Anderson localization casts clouds over adiabatic quantum optimization

Anderson lo calization casts clouds o v er adiabatic quan tum optimization Boris Al tshuler, 1, 2 , ∗ Hari Kr ovi, 2 , † and Jeremie R oland 2 , ‡ 1 Columbia University 2 NEC L ab or atories A meric a Inc. (Dated: Octob er 22, 2018) Abstract Understanding NP-complete problems is a cen tral topic in computer science. This is wh y adi- abatic quan tum optimization has attracted so m uch atten tion, as it provided a new approac h to tac kle NP-complete problems using a quan tum computer. The efficiency of this approac h is lim- ited by small spectral gaps betw een the ground and excited states of the quan tum computer’s Hamiltonian. W e sho w that the statistics of the gaps can b e analyzed in a nov el wa y , b orrow ed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson lo calization, exp onen tially small gaps app ear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately , adiabatic quantum optimization fails: the system gets trapp ed in one of the n umerous lo cal minima. ∗ Electronic address: bla@phys.colum bia.edu † Electronic address: hari.krovi@uconn.edu ‡ Electronic address: jroland@nec-labs.com 1 a. NP-c ompleteness. One of the central concepts in computational complexity theory is that of NP-completeness [1]. A computational problem b elongs to the class NP if its solution can b e v erified in a time at most polynomial in the input size N , i.e., the v erification requires not more than cN k computational steps, where c and k are indep endent of N . An NP-complete problem satisfies a second criterion: an y other problem in the class NP can b e reduced to it in p olynomial time. Remark ably , such problems exist, many of them b eing of a great practical imp ortance. The question of whether NP-complete problems are “easy to solv e”, or in other words whether they ma y b e solved in p olynomial time, is one of the most fundamen tal op en problems in computer science: this is the famous “P ? = NP” question [2]. It is commonly b eliev ed ho w ev er that it is not the case, i.e., that solving suc h a problem requires a computational time which is exp onen tial in N . b. A diab atic quantum optimization. The disco very of an efficien t (p olynomial time) quan tum algorithm for the factorization of large n umbers—a problem in NP but not b eliev ed to b e NP-complete—is a milestone in quantum computing [3], as no algorithm is known to solv e this problem efficien tly on a classical (non-quantum) computer. How ev er, this success w as not extended to NP-complete problems. That w as why the prop osal of F arhi et al . [4] to use adiabatic quan tum optimization (A QO) to solve NP-complete problems has attracted m uch attention since initial numerical sim ulations suggested suc h a p ossibilit y [5]. The basic idea of A QO is as follows: supp ose that the solution of a computational problem P can b e enco ded in the ground state (GS) of a Hamiltonian ˆ H P . T o implemen t A QO one needs to construct a ph ysical quan tum system that is gov erned b y a Hamiltonian ˆ H ( s ) = (1 − s ) ˆ H 0 + s ˆ H P where s is a tunable parameter, and ˆ H 0 is a Hamiltonian with a kno wn and easy-to-prepare ground state. The idea is to start with s = 0, initialize the system in the ground state of ˆ H (0) = ˆ H 0 and increase s with time as s = t/T . According to the Adiabatic Theorem [6], slo w enough v ariation of the parameter s = s ( t ) k eeps the system in the ground state of the Hamiltonian ˆ H ( s ( t )) at any time t . Therefore, if T is large enough, at t = T the system w ould find itself in the ground state of ˆ H (1) = ˆ H P and the problem w ould b e solv ed. This mo del has since b een shown to b e equiv alent to the standard (circuit) mo del of quan tum computing [7]. Of course, as long as the computational time T remains finite there is a non-zero probability that the system would undergo a Landau-Zener transition [6] and end up in an excited state. In order to maintain the excitation probability less than  , the adiabatic