Mean-Field Solution of the Weak-Strong Cluster Problem for Quantum Annealing with Stoquastic and Non-Stoquastic Catalysts

We study the weak-strong cluster problem for quantum annealing in its mean-field version as proposed by Albash [Phys. Rev. A 99 (2019) 042334] who showed by numerical diagonalization that non-stoquastic $XX$ interactions (non-stoquastic catalysts) remove the problematic first-order phase transition. We solve the problem exactly in the thermodynamic limit by analytical methods and show that the removal of the first-order transition is successfully achieved either by stoquastic or non-stoquastic $XX$ interactions depending on whether the $XX$ interactions are introduced within the weak cluster, within the strong cluster, or between them. We also investigate the case where the interactions between the two clusters are sparse, i.e. not of the mean-field all-to-all type. The results again depend on where to introduce the $XX$ interactions. We further analyze how inhomogeneous driving of the transverse field affects the performance of the system without $XX$ interactions and find that inhomogeneity in the transverse field removes the first-order transition if appropriately implemented.


Introduction
It is an interesting and important problem in quantum annealing [1][2][3][4][5][6][7] in its implementation as adiabatic quantum computing 8,9) whether or not the introduction of non-stoquastic interactions (non-stoquastic catalysts) enhances the performance compared to the case of the tradi- * takada@qa.iir.titech.ac.jp 1/25 arXiv:1912.09189v2 [quant-ph] 3 Mar 2020 tional formulation without non-stoquasticity. A stoquastic Hamiltonian can be represented as a matrix with non-positive off-diagonal elements in a product basis of local states, and can be simulated classically without the sign problem. 10,11) Introduction of non-stoquasticity into the Hamiltonian makes it difficult to classically simulate the system, 12) but it does not necessarily mean a speedup as compared to the case of a stoquastic Hamiltonian.
Numerical studies of finite-size systems indicate that the introduction of a non-stoquastic catalyst increases the success probability in a small subset of problem instances. [13][14][15] Analytical studies of the p-spin model (a mean-field-type p-body interacting ferromagnetic system) and the Hopfield model show that a non-stoquastic catalyst is effective to remove the firstorder phase transition, which exists in the original stoquastic Hamiltonian, leading to an exponential speedup compared to the traditional stoquastic case. [16][17][18][19] In a recent paper, Albash 20) introduced a mean-field version of the weak-strong cluster problem (also known as the largespin tunneling problem), 21,22) which was used to test the possibility of large-scale tunneling effects in the D-Wave quantum annealer. 23) Albash showed by numerical diagonalization of small-size systems that a non-stoquastic catalyst introduced between the two clusters in the problem eliminates the first-order transition that exists in the case without non-stoquasticity.
He also introduced a geometrically local Hamiltonian for which evidence was provided for a similar phenomenon. Under these circumstances, it is desirable to study more instances analytically toward the goal of understanding when and how non-stoquastic catalysts lead to (or do not lead to) increased performance, in particular given the ongoing efforts to implement non-stoquasticity at the hardware level. 24) We have carried out a comprehensive analytical study of the mean-field version of the weak-strong cluster problem formulated by Albash and its generalization to the case with sparse (not all-to-all) interactions between the clusters, the latter being closer to the realistic hardware implementation. We analytically confirm his numerical conclusion that the nonstoquastic catalyst introduced between the clusters with an appropriate amplitude removes the first-order transition. We have further found that the elimination of the first-order transition is possible even with a stoquastic catalyst if it is introduced in an appropriate way. We also study the effects of inhomogeneous driving of the transverse field in the original stoquastic problem, inhomogeneity meaning that the transverse field is driven more quickly in one of the clusters than in the other. We show that this protocol is effective to eliminate the first-order transition. Our results represent a complete solution to the weak-strong cluster problem with stoquastic or non-stoquastic catalysts in the mean-field framework. This paper is organized as follows. In Section 2, we solve the weak-strong cluster prob- 2/25 lem with dense interactions within the clusters and dense or sparse interactions between the clusters. Section 3 concludes the paper. Technical details are relegated to Appendices.

Weak-Strong Cluster Problem
We define the weak-strong cluster problem with mean-field-type dense interactions within and between the clusters as proposed in Ref. 20 and analyze it in Section 2.1. Then the case with sparse interactions between the clusters is solved in Section 2.2.

