Instance Independence of Single Layer Quantum Approximate Optimization Algorithm on Mixed-Spin Models at Infinite Size
aa r X i v : . [ qu a n t - ph ] F e b Instance Independence of Single Layer Quantum Approximate Optimization Algorithmon Mixed-Spin Models at Infinite Size
Jahan Claes
1, 2 and Wim van Dam
1, 3, 4 QC Ware Corporation, Palo Alto, CA USA Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Computer Science, University of California, Santa Barbara, CA USA Department of Physics, University of California, Santa Barbara, CA USA (Dated: February 25, 2021)This paper studies the application of the Quantum Approximate Optimization Algorithm (QAOA)to spin-glass models with random multi-body couplings in the limit of a large number of spins.We show that for such mixed-spin models the performance of depth 1 QAOA is independent ofthe specific instance in the limit of infinite sized systems and we give an explicit formula for theexpected performance. We also give explicit expressions for the higher moments of the expectedenergy, thereby proving that the expected performance of QAOA concentrates.
CONTENTS
I. Introduction 1II. Preliminaries and Notation 2III. Current Results and Prior Work 3A. Our Results on Mixed-Spin Sherrington-Kirkpatrick Model 3B. Relation to Prior Results 4IV. Derivation of Main Result 5A. Moment Generating Function 5B. Expectation when Couplings are Normal Distributed Variables 6C. General Form of Moments 7D. Evaluating Sums Over the Sketches ( n ++ , n + − , n − + , n −− ) 8E. Large n Limit 8F. Properties of f q and g q I. INTRODUCTION
The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm designed to giveapproximate solutions to optimization problems [1]. While QAOA can be proven to give the optimal answer in the limitwhere the number of QAOA layers p goes to infinity, rigorous results on the performance of QAOA with finite p aredifficult to obtain. In a recent paper, Farhi et al. [2] studied the application of the QAOA to the Sherrington-Kirkpatrick(SK) model, a spin-glass model with random all-to-all two-body couplings, in the limit of a large number of spins. Theydemonstrated that for fixed p , the performance of the QAOA is independent of the specific instance of the SK model andcan be predicted by explicit formulas. They demonstrate that the approximation ratio of the QAOA at p = 11 outperformsmany classical algorithms (although not the best classical algorithm [3]). In this work, we generalize the result of Farhiet al. to mixed-spin SK models, which allow for random all-to-all q -body couplings. We demonstrate that for p = 1,the performance of the QAOA is again independent of the specific instance, and we provide an explicit formula for theexpected performance. Our work provides a potential avenue to demonstrating the advantage of QAOA over classicallgorithms, as classical algorithms for mixed-spin SK models have an approximation ratio that is bounded away from 1[4, 5]. II. PRELIMINARIES AND NOTATION