condition requires that T ∼ 1  ∆ 2 , where ∆( s ) = E E S − E GS is the energy gap 2 b et w een the ground state and first excited state (ES) of the Hamiltonian ˆ H ( s ). Therefore A QO is not efficien t when ∆ is small. More precisely , the adiabatic quan tum approach to NP-complete problems can b eat kno wn classical algorithms (whic h require exponential time) pro vided that the minimal v alue of the gap scales as an inv erse p o wer of the problem size N . Previously , it w as shown that the gap can b ecome exp onen tially small under specific conditions, such as a bad choice of initial Hamiltonian [8, 9], or for sp ecifically designed hard instances [10, 11, 12]. In particular, it was recently argued that the presence of a first order phase transition could induce an exp onen tially small gap, and this effect was demonstrated for a particular instance of an NP-hard problem [13], and later for plante d instances of 3-SA T [14]. While these examples show that small gaps can o ccur for sp e cific instances of NP-complete problems, one could hop e that this is not the t ypical b eha vior, i.e., for randomly generated instances the gap could b e small only with very lo w probability . This hope follow ed from n umerical sim ulations [5, 15, 16] where the minim um gap seemed to decrease only polynomially for small instances, up to N = 124 for the latest sim ulations [17]. In this pap er we show that this scaling do es not p ersist for larger N . It turns out that as N → ∞ , the typical v alue of the minimal gap for random instances decays ev en faster than exp onen tially . As a result, the probability for AQO to yield a wrong solution in this limit tends to unity . c. A nderson lo c alization. The app earance of exp onen tially small sp ectral gaps can b e naturally attributed to the Anderson lo calization (AL) of the eigenfunctions of ˆ H ( s ) in the space of the solutions. Originally , AL implied that the wa v e function of a quantum particle in d -dimensional space ( d = 1 , 2 , 3 , . . . ) sub ject to a strong enough disorder p otential turns out to b e spatially lo calized in a small region and decays exp onen tially as a function of the distance from this region. Accordingly the probabilit y for the particle to tunnel through a large disordered region is suppressed exp onen tially . T o illustrate this, first note that the gap ∆ can not v anish at an y s unless there is a sp ecial symmetry reason. This is the famous Wigner-v on Neumann non-crossing rule [18]: the curves that describ e the s -dep endence of t wo eigenenergies do not cross on the ( E , s )-plane. This so-called level repulsion follows from the consideration of a reduced 2 × 2 Hamiltonian that describ es tw o anomalously close energy states and neglects the rest of the sp ectrum. Let E 1 and E 2 b e the diagonal matrix elemen ts of the Hamiltonian, and V 12 = V ∗ 21 b e its off-diagonal matrix elemen ts. W e then 3 find the energy gap to b e ∆ = E E S − E GS = p ( E 1 − E 2 ) 2 + | V 12 | 2 . (1) No w supp ose that E 1 ( s ) and E 2 ( s ) b ecome equal at s = s c , as depicted in Fig. 1. W e find that ∆ > 0 even for s = s c . This is known as a level anti-crossing. The minimal v alue of the energy gap is determined b y the off-diagonal matrix elemen t i.e., ∆ min = | V 12 | whic h is exp onen tially small under AL conditions. Accordingly the energy level repulsion betw een the lo calized states should b e exp onentially small in the spatial distance. Fig. 1 illustrates this situation sc hematically . At certain interv al of s close to s c the difference E 1 ( s ) − E 2 ( s ) is smaller or of the order of the tunneling matrix elemen t V 12 . It is the interv al where the an ti-crossing tak es place. Since V 12 dep ends exp onen tially on the distance b et w een the w ells, b oth the width of the anti-crossing interv al and the minimum gap turn out to b e exp onen tially small. The concept of AL w as introduced more than 50 years ago in order to describ e spin and charge transp ort in disordered solids [19]. Since then