivatives can be given. For instance, te erivatives wit respect to te electric el F are te permanent electric ipole, te static polarizability, an te static yperpolarizabilities of various orers: 2 E F =, 3 E F 2 = α, F 3 = β,... (3) were te subscript enotes ierentiation at F =. Te erivatives wit respect to nuclear coorinates R inclue te forces on te nuclei an te force constants, wile mixe erivatives can provie information suc as Raman intensities [2]. On a classical computer, an energy erivative can be evaluate eiter numerically or analytically, an we iscuss eac approac in turn.

3 2 Figure 1: Obtaining a numerical graient of a function e- ne on a -imensional space classically requires sampling te function + 1 times, once at te origin an once at a istance along eac of te axes. Sown above are te sample points for te cases = 1 troug = 3. Te quantum graient algoritm can evaluate many sample points in superposition, proucing te same calculate graient using only one call to te function. umerical erivative tecniques rely on computing te value of te energy at several iscrete points, an ten using tose values to estimate te true erivative. Te simplest tecnique is nite ierence, wic for te rst erivative in one imension is te familiar formula, µ E() E(). (4) In imensions, computing te graient requires at least +1 evaluations of te energy, once at te origin an once at a istance along eac axis (Fig. 1). Similarly, evaluating iger-orer erivatives requires te knowlege of te energy on a particular gri, wit at least n +1 points for te n t erivative. Wile numerical graient tecniques usually require minimal eort to implement, tey are occasionally susceptible to numerical instability, ue to te ill-poseness of numerical ierentiation in general [12]. Tis is particularly problematic wen using nite-precision aritmetic, were various rouning errors can accumulate an be amplie upon ivision by te small number. Te fact tat small errors in te evaluate function can lea to large errors in te erivative aects ab initio electronic structure metos insofar as tey usually involve long calculations wit many potential sources of error, incluing rouning an quarature. By contrast, analytic erivative tecniques are tose tat compute te erivative by irect evaluation of an analytic expression. Tey were introuce in quantum cemistry by Pulay [1], an ave since largely supplante numerical proceures. Tey are numerically more stable an, more importantly, tey are usually faster as well. Analytic graient formulas exist for just about all electronic structure tecniques an for most kins of perturbations. To illustrate te argument an establis te correct scaling, we will escribe te particularly simple case of erivatives of fully variational wavefunctions. We start by writing te molecular energy as a function E(µ, λ(µ)) of te external perturbation µ an te wavefunction parameters λ(µ). Tese parameters, suc as te conguration interaction coecients, completely escribe te electronic wavefunction. Altoug λ(µ) is a function of µ, for simplicity we will write only λ. Te energy is sai to be fully variational wit respect to λ if, for any given µ, λ assumes te value λ suc tat te variational conition ols: E(µ, λ) λ =, (5) were inicates λ = λ. In tat case we can write E(µ) = E (µ, λ ). For fully variational wavefunctions, te graient wit respect to µ is given by (µ) E(µ, λ) = E(µ, λ) µ µ + λ λ µ = E(µ, λ) = µ = λ H µ λ (6) were we ave use te variational conition an te Hellman-Feynman teorem. Since one nee not know te rst-orer wavefunction response λ µ, computing te graient is, to witin a small constant factor [6], as ar as computing te energy. Tat is, once λ is available, calculating te expectation value of te Hamiltonian as approximately te same computational cost as calculating its erivative. However, computing te secon erivative matrix oes require te knowlege of te rst-orer wavefunction response. In fact, as a irect consequence of Wigner's 2n + 1 rule of perturbation teory, one nees to know te rst n wavefunction responses in orer to calculate te (2n + 1) t erivative. Computing te responses often becomes te bottleneck, an it is wat leas to a iger asymptotic cost of iger-orer erivatives. Wile te graient requires about te same resources as te energy, te secon an tir erivatives require resources tat scale as O() times te cost of computing te energy (were is te number of egrees of freeom, i.e., te imension of µ) [6]. Tis scaling comes about because O() time is require to compute te matrix λ µ. Likewise, te scaling of te (2n+1) t erivative is O( n ), because te bottleneck becomes te computation of te n t orer wavefunction response. In oter wors, te computational cost of ning te n t analytical erivative is O( n/2 ), rougly a quaratic spee-up over te O( n ) numerical metos of te same egree. Te fact tat te scaling of erivative tecniques, bot numerical an analytical, epens on as meant tat tese tecniques are often restricte to small systems [29]. Tis is most acutely true of te molecular Hessian, wic is often beyon reac, even toug te graient is routinely accessible. We now sow tat if quantum computers were available, te cost of te iger erivatives woul no longer be proibitive. II. THE QUATUM ALGORITHM Te quantum algoritm for molecular properties is base on Joran's quantum graient estimation algoritm [13]. Joran's meto can numerically compute