The Quantum Approximate Optimization Algorithm (QAOA) [1] is a heuristic quantum algorithm for binary optimiza-tion. Given a cost function of n binary variables (spins) H ( z , . . . , z n ), QAOA seeks to produce a string z := ( z , . . . , z n )close to the minimum of H . We can view H as a Hamiltonian operator that is diagonal in the Z -basis. A depth- p QAOA circuit then consists of p repetitions of alternatively applying the Hamiltonian H and the mixing Hamiltonian B = X + · · · + X n to a uniform superposition as initial state, that is the product of + X single particle eigenstates.Explicitly, the depth- p QAOA state is given by | β , . . . , β p ; γ , . . . , γ p i := e − iβ p B e − iγ p H · · · e − iβ B e − iγ H · √ n X z ∈{± } n | z i . (1)A depth- p QAOA circuit is parameterized by the 2 p angles { γ i } , { β i } . For a given problem, these angles should beoptimized so that measuring in the Z -basis gives strings that make H as small as possible. In practice, this is typicallydone by minimizing the expectation value of the energy h H i := h β , . . . , β p ; γ , . . . , γ p | H | β , . . . , β p ; γ , . . . , γ p i . (2)For some problems, this minimization may be done analytically on a classical computer [1, 6–8]. Otherwise, the mini-mization can be performed by running the QAOA on a quantum computer repeatedly for a fixed set of angles, estimatingthe expectation value, and updating the angles according to classical minimization algorithms [1, 9, 10]. We note thatminimizing the energy expectation value is only one possible definition of “best” angles; in general minimizing the ex-pectation value and maximizing the probability of finding the optimal z (or maximizing the probability of H ( z , . . . , z n )falling below a certain threshold) do not coincide.It was recently demonstrated in [11] that in local optimization problems with cost functions drawn from from realisticrandom distributions (e.g. MAXCUT on random 3-regular graphs), the expectation value per spin is instance independent as n → ∞ . That is, for fixed angles { γ i } and { β i } h H/n i is the same for all problem instances. This implies that theangles { γ i } and { β i } do not need to be optimized for every problem instance, but can be optimized once and reused forevery problem drawn from the same distribution. The methods of [11] can also be used to derive concentration of measure results for local optimization problems. (While [11] did not explicitly address concentration of measure, it can be easilyderived from their methodology.) That is, in the limit as n → ∞ , the variance in the energy per spin goes to zero: h ( H/n ) i − h ( H/n ) i → n , every measurement of a QAOA state in the Z -basis gives strings withthe same energy per spin. In total, instance independence implies the QAOA angles do not be to be optimized frominstance to instance in order to minimize the expectation value, and concentration of measure implies that expectationvalue is the correct measure of the “best” angles.While instance independence (and concentration of measure) were initially derived for local cost functions, similarresults have also been derived for the Sherrington-Kirkpatrick (SK) model [12], a physics-inspired optimization problemwith cost function H = X ≤ j
1. Therefore, the QAOA angles for the SK model can be chosen on a classicalcomputer, and there are fixed performance guarantees in the limit n → ∞ . Ref. [3] demonstrated that at p = 11, QAOAoutperforms semidefinite programming for the SK model, but could not show that the QAOA matches the performanceof the Montanari algorithm. 