Anderson localization casts clouds over adiabatic quantum optimization
AAnderson localization casts clouds over adiabatic quantumoptimization
Boris Altshuler,
1, 2, ∗ Hari Krovi, † and Jeremie Roland ‡ Columbia University NEC Laboratories America Inc. (Dated: October 22, 2018)
Abstract
Understanding NP-complete problems is a central topic in computer science. This is why adi-abatic quantum optimization has attracted so much attention, as it provided a new approach totackle NP-complete problems using a quantum computer. The efficiency of this approach is lim-ited by small spectral gaps between the ground and excited states of the quantum computer’sHamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowedfrom the study of quantum disordered systems in statistical mechanics. It turns out that dueto a phenomenon similar to Anderson localization, exponentially small gaps appear close to theend of the adiabatic algorithm for large random instances of NP-complete problems. This impliesthat unfortunately, adiabatic quantum optimization fails: the system gets trapped in one of thenumerous local minima. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] D ec . NP-completeness. One of the central concepts in computational complexity theoryis that of NP-completeness [1]. A computational problem belongs to the class NP if itssolution can be verified in a time at most polynomial in the input size N , i.e., the verificationrequires not more than cN k computational steps, where c and k are independent of N . AnNP-complete problem satisfies a second criterion: any other problem in the class NP can bereduced to it in polynomial time. Remarkably, such problems exist, many of them being ofa great practical importance. The question of whether NP-complete problems are “easy tosolve”, or in other words whether they may be solved in polynomial time, is one of the mostfundamental open problems in computer science: this is the famous “P ? = NP” question [2].It is commonly believed however that it is not the case, i.e., that solving such a problemrequires a computational time which is exponential in N . b. Adiabatic quantum optimization. The discovery of an efficient (polynomial time)quantum algorithm for the factorization of large numbers—a problem in NP but not believedto be NP-complete—is a milestone in quantum computing [3], as no algorithm is known tosolve this problem efficiently on a classical (non-quantum) computer. However, this successwas not extended to NP-complete problems. That was why the proposal of Farhi et al . [4]to use adiabatic quantum optimization (AQO) to solve NP-complete problems has attractedmuch attention since initial numerical simulations suggested such a possibility [5].The basic idea of AQO is as follows: suppose that the solution of a computational problem P can be encoded in the ground state (GS) of a Hamiltonian ˆ H P . To implement AQO oneneeds to construct a physical quantum system that is governed by a Hamiltonian ˆ H ( s ) =(1 − s ) ˆ H + s ˆ H P where s is a tunable parameter, and ˆ H is a Hamiltonian with a known andeasy-to-prepare ground state. The idea is to start with s = 0, initialize the system in theground state of ˆ H (0) = ˆ H and increase s with time as s = t/T . According to the AdiabaticTheorem [6], slow enough variation of the parameter s = s ( t ) keeps the system in the groundstate of the Hamiltonian ˆ H ( s ( t )) at any time t . Therefore, if T is large enough, at t = T the system would find itself in the ground state of ˆ H (1) = ˆ H P and the problem would besolved. This model has since been shown to be equivalent to the standard (circuit) modelof quantum computing [7]. Of course, as long as the computational time T remains finitethere 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 (cid:15) ,the adiabatic condition requires that T ∼ (cid:15) ∆ , where ∆( s ) = E ES − E GS is the energy gap2etween the ground state and first excited state (ES) of the Hamiltonian ˆ H ( s ). ThereforeAQO is not efficient when ∆ is small. More precisely, the adiabatic