AL was found to b e relev ant for a v ariet y of physical situations. It also turned out to exist and mak e physical sense in a muc h broader class of spaces than R d . Below w e demonstrate that a phenomenon analogous to AL on the v ertices of the N -dimensional cub e naturally app ears in connection with AQO. d. Exact Cover 3. In order to explain the connection b et ween the AQO approac h to NP-complete problems and Anderson lo calization, w e pick a particular NP-complete problem known as Exact Cov er 3 (EC3), the same problem that was used for the early n umerical simulations of A QO [5]. Ho wev er, we b eliev e that this analysis can b e extended to any NP-complete problem. EC3 can b e formalized in the following wa y . Consider N bits x 1 , x 2 , . . . , x N whic h tak e v alues 0 or 1. An instance of EC3 consists of M triplets of bit indices ( i c , j c , k c ) (the clauses), where each clause is said to b e satisfied if and only if one of the corresp onding bits is 1 and the other tw o are 0. A solution of a particular instance of EC3 is an assignmen t of the bits x = ( x 1 , x 2 , . . . , x N ) which satisfies all of the clauses. This problem can b e assigned a cost function giv en by f ( x ) = P c ( x i c + x j c + x k c − 1) 2 : eac h solution has zero cost and all other assignments hav e a p ositiv e cost. W e consider a standard distribution of random instances, where an instance is built by picking the M clauses indep enden tly , each clause b eing obtained by picking 3 bit indices uniformly at random. The hardness of suc h random instances is characterized by the clauses-to-v ariables 4 FIG. 1: Sc hematic represen tation of a level anti-crossing. The energies of t w o quan tum states | Ψ 1 i and | Ψ 2 i lo calized in distan t wells can b e fine-tuned by applying a smo oth additional p oten tial. (a) Before the crossing, the ground state is | Ψ 2 i with energy close to E 2 ( s ), i.e., for s − < s c , w e ha ve that E 1 ( s − ) > E 2 ( s − ), so that | GS ( s − ) i = | Ψ 2 i . (b) After the crossing, the ground state b ecomes | Ψ 1 i with energy close to E 1 ( s ), i.e. for s + > s c , we hav e that E 1 ( s + ) < E 2 ( s + ), so that | GS ( s + ) i = | Ψ 1 i . The ground states b efore and after the crossing ha ve nothing to do with each other. A t a certain interv al of s close to s c , the anti-crossing takes place and the ground state is a linear combination of | Ψ 1 i and | Ψ 2 i . ratio α = M / N . There are tw o c haracteristic v alues of α : the clustering threshold α cl and the satisfiability threshold α s [20]. F or α < α cl , the densit y of the solutions is high and essen tially uniform, while for α > α cl the solutions b ecome clustered in the solution space with different clusters remote from each other (the distance b et ween tw o assignments is the so called Hamming distance which is defined as the n um b er of bits in which they differ). As α increases from α cl to α s , the clusters b ecome smaller and the distance b et w een them increases. F or α > α s , the probabilit y that the problem is satisfiable v anishes in the limit N , M → ∞ . It has been sho wn [21] that α s ≈ 0 . 6263. W e will b e in terested in instances with α close to α s , whic h only accept a few isolated solutions and are therefore hard to solv e. More precisely , kno wn classical algorithms can not solve suc h hard instances for a n umber of bits N more than a few thousands, so that this is the regime where an efficien t quantum 5 algorithm would b e particularly desirable. e. A diab atic quantum algorithm. In order to define an adiabatic quan tum algorithm for EC3, we need to choose ˆ H P and ˆ H 0 . The problem Hamiltonian ˆ H P for an EC3 instance can b e obtained from the ab ov e cost function b y first replacing x i b y the Ising v ariables σ ( i ) z = 1 − 2 x i = ± 1 and then substituting σ ( i ) z b y the Pauli Z operators ˆ σ ( i ) z , th us replacing the bits by qubits. The problem Hamiltonian b ecomes ˆ H P = M ˆ I − 1 2 N X i =1 B i ˆ σ ( i ) z + 1 4 N X i,j =1 J ij ˆ σ ( i ) z ˆ σ ( j ) z , (2) where B i is the n um b er of clauses that in volv e the bit i , J ij is the n um b er of clauses where the bits i and j participate together, and ˆ I is the iden tity op erator. F or ˆ H 0 , we make the con ven tional c hoice ˆ H 0 = − P i ˆ σ ( i ) x , which corresponds to spins in the magnetic field directed along x -axis (P auli X op erators). F or us it will also b e con venien t to mo dify the Hamiltonian ˆ H ( s ) as ˆ H QC ( λ ) = ˆ H P + λ ˆ H 0 . The parameter λ = 1 − s s c hanges adiabatically from λ = + ∞ at the b eginning t = 0 to λ = 0 at t = T . f. Conne ction to A nderson L o c alization. W e can no w see the relev ance of AL to the quan tum system describ ed b y ˆ H QC . Note that this Hamiltonian also describes a single quan- tum particle that is moving b et w een the vertices of an N -dimensional hypercub e. Indeed, eac h v ector σ = ( σ (1) z , σ (2) z , . . . , σ ( N ) z ), where σ ( i ) z = ± 1, determines a vertex of the h yp er- cub e, which is b o dy-cen tered at the origin of the N -dimensional space. Let | σ i denote the quan tum state of a particle lo calized at a site σ . The full set of these states forms a basis, in whic h the first term of the Hamiltonian is diagonal, while the second one describ es a hopping of this fictitious particle b etw een the nearest neighbors (n.n) ˆ H QC ( λ ) = X σ E P ( σ ) | σ ih σ | | {z } disorder + λ X σ , σ 0 n.n | σ ih σ 0 | . (3) Eac h on-site energy E P ( σ ) is nothing but the cost function f ( x ) of the corresp onding assign- men t σ . F or random instances, the on-site energies are obviously also random, introducing disorder in the Hamiltonian. Hence, Eq. (3) describ es the well known Anderson mo del, whic h was used to demonstrate the phenomenon of lo calization [19]. The only difference from more familiar situations is that lattices in d -dimensional space, which ha v e L d sites where L  1 is the system size, are substituted by the N -dimensional hypercub e with 2 N sites, where N  1. 6 FIG. 2: Schematic representation of the creation of a level anti-crossing. (a) Before adding the clause, w e ha ve tw o assignments which are b oth in the ground state at λ = 0 but due to the no- crossing rule, at λ > 0 we hav e E 1 ( λ ∗ ) − E 2 ( λ ∗ ) > 4. (b) By adding a clause satisfied by solution 1 but not solution 2, we create a lev el an ti-crossing since ˜ E 1 (0) < ˜ E 2 (0) but ˜ E 1 ( λ ∗ ) > ˜ E 2 ( λ ∗ ). Insets: (a) If the clause is violated by the wrong solution, then no anti-crossing app ears betw een these t wo levels. (b) How ev er, other lo w energy levels can create other an ti-crossings, leading to m ultiple small gaps. g. A nti-cr ossings in AQO. No w w e are ready to discuss the fundamen tal difficulties whic h A QO faces. W e will show that (i) the anti-crossings of the ground state with the first excited state happ en with high probabilit y and (ii) that the anti-crossing gaps in the limit N → ∞ are even less than exp onentially small. Let us start with the first statement. An EC3 instance with α < α s t ypically has sev eral solutions σ with E P ( σ ) = 0. If α is close to α s there are few solutions at a distance of order N of eac h other. The presence of m ultiple solutions imply that the ground state of ˆ H QC ( λ = 0) = ˆ H P is degenerate, but this do es not contradict the non-crossing rule: ˆ H P comm utes with each of the op erators ˆ σ ( i ) z , so it satisfies a sp ecial symmetry whic h is broken for λ > 0. Consider no w a particular instance with M − 1 clauses accepting tw o solutions σ 1 and σ 2 that are separated b y n ∼ N spin flips. When λ adiabatically c hanges from zero to a small but finite v alue the solutions ev olve into eigenstates of the Hamiltonian, | Ψ 1 , λ i and | Ψ 2 , λ i with the energies E 1 ( λ ) and E 2 ( λ ). According to the non-crossing rule, a degeneracy of these tw o states at a finite λ is improbable, i.e., the ˆ H 0 term in ˆ H QC splits the ground state degeneracy . This situation is sk etched in Fig. 2(a). Supp ose that E 2 ( λ ) < E 1 ( λ ), i.e. | Ψ 