2n this work, we study a generalization of the SK model, the mixed-spin SK model, that allows for polynomials of degree d in the binary variables instead of only degree-2 terms [15, 16]. This can serve as a model for nonlocal optimizationproblems with higher-order terms. The mixed-spin SK model is also known to have a ground state energy per spin thatis independent of the instance and can be computed explicitly [15, 17]. For a mixed-spin SK model with degree d = 3,the generalization of the Montanari algorithm [4] approaches a fixed approximation ratio of ∼ (0 . ± . p = 1QAOA applied to mixed-spin SK models. As part of our work, we derive an explicit formula for h H/n i in the limit n → ∞ ,implying that the QAOA angles for the mixed-spin SK models can be chosen on a classical computer. Our work can likelybe generalized to depth p ≥ III. CURRENT RESULTS AND PRIOR WORKA. Our Results on Mixed-Spin Sherrington-Kirkpatrick Model
The mixed-spin SK model (often called the mixed p -spin model, although we will not use this terminology to avoidconfusion with the p of QAOA) is given by [15, 16] H = n X j =1 J j Z j + 1 √ n X ≤ j
To put Equation 7 in perspective we explicitly calculate what it implies for two basic cases q = 2 and q = 3 and compareit with the literature on mixed-spin models.The standard Sherrington-Kirkpatrick model has q = 2, σ = 1, and σ q = 0 for all other q = 2. In this case ourEquation 7 tells us that E J ∼ N [ h H SK /n i ] = 2 γe − γ sin(2 β ) cos(2 β ) = γe − γ sin(4 β ) , (11)which agrees with previous work in [2] and [8].For the more general case q ≥ H = X q =1 σ q √ n q − X ≤ j < ··· 1) and for a fixed q the summation has (cid:0) nq (cid:1) terms. In [5] and elsewhere the different notation is usedwhere the Hamiltonian is defined by H ′ = X q =1 c q √ n q − X ≤ j ,...,j q ≤ n J j ··· j q Z j · · · Z j q , (13)with again J j ··· j q ∼ N (0 , n q terms and hence the indices j , . . . , j q can haverepeated values and permutations are treated distinctly.In the limit of large n one can show that the q -plets ( j , . . . j q ) with repeated values can be ignored as their relativecontribution decreases as a function n . For distinct indices there are q ! permutations to consider in H ′ , hence we havea sum of q ! standard normal distributions: J ′ j ...j q + · · · + J ′ j q ··· j , which is identical to a single normal distribution withvariance q !. As a result we re-express H ′ as H ′ = X q =1 c q √ q ! √ n q − X ≤ j < ··· 2, such that c = 1 / √ 2. It was shown that theexpected ground state energy-per-spin of this model equals − . ± . σ = √ E J ∼ N [ h H /n i ] = 3 γe − γ sin(2 β ) cos (2 β ) − γe − γ sin (2 β ) . (15)Numerical computations tell us that this expectation will be minimized by the angles β ≈ . γ ≈ − . E J [ h H /n i ] ≈ − . p = 1 QAOA can approximate the ground state energy bya factor of 0 . . 33 approximation factor had been reported earlier by Zhou et al.[19]4he proof in the current paper follows to large extent the framework of the earlier result by Farhi et al. [2], which reliedon manipulating the moment generating function E J (cid:10) e iλH/n i (cid:3) to extract expressions for the first and second momentsof ( H/n ). We use their method of simplifying the moment generating function, and their reorganization of the sum over z -strings into a sum over sketches (see Section IV B). We extend their proof technique by generalizing their form of themoment generating function to higher-spin models and demonstrating that it can still be written as a sum over sketches,developing careful power-counting methods to allow us to extract the relevant terms in the n → ∞ limit, and derivingidentities that allow us to explicitly evaluate the relevant sums. IV. DERIVATION OF MAIN RESULT In what follows, we will use the