quantum approach toNP-complete problems can beat known classical algorithms (which require exponential time)provided that the minimal value of the gap scales as an inverse power of the problem size N . Previously, it was shown that the gap can become exponentially small under specificconditions, such as a bad choice of initial Hamiltonian [8, 9], or for specifically designedhard instances [10, 11, 12]. In particular, it was recently argued that the presence of afirst order phase transition could induce an exponentially small gap, and this effect wasdemonstrated for a particular instance of an NP-hard problem [13], and later for planted instances of 3-SAT [14]. While these examples show that small gaps can occur for specific instances of NP-complete problems, one could hope that this is not the typical behavior,i.e., for randomly generated instances the gap could be small only with very low probability.This hope followed from numerical simulations [5, 15, 16] where the minimum gap seemed todecrease only polynomially for small instances, up to N = 124 for the latest simulations [17].In this paper we show that this scaling does not persist for larger N . It turns out that as N → ∞ , the typical value of the minimal gap for random instances decays even faster thanexponentially. As a result, the probability for AQO to yield a wrong solution in this limittends to unity. c. Anderson localization. The appearance of exponentially small spectral gaps can benaturally attributed to the Anderson localization (AL) of the eigenfunctions of ˆ H ( s ) in thespace of the solutions. Originally, AL implied that the wave function of a quantum particlein d -dimensional space ( d = 1 , , , . . . ) subject to a strong enough disorder potential turnsout to be spatially localized in a small region and decays exponentially as a function of thedistance from this region. Accordingly the probability for the particle to tunnel through alarge disordered region is suppressed exponentially. To illustrate this, first note that the gap∆ can not vanish at any s unless there is a special symmetry reason. This is the famousWigner-von Neumann non-crossing rule [18]: the curves that describe the s -dependence oftwo eigenenergies do not cross on the ( E, s )-plane. This so-called level repulsion followsfrom the consideration of a reduced 2 × E and E be the diagonal matrixelements of the Hamiltonian, and V = V ∗ be its off-diagonal matrix elements. We then3nd the energy gap to be∆ = E ES − E GS = (cid:112) ( E − E ) + | V | . (1)Now suppose that E ( s ) and E ( s ) become equal at s = s c , as depicted in Fig. 1. We findthat ∆ > s = s c . This is known as a level anti-crossing. The minimal value ofthe energy gap is determined by the off-diagonal matrix element i.e., ∆ min = | V | which isexponentially small under AL conditions. Accordingly the energy level repulsion betweenthe localized states should be exponentially small in the spatial distance. Fig. 1 illustratesthis situation schematically. At certain interval of s close to s c the difference E ( s ) − E ( s )is smaller or of the order of the tunneling matrix element V . It is the interval wherethe anti-crossing takes place. Since V depends exponentially on the distance between thewells, both the width of the anti-crossing interval and the minimum gap turn out to beexponentially small. The concept of AL was introduced more than 50 years ago in order todescribe spin and charge transport in disordered solids [19]. Since then AL was found to berelevant for a variety of physical situations. It also turned out to exist and make physicalsense in a much broader class of spaces than R d . Below we demonstrate that a phenomenonanalogous to AL on the vertices of the N -dimensional cube naturally appears in connectionwith AQO. d. Exact Cover 3. In order to explain the connection between the AQO approachto NP-complete problems and Anderson localization, we pick a particular NP-completeproblem known as Exact Cover 3 (EC3), the same problem that was used for the earlynumerical simulations of AQO [5]. However, we believe that this analysis can be extendedto any NP-complete problem. EC3 can be formalized in the following way. Consider N bits x , x , . . . , x N which take values 0 or 1. An instance of EC3 consists of M triplets of bitindices ( i c , j c , k c ) (the clauses), where each clause is said to be satisfied if and only if oneof the corresponding bits is 1 and the other two are 0. A solution of a particular instanceof EC3 is an assignment of the bits x = ( x , x , . . . , x N ) which satisfies all of the clauses.This problem can be assigned a cost function given by f ( x ) = (cid:80) c ( x i c + x j c + x k c − :each solution has zero cost and all other assignments have a positive cost. We considera standard distribution of random instances, where an instance is built by picking the M clauses independently, each clause being obtained by picking 3 bit indices uniformly atrandom. The hardness of such random instances is characterized by the clauses-to-variables4 IG. 1: Schematic representation of a level anti-crossing. The energies of two quantum states | Ψ (cid:105) and | Ψ (cid:105) localized in distant wells can be fine-tuned by applying a smooth additional potential.(a) Before the crossing, the ground state is | Ψ (cid:105) with energy close to E ( s ), i.e., for s − < s c , wehave that E ( s − ) > E ( s − ), so that | GS ( s − ) (cid:105) = | Ψ (cid:105) . (b) After the crossing, the ground statebecomes | Ψ (cid:105) with energy close to E ( s ), i.e. for s + > s c , we have that E ( s + ) < E ( s + ), so that | GS ( s + ) (cid:105) = | Ψ (cid:105) . The ground states before and after the crossing have nothing to do with eachother. At a certain interval of s close to s c , the anti-crossing takes place and the ground state is alinear combination of | Ψ (cid:105) and | Ψ (cid:105) . ratio α = M/N . There are two characteristic values of α : the clustering threshold α cl andthe satisfiability threshold α s [20]. For α < α cl , the density of the solutions is high andessentially uniform, while for α > α cl the solutions become clustered in the solution spacewith different clusters remote from each other (the distance between two assignments is theso called Hamming distance which is defined as the number of bits in which they differ).As α increases from α cl to α s , the clusters become smaller and the distance between themincreases. For α > α s , the probability that the problem is satisfiable vanishes in the limit N, M → ∞ . It has been shown [21] that α s ≈ . α close to α s , which only accept a few isolated solutions and are therefore hard to solve.More precisely, known classical algorithms can not solve such hard instances for a numberof bits N more than a few thousands, so that this is the regime where an efficient quantum5lgorithm would be particularly desirable. e. Adiabatic quantum algorithm. In order to define an adiabatic quantum algorithmfor EC3, we need to choose ˆ H P and ˆ H . The problem Hamiltonian ˆ H P for an EC3 instancecan be obtained from the above cost function by first replacing x i by the Ising variables σ ( i ) z = 1 − x i = ± σ ( i ) z by the Pauli Z operators ˆ σ ( i ) z , thus replacingthe bits by qubits. The problem Hamiltonian becomesˆ H P = M ˆ I − N (cid:88) i =1 B i ˆ σ ( i ) z + 14 N (cid:88) i,j =1 J ij ˆ σ ( i ) z ˆ σ ( j ) z , (2)where B i is the number of clauses that involve the bit i , J ij is the number of clauses wherethe bits i and j participate together, and ˆ I is the identity operator. For ˆ H , we makethe conventional choice ˆ H = − (cid:80) i ˆ σ ( i ) x , which corresponds to spins in the magnetic fielddirected along x -axis (Pauli X operators). For us it will also be convenient to modify theHamiltonian ˆ H ( s ) as ˆ H QC ( λ ) = ˆ H P + λ ˆ H . The parameter λ = − ss changes adiabaticallyfrom λ = + ∞ at the beginning t = 0 to λ = 0 at t = T . f. Connection to Anderson Localization. We can now see the relevance of AL to thequantum system described by ˆ H QC . Note that this Hamiltonian also describes a single quan-tum particle that is moving between the vertices of an N -dimensional hypercube. Indeed,each vector σ = ( σ (1) z , σ (2) z , . . . , σ ( N ) z ), where σ ( i ) z = ±