4 3 Classical Quantum Derivative umerical Analytical umerical µ + 1 O (1) 1 2 E µ O () 2 3 E O () 4 µ n E n + 1 O n/2 µ n 2 n 1 Table I: Time resources require by various tecniques of computing molecular properties, in terms of te cost of computing te energy. For example, te entry + 1 means tat computing te property requires +1 evaluations of te molecular energy, wile te entries in te Analytical column inicate comparable computational eort. E is te total electronic energy, µ is te external perturbation, an is te imension of µ. All te erivatives are evaluate at µ =. On classical computers, te numerical scalings correspon to te simplest nite-ierence sceme. Analytical tecniques are te ones tat evaluate te erivative irectly (te exponent n/2 comes from Wigner's 2n + 1 rule). On a quantum computer, te quantum graient estimation algoritm is use. It soul be note tat on a quantum computer, te number of evaluations of E neee for any erivative is inepenent of, an tus of te size of te system. te graient of any function F, given a black box (oracle) tat computes te value of F for an arbitrary input. In particular, te algoritm can evaluate te graient using a single query to F, regarless of te number of imensions of te omain of F. By contrast, te simplest classical nite-ierence sceme woul require +1 queries to F (see Fig. 1). Te spee-up is essentially acieve by being able to sample along all te imensions in superposition. We apply Joran's algoritm to te computation of molecular properties by specifying a way to compute te energy on a quantum computer as well as by outlining ow to obtain iger erivatives. In tis section, we escribe te algoritm, its application to quantum cemistry, an nally argue tat a return to numerical tecniques for molecular properties woul be justie if quantum computers became feasible. We assume tat te molecular energy is a smoot, boune function of te perturbation, E : [ 2, 2 ] [E min, E max ], were a small is cosen so tat E varies suciently slowly over te omain. For convenience, we express te perturbations in units suc tat is unitless an suc tat te bouns are te same along all of te axes. We also assume tat we ave a black box for E, wic, given a perturbation µ, outputs te energy E(µ). Te precise nature of te algoritm insie te black box is not important, so long as it can be implemente on a quantum computer. In particular, any classical tecnique of electronic-structure teory can be converte into a quantum algoritm [14]. In Sec. III, we will iscuss possible realizations of te black box, incluing te use of quantum simulation algoritms, wic oer a signicant improvement over classical ones. We begin by coosing te number n of bits of precision tat we esire in te nal graient. Joran's algoritm starts in an equal superposition on registers of n qubits eac (n qubits total) [14]: 1 1 k 1= 1 k = k 1 k = 1 k, (7) were = 2 n, te states k i are integers on n qubits represente in binary, an k is a -imensional vector of all te k 's. We use tis state as an input for te black box for E, wic will, for every integer-vector k in te superposition, appen a pase proportional to te energy E(µ) at perturbation µ = (k /2)/ (were is te vector (,,..., ) an serves to center te sampling region on te origin). To acieve maximum precision wit fewest qubits, one nees an estimate m of te largest magnitue of any of te rst erivatives of E. Ten, te energy evaluate by te black box is scale by a factor 2π/m. Because te black box operates on all te terms in te superposition at once, a single call results in te state 1 1 k [ 2πi exp m E exp k [ 2πi m ( ( E() + k )] (k /2) k )] (k /2) µ k. Te neglect of terms quaratic in an iger is a vali approximation for suciently small (te error cause by iger-orer terms is iscusse in [13] an is only polynomial). Te nal state is separable an equals e iφ() 1 k 1= [ 2πi exp m k 1 1 [ 2πi exp m k k = ] E µ 1 k 1 ] E µ k, (9) wit pase ( Φ() = 2π m E() 2m ) µ. (1) Applying te inverse quantum Fourier transform [14] to eac of te registers results in te graient e iφ() E m µ 1 E m µ = e iφ() m µ. (11) Te scaling factor /m ensures tat m µ is an integervector wit n bits of precision along eac imension. It soul be reiterate tat a single call to E was mae, as oppose to te + 1 tat woul be neee in te case of numerical ierentiation by nite ierence. (8)