2 , λ i is the unique ground state 7 of the Hamiltonian ˆ H QC ( λ ). If w e now add one more clause to the existing M − 1 ones, i.e. w e add a term ( x i M + x j M + x k M − 1) 2 to the cost function leading to Hamiltonian ˆ H P , b oth | σ 1 i and | σ 2 i remain eigenstates, but their eigenenergy can increase by either 1 or 4. With a non-zero probabilit y the last clause is satisfied by σ 1 but not b y σ 2 , i.e., ˜ E P ( σ 1 ) = 0 while ˜ E P ( σ 2 ) > 0, where ˜ E P ( σ ) is the cost function of the new instance. Accordingly | σ 1 i rather than | σ 2 i is the new ground state at λ = 0. At the same time | Ψ 2 , λ i can still remain the ground state at large enough λ if ˜ E 1 ( λ ) > ˜ E 2 ( λ ), as shown on Fig. 2(b). Suc h a situation corresp onds to the anti-crossing of | Ψ 1 , λ i and | Ψ 2 , λ i at certain λ , as previously describ ed in Fig. 1. Note that the addition of a clause to the cost function increases any eigenenergy of ˆ H QC ( λ ) by less than 4. T o satisfy the condition ˜ E 1 ( λ ) > ˜ E 2 ( λ ), it is th us sufficient to ac hieve a large enough splitting b et ween the eigenv alues of the instance with M − 1 clauses: E 1 ( λ ) − E 2 ( λ ) > 4. It turns out that if N  1, this happ ens when λ is small and one can use p erturbation theory in λ . h. Perturb ation the ory. T o demonstrate this, consider the eigenstate whic h in the limit λ → 0 evolv es to | σ i . At small λ its energy can b e expanded in a series E ( λ, σ ) = E P ( σ ) + ∞ X m =1 λ 2 m F ( m ) ( σ ) . (4) W e can show that each term in this sum scales linearly in N . F or the energy E P ( σ ) of an arbitrary assignmen t, we immediately hav e that 0 ≤ E P < M = αN . As for the co efficien ts F ( m ) ( σ ), the cluster expansion [22] of the Hamiltonian ˆ H QC implies that they ma y b e expressed as a sum of ∼ N statistically indep enden t terms, each b eing of order 1. The k ey element to pro ve this is that since M / N = α is constan t, with high probability eac h bit participates in a finite num b er of clauses as N → ∞ . As a result, all the co efficients B i and J ij in Eq. (2) are also finite: B i = P j J ij = O (1). In particular, when σ is a solution w e obtain F (1) ( σ ) = P i B − 1 i , which is therefore of order N . This statemen t is v alid for F ( m ) ( σ ) with arbitrary finite m > 1: all of them can b e presen ted as a finite sum of O ( N ) random terms, eac h one b eing of order unit y . Let us no w consider the p erturbativ e expansion for the energy splitting b et w een tw o solutions. Similarly to Eq. (4), w e obtain E 1 ( λ ) − E 2 ( λ ) = ∞ X m =2 λ 2 m F ( m ) 1 , 2 , (5) where F ( m ) 1 , 2 = F ( m ) ( σ 1 ) − F ( m ) ( σ 2 ) is a sum of O ( N ) terms of order 1. Each of the terms is random with a zero mean and hence the sums F ( m ) 1 , 2 a verages to zero if N is large. Therefore, 8 FIG. 3: Statistics of the square of the difference in energies of tw o solutions up to fourth order i.e. ( F (2) 1 , 2 ) 2 . Linear fits confirm that the square of the energy difference scales as O ( N ). Inset: Statistics of the sixth order correction of the splitting ( F (3) 1 , 2 ) 2 . Each data p oin t is obtained from 2500 random instances of EC3 with α ≈ 0 . 62. Linear fits for the mean yield f (2) ≈ 0 . 18 and f (3) ≈ 0 . 65. it is ( F ( m ) 1 , 2 ) 2 rather than F ( m ) 1 , 2 whic h is prop ortional to N . W e thus arrive to the conclusion that | E 1 ( λ ) − E 2 ( λ ) | = √ N X m λ 2 m f ( m ) , (6) where the co efficien ts f ( m ) = O (1) can b e ev aluated b y the cluster expansion [22]. W e hav e seen that F (1) ( σ ) = P i B − 1 i for an y solution σ , so that F (1) 1 , 2 = 0. How ev er terms with m > 1 do not v anish, making the splitting finite. On Fig. 3, we show the results of the statistical analysis of the n umerical calculations of the co efficients ( F (2) 1 , 2 ) 2 and ( F (3) 1 , 2 ) 2 , with linear fits confirming their scaling O ( N ). F or small λ , we can restrict ourselv es to the leading term ( m = 2) in Eq. (6). Accordingly in the N → ∞ limit, the splitting | E 1 ( λ ) − E 2 ( λ ) | exceeds 4 as long as λ > λ ∗ , with λ ∗ = √ 2 ( f (2) ) − 1 / 4 N − 1 / 8 , (7) and λ ∗  1 so that we can neglect higher orders, λ ∗  1 (the v alidit y of this appro ximation will b e discussed in the next paragraph). F rom Eq. (7), it follows that the an ti-crossing probabilit y for the instance with M clauses is finite pro vided that λ ≥ λ ∗ ∼ N − 1 / 8 . How big is the gap ∆ of suc h an an ti-crossing? As explained ab o ve, we can ev aluate the gap b y 9 considering the matrix elemen t V 12 b et w een the states | Ψ 1 , λ i and | Ψ 2 , λ i corresp onding to the t w o assignments, at the v alue λ where the anti-crossing o ccurs. Note that if the tw o assignmen ts σ 1 and σ 2 satisfying the ( M − 1) clauses are separated by a distance (n umber of flips) n , this matrix element only app ears at the n -th order of the perturbation theory , i.e. it is prop ortional to λ n : V 12 = λ n X tr  Π n k =1 E P ( σ ( k ) tr )  − 1 + O ( λ n +1 ) (8) where the sum is ov er all ”tra jectories” tr - all p ossible orders of the n spin flips needed to transform σ 1 in to σ 2 , σ ( k ) tr is the assignment along a particular tra jectory that app ears after k flips and E P ( σ ( k ) tr ) is the cost function of this assignment. Therefore we can estimate the matrix elemen t and th us the an ti-crossing gap as V 12 < w ( n ) λ n . The prefactor w ( n ) reflects the fact that man y ( ∼ n !) tra jectories contribute to the sum in Eq. (8). F or a t ypical tra jectory E P ( σ ( k ) tr ) = O ( k ) for k < n/ 2 and E P ( σ ( k ) tr ) = O ( n/ 2 − k ) for k > n/ 2. As a result the pro duct of E P ( σ ( k ) tr ) in Eq. (8) is also ∼ n !. The factorials th us cancel eac h other and w ( n ) can not increase faster than A n with some constan t A ∼ 1. Therefore, V 12 < ( Aλ ) n . Combining this with Eq. (7), we see that an anti-crossing at λ close to λ ∗ yields the minimum gap as small as ∆ min ∼ exp[ − ( n/ 8) ln( N / N 0 )], where N 0 = 16 A 8 ( f (2) ) − 2 = O (1). W e can estimate the distance n b et ween the assignmen ts as v ( α ) N , where v ( α ) ≈ (4 / 9)(1 − exp( − 3 α )), and obtain the final form of the minimal gap estimation ∆ min ∼ exp[( − v ( α ) N / 8) ln( N / N 0 )] . (9) One can see that as N → ∞ , the gap indeed decreases even faster than an exp onen tial - statemen t (ii). This implies that the adiabatic computation time exceeds exp( N ). In Fig. 4, w e hav e plotted an an ti-crossing for a particular instance with N = 200 generated during our numerical simulations. The figure sho ws tw o energy levels (estimated b y fourth order p erturbation theory) corresponding to assignments separated by 60 bit flips, and crossing at λ ≈ 0 . 51. i. Applic ability of the p erturb ation the ory. Our main result - the estimation of the minimal gap (Eq. (9)), is based on the p erturbativ e expansion for the energies (Eq. (4)) and the matrix element V 12 (Eq. (8)). Is the p erturbation theory in λ alw a ys applicable? At first sigh t Eq. (8) b ecomes meaningless if E P = 0 for any of the intermediate assignments σ ( k ) tr . In this case there is an a v oided crossing b et w een the states corresponding to the assignmen ts 10 FIG. 4: Sim ulation of a lev el anti-crossing for a random instance with N = 200 bits and α ≈ 0 . 62, obtained by fourth order p erturbation theory . The figure sho ws the energies of tw o assignmen ts after adding a clause to the final Hamiltonian. The added clause is satisfied b y assignment 1 but not b y assignment 2. The figure shows a level crossing similar to the carto on in Fig. 2. Inset: T o mak e the crossing more apparen t, we plotted the energy differences E 1 − E 2 and E 2 − E 1 . The crossing o ccurs at λ ≈ 0 . 