following conventions: • A Z -basis state | z i is specified by a string z = ( z , z , . . . , z n ) of ± z ∈ {± } n for this. • The XOR of two bitstrings z and z ′ is given by the componentwise product zz ′ := ( z z ′ , z z ′ , . . . , z n z ′ n ) ∈ {± } n . • For a set S ⊆ { , . . . , n } , we denote the product of the bits in S as Q i ∈ S z i =: z S ; with this convention we thus have Z S | z i = z S | z i . • The uniform superposition over all strings z is denoted by | Σ i := 1 √ n X z ∈{± } n | z i . (16)This is in contrast to the usual convention in quantum information, in which a Z -basis state is specified by a string z of 0s and 1s and the XOR of two strings is given by componentwise addition modulo 2. We choose our notation to beconsistent with [2] and to simplify certain expressions in our derivation. A. Moment Generating Function Following [2], to evaluate E J [ h H/n i ] and E J [ h ( H/n ) i ] we use the moment-generating function E J [ h e iλH/n i ]. From themoment-generating function, we can find the moments via E J ∼ N [ h ( H/n ) m i ] = ( − i∂ λ ) m E J ∼ N hD e iλH/n Ei(cid:12)(cid:12)(cid:12) λ =0 (17)We can simplify the expectation inside the moment-generating function by inserting three complete sets of states: h e iλH/n i = h Σ | e iγH e iβB e iλH/n e − iβB e − iγH | Σ i (18)= X z,z ′ ,z ′′ ∈{± } n h Σ | z ih z | e iγH e iβB | z ′′ ih z ′′ | e iλH/n e − iβB e − iγH | z ′ ih z ′ | Σ i (19)= 12 n X z,z ′ ,z ′′ ∈{± } n exp (cid:18) iγ [ H ( z ) − H ( z ′ )] + i λH ( z ′′ ) n (cid:19) h z | e iβB | z ′′ ih z ′′ | e − iβB | z ′ i (20)= 12 n X z,z ′ ,z ′′ ∈{± } n exp i d X q =1 n − q X | S | = q J S (cid:20) γ ( z S − z ′ S ) + λz ′′ S n (cid:21) h z | e iβB | z ′′ ih z ′′ | e − iβB | z ′ i (21)= 12 n X z,z ′ ,z ′′ ∈{± } n exp i d X q =1 n − q X | S | = q z ′′ S J S (cid:20) γ ( z S − z ′ S ) + λn (cid:21) h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i , (22)where in the last step we used that h z | e iβB | z ′ i = h zz ′ | e iβB | (+1) n i and made the replacements z zz ′′ and z ′ z ′ z ′′ .5 . Expectation when Couplings are Normal Distributed Variables We will now treat the J S couplings as a random variable and consider the expectation E J of the energy. We assumethat the distribution is symmetric with respect to +1 ↔ − 1, such that we can replace z ′′ S J S by J S to get E J ∼ N h h e iλH/n i i = 12 n X z,z ′ ,z ′′ ∈{± } n E J ∼ N exp i d X q =1 n − q X | S | = q z ′′ S J S (cid:20) γ ( z S − z ′ S ) + λn (cid:21) h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i (23)= X z,z ′ ∈{± } n E J ∼ N exp i d X q =1 n − q X | S | = q J S (cid:20) γ ( z S − z ′ S ) + λn (cid:21) h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i . (24)When we further assume that the J S variables are independent between different S we can continue by E J ∼ N h h e iλH/n i i = X z,z ′ ∈{± } n d Y q =1 Y | S | = q E J S (cid:20) exp (cid:18) i · n − q J S (cid:20) γ ( z S − z ′ S ) + λn (cid:21)(cid:19)(cid:21) h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i . (25)Next we assume that the J S are normally distributed with a standard deviation that is the same for all sets S of thesame size, that is J S ∼ N (0 , σ | S | ). We note that taking the expectation value of a Gaussian random variable J withstandard deviation σ gives E J ∼ N (0 ,σ ) (cid:2) e icJ (cid:3) = 1 σ √ π Z e − J / σ + icJ = e − c σ / (26)so that our overall expression becomes E J ∼ N h h e iλH/n i i = X z,z ′ ∈{± } n exp − d X q =1 σ q n q − X | S | = q (cid:20) γ ( z S − z ′ S ) + 2 γλ ( z S − z ′ S ) n + λ n (cid:21) h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i . (27)To do the sum over z, z ′ , we claim that the summand does not depend on all 2 n spin values of z and z ′ . Instead, itis only a function of the four integer values ( n ++ , n + − , n − + , n −− ), where n ss ′ is defined to be the number of positions k ∈ { , . . . , n } with ( z k , z ′ k ) = ( s, s ′ ). Note that only three of these variables are actually independent, as we alwayshave n ++ + n + − + n − + + n −− = n . As these numbers summarize the crucial information of the strings, we will referto ( n ++ , n + − , n − + , n −− ) as the sketch of ( z, z ′ ). Writing the summand in terms of the sketch rather than ( z, z ′ ) wasintroduced in [2]; here we establish that we can still write the summand in terms of the sketch for the mixed-spin SKmodel. To start, it is straightforward to verify that h z | e iβB | (+1) n ih (+1) n | e − iβB | z ′ i = Y ss ′ ∈{±} Q n ss ′ ss ′ , with Q ++ := cos ( β ) Q −− := sin ( β ) Q − + = − Q + − := i sin( β ) cos( β ) . (28)6e can also write explicit combinatorial formulas for the sums in the exponential: X | S | = q ( z S − z ′ S ) = q X a =0 ( − q − a (cid:20)(cid:18) n ++ + n + − a (cid:19) (cid:18) n − + + n −− q − a (cid:19) − (cid:18) n ++ + n − + a (cid:19) (cid:18) n + − + n −− q − a (cid:19)(cid:21) (29)= q X a =0 ( − q − a (cid:20)(cid:18) n +( n + − − n − + )+( n ++ − n −− )2 a (cid:19) (cid:18) n − ( n + − − n − + ) − ( n ++ − n −− )2 q − a (cid:19) − (cid:18) n − ( n + − − n − + )+( n ++ − n −− )2 a (cid:19) (cid:18) n +( n + − − n − + ) − ( n ++ − n −− )2 q − a (cid:19)(cid:21) (30)=: f q ( n + − − n − + , n ++ − n −− ) (31) X | S | = q ( z S − z ′ S ) = 4 X a odd (cid:18) n + − + n − + a (cid:19) (cid:18) n ++ + n −− q − a (cid:19) (32)= 4 X a odd (cid:18) n + − + n − + a (cid:19) (cid:18) n − ( n + − + n − + ) q − a (cid:19) (33)=: g q ( n + − + n − + ) . (34)Therefore, the summand indeed depends only on ( n ++ , n + − , n − + , n −− ) and the number of ways to assign the n positionsinto four groups of these sizes is the multinomial n ! / ( n ++ ! n + − ! n − + ! n −− !). To condense our notation we will use { n ∗ } todenote the set of sketches ( n ++ , n + − , n − + , n −− ), allowing us to use the shorthand X { n ∗ } (cid:18) nn ∗ (cid:19) F ( n ∗ ) := X ( n ++ ,n + − ,n − + ,n −− ) ∈ N n ++ + n + − + n − + + n −− = n (cid:18) nn ++ , n + − , n − + , n −− (cid:19) F ( n ++ , n + − , n − + , n −− ) . (35)Note that this summation has (cid:0) n +33 (cid:1) terms. We thus have E J ∼ N h h e iλH/n i i = X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q =1 (cid:20) γ g q ( n ∗ ) + 2 γλ f q ( n ∗ ) n + λ ( nq ) n (cid:21) σ q n q − ! Y ss ′ ∈{±} Q n ss ′ ss ′ . (36)Eq. 36 is the form of the moment-generating function we will use to evaluate E J [ h H/n i ] and E J (cid:2) h ( H/n ) i (cid:3) . C. General Form of Moments Using Eq. 17 combined with the form of the moment-generating function given in Eq. 36, we can write the first momentas: E J ∼ N [ h H/n i ] = iγ d X q =1 σ q n q X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q =1 γ g q ( n ∗ ) σ q n q − ! f q ( n ∗ ) Y ss ′ ∈{±} Q n ss ′ ss ′ , (37)and the second moments as: E J ∼ N (cid:2) h ( H/n ) i (cid:3) = d X q =1 (cid:18) nq (cid:19) σ q n q +1 − γ d X q,q ′ =1 σ q σ q ′ n q + q ′ X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q ′′ =1 γ g q ′′ ( n ∗ ) σ q ′′ n q ′′ − f q ( n ∗ ) f q ′ ( n ∗ ) Y ss ′ ∈{±} Q n ss ′ ss ′ . (38)The explicit expression for f q (Eq. 31) shows that f q is a degree- q polynomial in the variables ( n + − − n − + ), ( n ++ − n −− ),and n , so we can expand f q as f q ( n ∗ ) = X a + b + c ≤ q f abcq ( n + − − n − + ) a ( n ++ − n −− ) b n c (39)where the f abcq are constants independent of n . In terms of this expansion, we have E J ∼ N [ h H/n i ] = iγ d X q =1 σ q X a + b + c ≤ q f abcq n q − c X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q ′ =1 g q ′ ( n ∗ ) γ σ q ′ n q ′ − ( n + − − n − + ) a ( n ++ − n −− ) b Y ss ′ ∈{±} Q n ss ′ ss ′ (40)7nd E J ∼ N (cid:2) h ( H/n ) i (cid:3) = d X q =1 (cid:18) nq (cid:19) σ q n q +1 − γ d X q,q ′ =1 σ q σ q ′ X a + b + c ≤ qa ′ + b ′ + c ′ ≤ q ′ f abcq n q − c f a ′ b ′ c ′ q ′ n q ′ − c ′ X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q ′′ =1 g q ′′ ( n ∗ ) γ σ q ′′ n q ′′ − (41) × ( n + − − n − + ) a + a ′ ( n ++ − n −− ) b + b ′ Y ss ′ ∈{±} Q n ss ′ ss ′ . Ref. [2] could explicitly evaluate these terms for the small values of a and b relevant for the two-body SK model, usingconcise expressions for f and g . However, to get explicit formulas beyond q = 2 requires carefully counting powers of n to establish which terms survive in the n → ∞ limit, and using the general expressions for f q and g q (Eqs. 31 and 34)to derive explicit forms of the leading-order terms. Our derivation thus goes beyond a simple generalization of [2] in thetechniques we must use to tame this sum. D. Evaluating Sums Over the Sketches ( n ++ , n + − , n − + , n −− ) We see that evaluating both moments reduces to repeatedly evaluating terms of the form T abξ := 1 n ξ X { n ∗ } (cid:18) nn ∗ (cid:19) exp − d X q =1 γ g q ( n + − + n − + ) σ q n q − ! ( n + − − n − + ) a ( n ++ − n −− ) b Y ss ′ ∈{±} Q n ss ′ ss ′ . (42)We can organize the sum over { n ∗ } by first summing over the variable t = ( n + − + n − + ) and then summing over theremaining degrees of freedom n + − and n ++ . In this case, the term becomes T abξ = 1 n ξ n X t =0 (cid:18) nt (cid:19) exp − d X q =1 γ g q ( t ) σ q n q − ! X n + − + n − + = t (cid:18) tn + − (cid:19) ( n + − − n − + ) a Q n + − + − Q n − + − + | {z } A t (43) × X n ++ + n −− = n − t (cid:18) n − tn ++ (cid:19) ( n ++ − n −− ) b Q n ++ ++ Q n −− −− | {z } B t (44)where we have defined A t and B t for later reference. E. Large n Limit In general, the summations for A t and B t can be evaluated exactly via the identity (which is a generalization of theidentities given in [2] for the case of a = 1 , X n + n = u (cid:18) sn (cid:19) ( n − n ) a Q n Q n = ( x∂ x − y∂ y ) a ( x + y ) u (cid:12)(cid:12)(cid:12)(cid:12) x = Q y = Q (45)As we plan to take the limit n → ∞ , we will only keep track of the terms in A t and B t that are relevant in this limit.For A t , we have A t = ( x∂ x − y∂ y ) a ( x + y ) t (cid:12)(cid:12)(cid:12)(cid:12) x = Q + − y = Q − + (46)Noting that ( Q + − + Q − + ) = 0 (see Eq. 28), we see that this term can only be nonzero when a ≥ t . In the particular caseof a = t , the only nonzero terms occur when all a derivatives hit ( x + y ) t , and we have A t = a !( Q + − − Q − + ) a = ( − i ) a sin a (2 β ) . (47)8herefore, in total we find A t = δ a,t a !