5 4 Overall, te graient estimation algoritm prouces te transformation e iφ() m µ. (12) We can compute te Hessian (an iger erivatives) by iterating tis algoritm. If, instea of making a call to E(µ), te algoritm sougt E(µ ν) from te black box, we woul perform, at te cost of tis single aitional subtraction, e iφ(ν) m µ, (13) ν wit global pase ( Φ(ν) = 2π m E(ν) 2m ) µ. (14) ν Because we will be using tis proceure as a subroutine, it is important to remove (or uncompute) te global pase, wic woul oterwise become a relative pase. One aitional evaluation of E (at ν) suces for tis uncomputation. Overall, tis supplies anoter black box, wic, given ν, yiels m µ using only two calls to ν te original black box for E. One can use te graient algoritm wit tis new black box, proucing te state e iφ(2) () m (2) 2 E µ 2, (15) wic is a two-imensional array of 2 quantum registers containing all te elements of te Hessian matrix of E. In aition, m (2) is an estimate for te magnitue of te largest secon erivative, an te pase is ( Φ (2) () = 2π m (2) µ ) 2m 2 E (2) µ 2. (16) Computing iger erivatives woul require aitional factors of two in te number of require black box calls, cause by te nee to uncompute a global pase at eac step (tis problem is a common feature wen it comes to recursing quantum algoritms [15]). Hence, evaluating te n t erivative requires 2 n 1 queries to E, wic, altoug exponential in n, is muc better tan n + 1, wic is te minimum number of function queries require to compute te erivative by classical nite ifference. We stress tat te number of calls to E is inepenent of, an tus of te size of te system, for te erivative of any orer. One coul remark tat te quantum graient algoritm is a numerical approac an tat terefore, just like classical numerical tecniques, it woul be aecte by numerical instability. Tis implies tat te quantum graient algoritm cannot be use iniscriminately for problems tat feature errors tat cannot be controllably reuce troug aitional computational eort. For example, nite-precision aritmetic presents te same problems to quantum computers as it oes to classical ones, but te rouning errors can be brougt uner control by using more igits of precision (as on classical computers). Quantum cemistry tecniques migt present numerical problems as well, insofar as tey contain uncontrolle sources of error. However, if te tecnique for computing te energy is numerically exact, tat is to say, if te error in te energy can be controllably reuce below any level, te magnitue of te numerical error in te calculate erivative can likewise be mae arbitrarily small. Fortunately, quantum computers woul make it possible to eciently evaluate te numerically exact molecular energy, meaning tat numerical instability will not be a problem. We turn to te topic of molecular energy evaluation next. III. THE BLACK BOX FOR THE EERGY Te application of Joran's graient algoritm to cemical problems requires tat tere be a black box tat can compute te value of te groun-state molecular energy at any value of te perturbation µ in te neigboroo of µ =. Furtermore, to avoi numerical artifacts, tis black box soul be numerically exact, allowing te error in te energy to be controllably reuce troug aitional computational work. Te problem of exact classical electronic structure metos is tat tey generally ave a computational cost tat scales exponentially wit te size of te system. Altoug tese classical algoritms coul also be use as subroutines in te quantum graient algoritm, tere are quantum electronic-structure algoritms tat coul avoi te exponential scaling in many cases. In particular, we ave recently escribe a quantum full CI algoritm [8] for computing te molecular groun state energy to a given precision in O ( M 5) time [16], were M is te number of basis functions. Tis algoritm coul be easily recast as a subroutine tat woul function as te black box for te energy. Several moications woul ave to be mae, incluing te irect computation of te overlap integrals on te quantum computer, renering it possible to introuce te perturbation µ into te calculation. everteless, a quantum computer running te quantum FCI algoritm coul be use to obtain a molecular property of a system wit basis size M in O ( M 5) time, a ramatic improvement over te possibilities of classical computers. A more recent evelopment is te real-space cemical ynamics simulation algoritm [1, 17], base on Zalka's earlier work [18]. It is known tat simulating, to a given precision, te exact ynamics of a system of P particles interacting uner a pairwise interaction requires at most O(P 2 ) time an O(P ) space, in contrast to te classical exponential cost. If an eigenstate of te system Hamiltonian were prepare as te initial state [19],