1, determines a vertex of the hyper-cube, which is body-centered at the origin of the N -dimensional space. Let | σ (cid:105) denote thequantum state of a particle localized at a site σ . The full set of these states forms a basis,in which the first term of the Hamiltonian is diagonal, while the second one describes ahopping of this fictitious particle between the nearest neighbors (n.n)ˆ H QC ( λ ) = (cid:88) σ E P ( σ ) | σ (cid:105)(cid:104) σ | (cid:124) (cid:123)(cid:122) (cid:125) disorder + λ (cid:88) σ , σ (cid:48) n.n | σ (cid:105)(cid:104) σ (cid:48) | . (3)Each on-site energy E P ( σ ) is nothing but the cost function f ( x ) of the corresponding assign-ment σ . For random instances, the on-site energies are obviously also random, introducingdisorder in the Hamiltonian. Hence, Eq. (3) describes the well known Anderson model,which was used to demonstrate the phenomenon of localization [19]. The only differencefrom more familiar situations is that lattices in d -dimensional space, which have L d siteswhere L (cid:29) N -dimensional hypercube with 2 N sites, where N (cid:29)
1. 6
IG. 2: Schematic representation of the creation of a level anti-crossing. (a) Before adding theclause, we have two assignments which are both in the ground state at λ = 0 but due to the no-crossing rule, at λ > E ( λ ∗ ) − E ( λ ∗ ) >
4. (b) By adding a clause satisfied by solution1 but not solution 2, we create a level anti-crossing since ˜ E (0) < ˜ E (0) but ˜ E ( λ ∗ ) > ˜ E ( λ ∗ ).Insets: (a) If the clause is violated by the wrong solution, then no anti-crossing appears betweenthese two levels. (b) However, other low energy levels can create other anti-crossings, leading tomultiple small gaps. g. Anti-crossings in AQO. Now we are ready to discuss the fundamental difficultieswhich AQO faces. We will show that (i) the anti-crossings of the ground state with thefirst excited state happen with high probability and (ii) that the anti-crossing gaps in thelimit N → ∞ are even less than exponentially small. Let us start with the first statement.An EC3 instance with α < α s typically has several solutions σ with E P ( σ ) = 0. If α isclose to α s there are few solutions at a distance of order N of each other. The presenceof multiple solutions imply that the ground state of ˆ H QC ( λ = 0) = ˆ H P is degenerate, butthis does not contradict the non-crossing rule: ˆ H P commutes with each of the operatorsˆ σ ( i ) z , so it satisfies a special symmetry which is broken for λ >
0. Consider now a particularinstance with M − σ and σ that are separated by n ∼ N spin flips. When λ adiabatically changes from zero to a small but finite value the solutionsevolve into eigenstates of the Hamiltonian, | Ψ , λ (cid:105) and | Ψ , λ (cid:105) with the energies E ( λ ) and E ( λ ). According to the non-crossing rule, a degeneracy of these two states at a finite λ isimprobable, i.e., the ˆ H term in ˆ H QC splits the ground state degeneracy. This situation issketched in Fig. 2(a). Suppose that E ( λ ) < E ( λ ), i.e. | Ψ , λ (cid:105) is the unique ground state7f the Hamiltonian ˆ H QC ( λ ). If we now add one more clause to the existing M − x i M + x j M + x k M − to the cost function leading to Hamiltonian ˆ H P , both | σ (cid:105) and | σ (cid:105) remain eigenstates, but their eigenenergy can increase by either 1 or 4. Witha non-zero probability the last clause is satisfied by σ but not by σ , i.e., ˜ E P ( σ ) = 0 while˜ E P ( σ ) >
0, where ˜ E P ( σ ) is the cost function of the new instance. Accordingly | σ (cid:105) ratherthan | σ (cid:105) is the new ground state at λ = 0. At the same time | Ψ , λ (cid:105) can still remain theground state at large enough λ if ˜ E ( λ ) > ˜ E ( λ ), as shown on Fig. 2(b). Such a situationcorresponds to the anti-crossing of | Ψ , λ (cid:105) and | Ψ , λ (cid:105) at certain λ , as previously describedin Fig. 1. Note that the addition of a clause to the cost function increases any eigenenergyof ˆ H QC ( λ ) by less than 4. To satisfy the condition ˜ E ( λ ) > ˜ E ( λ ), it is thus sufficient toachieve a large enough splitting between the eigenvalues of the instance with M − E ( λ ) − E ( λ ) >