( − i ) a sin a (2 β ) + · · · (48)where the dots represents terms proportional to some δ t,α with α < a .For B t , we have B t = ( x∂ x − y∂ y ) b ( x + y ) n − t (cid:12)(cid:12)(cid:12)(cid:12) x = Q ++ y = Q −− (49)If we focus on only the highest-order terms in n , these occur when all a derivatives hit the ( x + y ) n − t term. Thus, B t = n b ( Q ++ − Q −− ) b ( Q ++ + Q −− ) n − t − b + O ( n b − ) (50)= n b cos b (2 β ) + O ( n b − ) . (51)Plugging our Eqs. 48 and 51 for A t and B t into Eq. 44 for T abξ gives T abξ = 1 n ξ n X t =0 (cid:18) nt (cid:19) exp − d X q =1 γ g q ( t ) σ q n q − ! [ δ a,t a !( − i ) a sin a (2 β ) + · · · ] (cid:2) n b cos b (2 β ) + O ( n b − ) (cid:3) (52)= 1 n ξ (cid:18) na (cid:19) exp − d X q =1 γ g q ( a ) σ q n q − ! a !( − i ) a sin a (2 β ) (cid:2) n b cos b (2 β ) + O ( n b − ) (cid:3) + · · · (53)(54)where the dots represent terms with t replaced by some α with α < a . We note that in the limit n → ∞ , we have( na ) = n a /a ! + O ( n a − ), while ( na − ) = O ( n a − ). Therefore, we have T abξ = 1 n ξ − a − b ( − i ) a exp − d X q =1 γ g q ( a ) σ q n q − ! sin a (2 β ) cos b (2 β ) + O (cid:18) n ξ − a − b − (cid:19) (55)We see that in the limit as n → ∞ , this becomes T abξ → ( ( − i ) a exp (cid:16) − P dq =1 γ σ q lim n →∞ h g q ( a )2 n q − i(cid:17) sin a (2 β ) cos b (2 β ) , a + b = ξ , a + b < ξ . (56) F. Properties of f q and g q To complete the evaluation of the moments, we need expressions for the relevant terms in f q and g q . While our explicitformulas for f q and g q involve many terms, only a small fraction these terms survive in the n → ∞ limit.To start, we note from Eq. 56 that the only relevant terms in g q are those of degree at least ( q − 1) in n . Starting fromthe definition of g q (Eq. 34), elementary algebra gives g q ( n + − + n − + ) = 4( n + − + n − + ) n q − ( q − O ( n q − ) . (57)Therefore, we have lim n →∞ g q ( a )2 n q − = 2 a ( q − . (58)In addition, we have seen above that the only relevant terms in f q are those f abcq with ( a + b + c ) = q . We can findthese terms starting from the explicit formula for f q (Eq. 31) by keeping only terms of degree q in combined powers of( n + − − n − + ), ( n ++ − n −− ), and n . This gives f q ( n + − − n − + , n ++ − n −− ) = X a odd a ≤ q a !( q − a )! ( n + − − n − + ) a ( n ++ − n −− ) q − a + (lower order terms) . (59)9ote, there are no terms ( n + − − n − + ) a ( n ++ − n −− ) b n c with c = 0 and ( a + b + c ) = q in f q . In terms of the expansionEq. 39 of f q , Eq. 59 says that f abcq = a ! b ! , a + b = q, c = 0 , a odd0 , a + b = q, c = 0 , a even0 , a + b + c = q, c = 0 . (60)With these properties of f q and g q , we now have sufficient information to evaluate our moments. G. Evaluating the First Moment To evaluate the first moment, we simplify Eq. 40 using the definition of T abξ (Eq. 42). Using our explicit expression for T abǫ in the limit n → ∞ (Eq. 56) as well as our expression for the limit of g q as n → ∞ (Eq. 58), we have E J ∼ N [ h H/n i ] = iγ X q σ q X a + b + c ≤ q f abcq T abq − c (61) → iγ X q σ q X a odd a ≤ q a !( q − a )! ( − i ) a exp − γ a X q ′ σ q ′ ( q ′ − sin a (2 β ) cos q − a (2 β ) (62)which is precisely what we claimed in Eq. 7. H. Evaluating the Second Moment To evaluate the second moment, we simplify Eq. 41 using the definition of T abξ (Eq. 42). Again using our explicitexpression for T abǫ in the limit n → ∞ (Eq. 56) and our expression for the limit of g q as n → ∞ (Eq. 58), we have E J ∼ N (cid:2) h ( H/n ) i (cid:3) = d X q =1 (cid:18) nq (cid:19) σ q n q +1 − γ d X q,q ′ =1 σ q σ q ′ X a + b + c ≤ qa ′ + b ′ + c ′ ≤ q ′ f abcq f a ′ b ′ c ′ q ′ T ( a + a ′ )( b + b ′ ) q − c + q ′ − c ′ (63) → − γ d X q,q ′ =1 σ q σ q ′ X a,a ′ odd a ≤ qa ′ ≤ q ′ a !