6 5 te ynamics woul only apply a pase to te wavefunction. Tis pase coul be rea out by te pase estimation algoritm [14, 2, 21], forming te require energy black box. Altoug te large prefactor of tis algoritm woul make it slower for small molecules tan te equivalent quantum FCI calculation, it benets from a superior asymptotic scaling as well as from te fact tat only minimal moications woul nee to be mae to insert te perturbation µ into te calculation. For example, simulations wit ierent nuclear coorinates procee in exactly te same way, wile an electromagnetic el requires only a small moication of te simulate Hamiltonian [18]. It soul be remarke tat current quantum algoritms for energy estimation, suc as te ones mentione above, rely on quantum pase estimation, wic as been criticize as inecient [22] because its cost grows exponentially wit te number of bits of precision sougt. Tis coul be signicant for graient estimation, wic may require precise energy evaluations to avoi numerical errors. To estimate te cost, we note tat if te graient is esire to n bits of precision (as in Eq. 11), te black box soul evaluate te energy to [ ] Emax E min 2π n E = log 2 (m/2 n n + log )(θ/2π) 2 θ (17) bits of precision [13], were cos 2 θ is te esire success probability of te algoritm. For example, wit θ = π/8, te algoritm succees 85% of te time an requires four more igits of precision in te energy tan is esire in te graient. Te four aitional igits present only a constant overea, meaning tat te computation of any molecular property at any precision is, up to a constant factor, as ar as computing te energy of te same molecule at te same precision. Finally, a limitation of current quantum simulation algoritms is tat tey are generally spin-free an nonrelativistic, wic limits te ability to compute erivatives suc as inirect spin-spin coupling. IV. EWTO'S METHOD AD GEOMETRY OPTIMIZATIO Peraps te single most common use of molecular erivatives is molecular geometry optimization. We can terefore use it to illustrate some of te avantages of a quantum algoritm over a classical one, incluing a quantum version of ewton's meto, wic oers an aitional quaratic speeup over its classical counterpart. A simple way for ning te locally optimal geometry is to perform te stanar ewton iterations, ( R n+1 = R n R Rn ) ( 2 1 E R 2, (18) Rn) until convergence is reace. Here, R n are te nuclear coorinates at te n t iteration, an R Rn an 2 E Rn R 2 are, respectively, te graient an Hessian of E wit respect to nuclear isplacement (te molecular graient an te molecular Hessian). If a quantum computer were use to compute te erivatives, one woul require exactly 3 calls to a black box for E per iteration: one for te graient an two for te Hessian. A classical approac, on te oter an, woul be muc slower, requiring at least function calls for nite ierence, an approximately O() eort in te analytical case [3]. For large molecules wit large, tis savings coul prove signicant, even if eac energy evaluation takes muc longer on a quantum macine tan on a classical computer. Tere are many classical tricks available for speeing up te convergence of ewton's meto if te initial guess is not close to a local minimum, in wic case te usual ewton step migt be inappropriately large. Tecniques suc as trust regions an level sifts [23] are still available to quantum computers, or tey can be implemente as classical post-processing. In aition, we remark tat ewton's meto is te rst in te class of Houseoler metos, wic oer a rate of convergence of l + 1, provie tat erivatives up to orer l + 1 exist an can be calculate. A quantum computer coul be use to accelerate Houseoler metos of any egree, requiring l+1 m=1 2m 1 = 2 l+1 1 calls to te black box for orer-l Houseoler optimization meto. Altoug exponential in l, tis expression is inepenent of system imension. Of course, ewton's meto is only useful for local minimization, an we are often intereste in global optimization. Here, we can use a quantum version of te multistart tecnique, calle te quantum basin opper [24, 25, 26, 27]. A number of points is selecte at ranom, an eac is followe, using a local searc, to its local basin (if a quantum version of ewton's meto is use for te local searc, suc as te one we propose above, we can get te usual quaratic convergence). Ten, te minima of all te basins are compare an te least one cosen as te global minimum. Quantum computers coul a a quaratic spee-up to suc a multistart tecnique, since te resulting local minima form an unstructure atabase tat can be searce using Grover's algoritm [11, 14] wit a quaratic spee-up. As Dürr an Høyer pointe out [28], a Grover searc can n te minimum of an unstructure atabase wit O( K log K) calls to te atabase (were K is te number of atabase entries, i.e. multistart points), as oppose to te classically require O(K log K) queries. V. COCLUSIO We ave sown tat Joran's quantum graient estimation algoritm can be applie to te estimation of time-inepenent, non-relativistic molecular properties.