4. It turns out that if N (cid:29)
1, this happens when λ is small and one canuse perturbation theory in λ . h. Perturbation theory. To demonstrate this, consider the eigenstate which in the limit λ → | σ (cid:105) . At small λ its energy can be expanded in a series E ( λ, σ ) = E P ( σ ) + ∞ (cid:88) m =1 λ m F ( m ) ( σ ) . (4)We can show that each term in this sum scales linearly in N . For the energy E P ( σ ) ofan arbitrary assignment, we immediately have that 0 ≤ E P < M = αN . As for thecoefficients F ( m ) ( σ ), the cluster expansion [22] of the Hamiltonian ˆ H QC implies that theymay be expressed as a sum of ∼ N statistically independent terms, each being of order 1.The key element to prove this is that since M/N = α is constant, with high probability eachbit participates in a finite number of clauses as N → ∞ . As a result, all the coefficients B i and J ij in Eq. (2) are also finite: B i = (cid:80) j J ij = O (1). In particular, when σ is a solution weobtain F (1) ( σ ) = (cid:80) i B − i , which is therefore of order N . This statement is valid for F ( m ) ( σ )with arbitrary finite m >
1: all of them can be presented as a finite sum of O ( N ) randomterms, each one being of order unity. Let us now consider the perturbative expansion forthe energy splitting between two solutions. Similarly to Eq. (4), we obtain E ( λ ) − E ( λ ) = ∞ (cid:88) m =2 λ m F ( m )1 , , (5)where F ( m )1 , = F ( m ) ( σ ) − F ( m ) ( σ ) is a sum of O ( N ) terms of order 1. Each of the terms israndom with a zero mean and hence the sums F ( m )1 , averages to zero if N is large. Therefore,8 IG. 3: Statistics of the square of the difference in energies of two solutions up to fourth orderi.e. ( F (2)1 , ) . 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 , ) . Each data point is obtained from2500 random instances of EC3 with α ≈ .
62. Linear fits for the mean yield f (2) ≈ .
18 and f (3) ≈ . it is ( F ( m )1 , ) rather than F ( m )1 , which is proportional to N . We thus arrive to the conclusionthat | E ( λ ) − E ( λ ) | = √ N (cid:88) m λ m f ( m ) , (6)where the coefficients f ( m ) = O (1) can be evaluated by the cluster expansion [22]. We haveseen that F (1) ( σ ) = (cid:80) i B − i for any solution σ , so that F (1)1 , = 0. However terms with m > F (2)1 , ) and ( F (3)1 , ) , with linear fitsconfirming their scaling O ( N ). For small λ , we can restrict ourselves to the leading term( m = 2) in Eq. (6). Accordingly in the N → ∞ limit, the splitting | E ( λ ) − E ( λ ) | exceeds4 as long as λ > λ ∗ , with λ ∗ = √ f (2) ) − / N − / , (7)and λ ∗ (cid:28) λ ∗ (cid:28) M clauses is finite provided that λ ≥ λ ∗ ∼ N − / . Howbig is the gap ∆ of such an anti-crossing? As explained above, we can evaluate the gap by9onsidering the matrix element V between the states | Ψ , λ (cid:105) and | Ψ , λ (cid:105) corresponding tothe two assignments, at the value λ where the anti-crossing occurs. Note that if the twoassignments σ and σ satisfying the ( M −
1) clauses are separated by a distance (numberof flips) n , this matrix element only appears at the n -th order of the perturbation theory,i.e. it is proportional to λ n : V = λ n (cid:88) tr (cid:16) Π nk =1 E P ( σ ( k ) tr ) (cid:17) − + O ( λ n +1 ) (8)where the sum is over all ”trajectories” tr - all possible orders of the n spin flips neededto transform σ into σ , σ ( k ) tr is the assignment along a particular trajectory that appearsafter k flips and E P ( σ ( k ) tr ) is the cost function of this assignment. Therefore we can estimatethe matrix element and thus the anti-crossing gap as V < w ( n ) λ n . The prefactor w ( n )reflects the fact that many ( ∼ n !) trajectories contribute to the sum in Eq. (8). For atypical trajectory E P ( σ ( k ) tr ) = O ( k ) for k < n/ E P ( σ ( k ) tr ) = O ( n/ − k ) for k >n/