( q − a )! 2 a ′ !( q ′ − a ′ )! ( − i ) a + a ′ × exp − γ ( a + a ′ ) d X q ′′ =1 σ q ′′ ( q ′′ − sin a + a ′ (2 β ) cos q − a + q ′ − a ′ (2 β ) (64)= iγ d X q =1 σ q X a odd a ≤ q a !( q − a )! ( − i ) a exp − γ a d X q ′′ =1 σ q ′′ ( q ′′ − sin a (2 β ) cos q − a (2 β ) (65)= lim n →∞ E J ∼ N [ h H/n i ] (66)which is precisely what we claimed in Eq. 8. I. Evaluating Higher Moments The proof technique used above also applies to higher moments. When computing the m th moment by taking derivativesof the moment-generating function according to Eq. 17, the only terms that survive in the limit n → ∞ are the termswhere all derivatives hit the 2 γλf q /n term in the exponential, so that the expression for the moment becomes E J ∼ N [ h ( H/n ) m i ] = ( iγ ) m d X q ,...,q m =1 σ q · · · σ q m X a + b + c ≤ q ...a m + b m + c m ≤ q m f a b c q · · · f a m b m c m q m T ( a + ··· + a m )( b + ··· + b m )( q − c )+ ··· +( q m − c m ) + O (cid:18) n (cid:19) (67)10oting from Eqs. 56 and 58 that T ( a + ··· + a m )( b + ··· + b m )( q − c )+ ··· +( q m − c m ) = T a b q − c · · · T a m b m q m − c m + O (cid:18) n (cid:19) , (68)we can write the m th moment as E J ∼ N [ h ( H/n ) m i ] = ( iγ ) m d X q ,...,q m =1 σ q · · · σ q m X a + b + c ≤ q ··· a m + b m + c m ≤ q m f a b c q · · · f a m b m c m q m T a b q − c . . . T a m b m q m − c m + O (cid:18) n (cid:19) (69)= iγ d X q =1 σ q X a + b + c ≤ q f abcq T aq − c m + O (cid:18) n (cid:19) (70)= E J ∼ N [ h H/n i ] m + O (cid:18) n (cid:19) . (71)Thus, we find that for all m , lim n →∞ E J ∼ N [ h ( H/n ) m i ] = lim n →∞ E J ∼ N [ h H/n i ] m . (72) V. DISCUSSION AND CONCLUSION In this work, we have derived explicit formulas to quantify the performance of p = 1 QAOA on mixed-spin models inthe n → ∞ limit. We demonstrated both concentration of measure and instance independence for arbitrary mixed-spinmodels, which imply that the expectation value of the energy per spin is independent of the specific model specificationand that measurements of the QAOA state are guaranteed to give energies close to the expectation value. Our explicitformula for the expectation value of the energy for arbitrary mixed-spin models allows us to find the optimal angles on aclassical computer.There are two obvious open questions raised by this work. First, the approach of this paper can probably be combinedwith the methods of [2] to generalize our work to depth p > p . This is a particularly interesting route of research, since it is known that in case of σ q ∝ δ q, ,Montanari’s classical algorithm does not approach the optimal solution [5], so that at sufficient depth p the QAOA has achance of outperforming the best known classical algorithm. Higher-spin models may even provide a more direct route,since the Montanari algorithm may perform correspondingly worse [4, 5]. While the generalization to higher p is likelypossible, it is a nontrivial extension of this paper, and we leave it for future work.Second, it remains an open question what to what extent results on the random models can be used to find optimalangles for realistic binary optimization problems. One hypothetical approach to finding QAOA angles for a single instanceof an n -spin optimization problem would be the procedure:1. 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