7 6 Doing so requires a quantum electronic-structre black box, for wic known quantum simulations algoritms are well suite. Te quantum algoritm oers a speeup from te classical cost of O ( n/2 ) for analytical erivatives to te quantum query complexity of 2 n 1. Tat is, te number of energy calculations require on te quantum computer is inepenent of, an tus of te system size, wic coul oer a signicant avantage for te computation of properties of large systems. In particular, it woul make te molecular Hessian of any molecule only twice as expensive as its molecular graient, enabling a fast, local geometry optimization using ewton's meto. Finally, global optimization coul combine te local ewton's meto wit Grover searc to oer an aitional quaratic spee-up over te classical multi-start tecnique. Acknowlegments We acknowlege support from te Army Researc Of- ce uner contract W911F I.K. tanks te Joyce an Zlatko Balokovi Scolarsip. [1] P. Pulay, Mol. Pys. 17, 197 (1969). [2] P. Pulay, Av. Cem. Pys. 69, 241 (1987). [3] T. Helgaker an P. Jørgensen, Av. Quantum Cem. 19, 183 (1988). [4] Y. Yamaguci, J. D. Goar, Y. Osamura, an H. Scaefer, A ew Dimension to Quantum Cemistry: Analytic Derivative Metos in Ab Initio Molecular Electronic Structure Teory (Oxfor University Press, ew York, 1994). [5] R. Separ, in Moern Electronic Structure Teory, eite by D. Yarkony (Worl Scientic, Singapore, 1995), pp [6] P. Pulay, in Moern Electronic Structure Teory, eite by D. Yarkony (Worl Scientic, Singapore, 1995), pp [7] T. Helgaker, in Te Encyclopeia of Computational Cemistry, eite by P. v. R. Scleyer,. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Scaefer III, an P. R. Screiner (Wiley, ew York, 1998), pp [8] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, an M. Hea- Goron, Science 39, 174 (25). [9] H. Wang, S. Kais, A. Aspuru-Guzik, an M. R. Homann, Pys. Cem. Cem. Pys. 1, 5388 (28). [1] I. Kassal, S. P. Joran, P. J. Love, M. Moseni, an A. Aspuru-Guzik, Proc. atl. Aca. Sci. 15, (28). [11] L. Grover, Proceeings of te 28t Annual ACM Symposium on te Teory of Computing pp (1996). [12] A.. Tikonov an V. Y. Arsenin, Solutions of Ill-Pose Problems (Winston, Wasington, 1977). [13] S. P. Joran, Pys. Rev. Lett. 95, 551 (25). [14] M. A. ielsen an I. L. Cuang, Quantum Computation an Quantum Information (Cambrige University Press, ew York, 2). [15] S. Aaronson, Quantum Inform. Comput. 3, 165 (23). [16] B. P. Lanyon, J. D. Witel, G. G. Gillet, M. E. Goggin, M. P. Almeia, I. Kassal, J. D. Biamonte, M. Moseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, an A. G. Wite, arxiv: (29). [17] D. A. Liar an H. Wang, Pys. Rev. E 59, 2429 (1999). [18] C. Zalka, Proc. Roy. Soc. A 454, 313 (1998). [19]. J. War, I. Kassal, an A. Aspuru-Guzik, J. Cem. Pys. 13, (29). [2] A. Y. Kitaev, quant-p/ (1995). [21] D. S. Abrams an S. Lloy, Pys. Rev. Lett. 83, 5162 (1999). [22] K. R. Brown, R. J. Clark, an I. L. Cuang, Pys. Rev. Lett. 97, 554 (26). [23] R. Fletcer, Practical Metos of Optimization (Wiley, ew York, 2), 2n e. [24] D. Bulger, quant-p/57193 (25). [25] D. Bulger, J. Optim. Teory Appl. 133, 289 (27). [26] W. P. Baritompa, D. Bulger, an G. Woo, SIAM J. Optim. 15, 117 (25). [27] J. Zu, Z. Huang, an S. Kais, arxiv: (29). [28] C. Dürr an P. Høyer, quant-p/96714 (1996). [29] For some classes of useful properties, is inepenent of system size. For example, if te perturbation is te electric el, ten = 3, an inee tere are classical tecniques for computing electrical properties of large molecules. Te quantum spee-up is terefore most pronounce in cases were varies wit system size, as it oes wenever tere is ierentiation wit respect to nuclear cooriates. [3] Quasi-ewton metos (suc as te BroyenFletcerGolfarbSanno meto), or even simpler metos suc as simple graient escent, can remove te nee to compute te molecular Hessian at eac step, or at all. Wile suc scemes are useful an wiely applie, we o not iscuss tem ere because tey are typically slower an, for a given number of steps, less accurate tan ewton's meto. Wile tey oer a classical computational avantage, on a quantum computer tat avantage woul be erase by te ability to rapily compute te exact Hessian at eac step.