2. As a result the product of E P ( σ ( k ) tr ) in Eq. (8) is also ∼ n !. The factorials thuscancel each other and w ( n ) can not increase faster than A n with some constant A ∼ V < ( 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 )], where N = 16 A ( f (2) ) − = O (1). We can estimate the distance n between the assignments as v ( α ) N , where v ( α ) ≈ (4 / − exp( − α )), and obtain the final form of the minimal gapestimation ∆ min ∼ exp[( − v ( α ) N/
8) ln(
N/N )] . (9)One can see that as N → ∞ , the gap indeed decreases even faster than an exponential -statement (ii). This implies that the adiabatic computation time exceeds exp( N ). In Fig. 4,we have plotted an anti-crossing for a particular instance with N = 200 generated duringour numerical simulations. The figure shows two energy levels (estimated by fourth orderperturbation theory) corresponding to assignments separated by 60 bit flips, and crossingat λ ≈ . i. Applicability of the perturbation theory. Our main result - the estimation of theminimal gap (Eq. (9)), is based on the perturbative expansion for the energies (Eq. (4)) andthe matrix element V (Eq. (8)). Is the perturbation theory in λ always applicable? At firstsight Eq. (8) becomes meaningless if E P = 0 for any of the intermediate assignments σ ( k ) tr .In this case there is an avoided crossing between the states corresponding to the assignments10 IG. 4: Simulation of a level anti-crossing for a random instance with N = 200 bits and α ≈ . E − E and E − E . Thecrossing occurs at λ ≈ .
51, and the corresponding assignments are at distance n = 60 from eachother. σ and σ ( k ) tr (such as in Eq. (1)) and formally perturbation theory fails in the vicinity of thisanti-crossing point. This apparent difficulty can be overcome by considering only a finitetime T for the evolution. This is equivalent to adding imaginary parts iη ≈ i/T to theenergies. For the AQO algorithm, it is the computation time T that determines η . Since weare considering the N → ∞ limit, we have that T → ∞ and thus η →
0. This is the limitthat was shown to be relevant for the localization problem [19, 23]. The celebrated discoveryof Anderson was that if the limit η → N ) tends to infinity,and λ is small enough i.e., λ < λ cr , the spectrum of the Hamiltonian described in Eq. (3)remains discrete (all states are localized) and thus the second term in Eq. (3) (the kineticenergy term) can be treated perturbatively. As soon as λ > λ cr , there appears a strip ofextended states in the middle of the energy band which widens as λ increases further. Stateswithin this strip are not perturbative because the number of the trajectories connecting twopoints in a d -dimensional space (for finite d ) increases exponentially with distance. Thelarge number of terms in the expansions like Eq. (8) overwhelms the smallness of λ n and11he perturbation series thus diverges for λ > λ cr . For a d -dimensional space, the criticalvalue λ cr is believed to be (in our units) of the order of λ cr ∼ / log d [24, 25]. We have seenthat the AQO algorithm for problems like EC3 can be mapped to the Anderson model onan N -dimensional hypercube. Then, the number of trajectories increases with the length n as n ! ∼ n n e − n , i.e. even faster than an exponential. However, as we already mentioned, the n n factor cancels with the same factor in the products of the energy in the denominatorsof Eq. (8). Accordingly, λ cr can still be estimated as λ cr ∼ / log N , which, together withEq. (7), implies that anti-crossings appear for λ ∗ (cid:28) λ cr when N (cid:29)