51, and the corresp onding assignmen ts are at distance n = 60 from each other. σ 1 and σ ( k ) tr (suc h as in Eq. (1)) and formally p erturbation theory fails in the vicinit y of this an ti-crossing p oin t. This apparen t difficult y can be o vercome b y considering only a finite time T for the ev olution. This is equiv alen t to adding imaginary parts iη ≈ i/T to the energies. F or the AQO algorithm, it is the computation time T that determines η . Since we are considering the N → ∞ limit, we hav e that T → ∞ and thus η → 0. This is the limit that w as sho wn to be relev an t for the localization problem [19, 23]. The celebrated disco v ery of Anderson was that if the limit η → 0 is tak en after the v olume (here N ) tends to infinity , and λ is small enough i.e., λ < λ cr , the sp ectrum of the Hamiltonian describ ed in Eq. (3) remains discrete (all states are lo calized) and th us the second term in Eq. (3) (the kinetic energy term) can be treated perturbatively . As so on as λ > λ cr , there appears a strip of extended states in the middle of the energy band whic h widens as λ increases further. States within this strip are not p erturbative b ecause the n umber of the tra jectories connecting tw o p oin ts in a d -dimensional space (for finite d ) increases exp onen tially with distance. The large n um b er of terms in the expansions lik e Eq. (8) o v erwhelms the smallness of λ n and 11 the p erturbation series th us div erges for λ > λ cr . F or a d -dimensional space, the critical v alue λ cr is b eliev ed to b e (in our units) of the order of λ cr ∼ 1 / log d [24, 25]. W e ha ve seen that the A QO algorithm for problems like EC3 can b e mapp ed to the Anderson mo del on an N -dimensional h yp ercub e. Then, the n umber of tra jectories increases with the length n as n ! ∼ n n e − n , i.e. ev en faster than an exp onen tial. How ev er, as w e already mentioned, the n n factor cancels with the same factor in the pro ducts of the energy in the denominators of Eq. (8). Accordingly , λ cr can still b e estimated as λ cr ∼ 1 / log N , which, together with Eq. (7), implies that an ti-crossings app ear for λ ∗  λ cr when N  1. Moreo ver, at λ < λ cr all of the states are supp osed to b e lo calized. The AQO algorithm inv olv es only lo w energy states, whic h remain lo calized muc h longer than the middle-band states with the energies ∼ N . Therefore, it is quite likely that the exp onen tially small gaps app ear even at λ ∼ 1. j. Conclusions. W e finish our discussion with the follo wing observ ation. W e monitored t wo assignments that satisfied M − 1 clauses and added an extra clause to create a small gap at finite λ . Of course, for randomly selected clauses this happ ens only with a finite probabilit y and the situation sketc hed in the inset in Fig. 2(a) is also p ossible. One could th us hop e [14] that the AQO algorithm can find the solution with a sizable probability . Unfortunately , this is not the case. Indeed, let us adopt the most conserv ativ e limitation on the perturbative approac h λ cr ∼ 1 / log N and consider the spectrum at λ ∗  λ cr ∼ 1 / log N . According to Eq. (6) all states in the energy in terv al [0 ,  ] with  ∼ √ N λ 4  1 hav e similar c hances to evolv e in to the ground state at λ = 0. This means that t ypically the ground state undergo es ν (  ) an ti-crossings (participates in ν (  ) an ti-crossing gaps) as the parameter ev olves from 0 to λ (see the inset of Fig. 2(b)). Here ν (  ) is the n um b er of states, whose energies at the giv en λ differ from the ground state energy by less than  . T aking into accoun t that ν (  ) increases with  exp onen tially and that the probability to completely a void an ti-crossings (the probability to hav e a gap of size  separating the ground state from the rest of the sp ectrum) is exp onen tially small in ν (  ) w e conclude that this probabilit y is indeed negligible. Therefore, these findings suggest that there is no c hance of obtaining the solution of the problem in p olynomial time using the AQO algorithm for random instances of the Exact Cov er 3 problem. W e also believe that the methods described in this article can b e applied to other similar NP-complete problems, such as 3-SA T. 12 Ac kno wledgments W e thank A. Childs, E. F arhi, J. Goldstone, S. Gutmann, M. R¨ otteler and A. P . Y oung for in teresting discussions. 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