1. Moreover, at λ < λ cr all of the states are supposed to be localized. The AQO algorithm involves only low energystates, which remain localized much longer than the middle-band states with the energies ∼ N . Therefore, it is quite likely that the exponentially small gaps appear even at λ ∼ j. Conclusions. We finish our discussion with the following observation. We monitoredtwo assignments that satisfied M − λ . Of course, for randomly selected clauses this happens only with a finiteprobability and the situation sketched in the inset in Fig. 2(a) is also possible. One couldthus hope [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 conservative limitation onthe perturbative approach λ cr ∼ / log N and consider the spectrum at λ ∗ (cid:28) λ cr ∼ / log N .According to Eq. (6) all states in the energy interval [0 , (cid:15) ] with (cid:15) ∼ √ N λ (cid:29) λ = 0. This means that typically the groundstate undergoes ν ( (cid:15) ) anti-crossings (participates in ν ( (cid:15) ) anti-crossing gaps) as the parameterevolves from 0 to λ (see the inset of Fig. 2(b)). Here ν ( (cid:15) ) is the number of states, whoseenergies at the given λ differ from the ground state energy by less than (cid:15) . Taking intoaccount that ν ( (cid:15) ) increases with (cid:15) exponentially and that the probability to completelyavoid anti-crossings (the probability to have a gap of size (cid:15) separating the ground state fromthe rest of the spectrum) is exponentially small in ν ( (cid:15) ) we conclude that this probability isindeed negligible. Therefore, these findings suggest that there is no chance of obtaining thesolution of the problem in polynomial time using the AQO algorithm for random instancesof the Exact Cover 3 problem. We also believe that the methods described in this articlecan be applied to other similar NP-complete problems, such as 3-SAT.12 cknowledgments We thank A. Childs, E. Farhi, J. Goldstone, S. Gutmann, M. R¨otteler and A. P. Youngfor interesting discussions. We thank the
High Availability Grid Storage department of NECLaboratories America for giving us access to their cloud to run our numerical simulations.This research was supported in part by US DOE contract No. AC0206CH11357. [1] M. R. Garey and D. S. Johnson,
Computers and Intractability: A Guide to the Theory ofNP-Completeness (W. H. Freeman & Co., New York, NY, USA, 1979).[2] S. Arora and B. Barak,
Computational Complexity: A Modern Approach (Cambridge Univer-sity Press, 2009).[3] P. W. Shor, SIAM Journal on Computing , 1484 (1997).[4] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser (2000), e-print arXiv:quant-ph/0001106.[5] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science , 472(2001).[6] A. Messiah, M´ecanique Quantique (Dunod, Paris, 1959).[7] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, and S. Lloyd, SIAM Journal on Computing , 166 (2007).[8] M. ˇZnidariˇc and M. Horvat, Physical Review A , 022329 (2006).[9] E. Farhi, J. Goldstone, S. Gutmann, and D. Nagaj, International Journal of Quantum Infor-mation , 503 (2008).[10] W. van Dam, M. Mosca, and U. Vazirani, in Proceedings of the 42nd Annual IEEE Symposiumon the Foundations of Computer Science (IEEE, New York, 2001), pp. 279–287.[11] W. van Dam and U. Vazirani (2003), unpublished manuscript.[12] B. Reichardt, in
Proceedings of the 36th Annual ACM Symposium on Theory of Computing (IEEE, New York, 2004), pp. 279–287.[13] M. H. S. Amin and V. Choi (2009), e-print arXiv:0904.1387.[14] E. Farhi, J. Goldstone, D. Gosset, S. Gutmann, H. B. Meyer, and P. Shor (2009), e-printarXiv:0909.4766.[15] T. Hogg, Physical Review A , 022314 (2003).
16] M. C. Ba˜nuls, R. Or´us, J. I. Latorre, A. P´erez, and P. Ruiz-Femen´ıa, Physical Review A ,022344 (2006).[17] A. P. Young, S. Knysh, and V. N. Smelyanskiy, Physical Review Letters , 170503 (2008).[18] J. von Neuman and E. Wigner, Zeitschrift f¨ur Physik , 467 (1929).[19] P. W. Anderson, Physical Review , 1492 (1958).[20] G. Biroli, R. Monasson, and M. Weigt, European Physical Journal B , 551 (2000).[21] J. Raymond, A. Sportiello, and L. Zdeborov´a, Physical Review E , 011101 (2007).[22] G. E. Mayer and M. Goeppert-Mayer, Statistical Mechanics (Wiley, New York, 1948).[23] P. W. Anderson,
Local Moments and Localized States (World Scientific, 1992), Nobel Lectures,Physics 1971-1980.[24] K. Efetov, Soviet Physics - Journal of Experimental and Theoretical Physics , 199 (1988).[25] R. Abou-Chacra, D. J. Thouless, and P. W. Anderson, Journal of Physics C , 1734 (1972)., 1734 (1972).