Weak-strong cluster problem with dense interactions between clusters
We first study the weak-strong cluster problem with dense interactions between the two clusters. The model has two subsystems (clusters) with the problem Hamiltonian where N is the total number of spins (qubits) in the system andσ ar = (σ x ar ,σ y ar ,σ z ar ) is the Pauli operator at (a, r), with a (= 1, 2) representing the cluster index and r = 1, . . . , N/2 the site index within each cluster. We set the strengths of longitudinal magnetic fields to h 1 = 1 and h 2 = −0.49. Notice that the strong longitudinal field h 1 and the weak one h 2 point in the opposite directions. We refer to the first subsystem a = 1 as the strong cluster and the second subsystem a = 2 as the weak cluster. The structure of the problem is schematically depicted in Fig. 1. The ground state ofĤ p is the eigenstate ofσ z 1r andσ z 2r with eigenvalues σ z 1r = σ z 2r = +1 (all spins pointing up, to be called 'state A'), while a metastable state exists with eigenvalues σ z 1r = +1 and σ z 2r = −1 (spins in the strong cluster pointing up and those in the weak cluster pointing down, to be called 'state B'). The conventional quantum annealing with a uniform transverse field, in which the Hamiltonian is stoquastic, has a first-order phase transition when state A and state B exchange their (meta)stability, meaning a large-scale spin flip in the weak cluster. 23) Let us construct a quantum annealing Hamiltonian for this problem, generalizing the formulation in Ref. 20. Using the magnetization operatorsm a = (2/N) rσar , the Hamiltonian H(s) is defined asĤ where s ∈ [0, 1] denotes the dimensionless time. Suppose that γ 1 (s) and γ 2 (s) are monotonically increasing functions which satisfy γ 1 (0) = γ 2 (0) = 0 and γ 1 (1) = γ 2 (1) = 1, and ξ 11 , We can achieve inhomogeneous driving of the transverse field by choosing different functions for γ 1 (s) and γ 2 (s). As for the XX interactions, non-zero ξ ab makes the corresponding term non-vanishing except at the beginning and the end of annealing. When 0 < s < 1, the HamiltonianĤ(s) is non-stoquastic for ξ 11 < 0, ξ 22 < 0, or ξ 12 < 0 and stoquastic for ξ 11 , ξ 22 , ξ 12 ≥ 0.
For the moment, we assume γ 1 (s) = γ 2 (s) = s (homogeneous field driving) and focus on effects of the XX interactions. First consider the case of the XX interaction between the clusters. Sincem a is the sum of a large number of spins, it reduces to a classical variable in the thermodynamic limit N → ∞, which significantly facilitates the analysis. As a consequence, we can calculate the magnetization m a for each cluster in the ground state as detailed in  We next calculate the energy gap between the ground state and the first excited state, which can be achieved by evaluating quantum fluctuations around the classical limit 17,25) as detailed in Appendix A.2. We show the resulting energy gaps ∆ a (a = 1, 2) for ξ 11 = ξ 22 = 0 and ξ 12 = ξ = 0, −4, −10 in Fig. 3. Here, ∆ a denote the energy gaps created by the quasiparticle excitationsb † a above the classical ground state. We calculated ∆ a by numerically diagonalizing the four-dimensional matrix E defined as Eq. (A·35) and multiplying the nonnegative eigenvalues ε a by four (see Eq. (A·41)). The smaller gap ∆ 1 is equal to the energy gap between the ground and first excited states of the HamiltonianĤ(s) except at the firstorder transition point. The correct energy gap at a first-order transition is exponentially small as a function of the system size N, 20) which cannot be evaluated by our method since our method gives the energy gap in the thermodynamic limit (see Appendix A.2). In general, the energy gaps ∆ a calculated by our method are discontinuous at first-order transitions due to the discontinuity of the magnetizations m a , although we cannot clearly see a discontinuous jump in the lower of the two gaps ∆ 1 for ξ = 0 at least in our precision whereas the other ∆ 2 shows discontinuity. In the case of ξ = −4, the energy gaps ∆ a are continuous because there is no first-order transition. Now we derive the minimum gap min s ∆ 1 for the range of ξ where there is no first-order 5/25  Let us move on to the case of the XX interaction in each cluster, which was not covered in Ref. 20. We show the magnetization in the weak cluster m z 2 for (ξ 11 , ξ 22 , ξ 12 ) = (ξ, 0, 0), (0, ξ, 0) as functions of s and ξ in Fig. 5. We find that the non-stoquastic XX interaction in the strong cluster or the stoquastic XX interaction in the weak cluster removes the first-order transition, while the other types of intracluster XX interaction do not.
We can interpret these results as follows. In the case of the non-stoquastic XX interaction in the strong cluster, |m x 1 | becomes smaller and m z 1 larger, which makes m z 2 larger thanks to the 6/25 ferromagnetic coupling between the clusters. In the case of the stoquastic XX interaction in the weak cluster, m x 2 becomes larger and |m z 2 | smaller. Both of these types of XX interaction prevent m z 2 from being a large negative value due to the longitudinal field h 2 , which reduces the possibility of a jump in m z 2 . We also found that the first-order transition cannot be removed in the case where the XX interaction is proportional to 1 2 (m x 1 +m x 2 ) 2 (i.e., (ξ 11 , ξ 22 , ξ 12 ) = (ξ/2, ξ/2, ξ)) regardless of the sign of the coefficient ξ, which is shown in Appendix C. The result in the non-stoquastic case ξ < 0 is in agreement with the numerical consequence given in Appendix F of Ref. 20.
We next consider the problem in which the transverse field is driven inhomogeneously and there is no XX interaction. Then, the Hamiltonian is stoquastic. We show the magnetization in the weak cluster m z 2 for (γ 1 (s), γ 2 (s)) = (γ(s), s), (s, γ(s)) in Fig. 6, where the increasing function γ(s) can be chosen arbitrarily as long as γ(0) = 0 and γ(1) = 1. Notice that the value of m z 2 is indefinite at s = 0 and γ 2 = 1, where neither magnetic field nor interaction is applied to the weak cluster. We find that the weaker transverse field in the strong cluster and the stronger transverse field in the weak cluster can remove the first-order transition in the process of quantum annealing. The mechanism for removing the first-order transition is similar to the case of the non-stoquastic XX interaction in the strong cluster or the stoquastic XX interaction in the weak cluster.

Weak-strong cluster problem with sparse interactions between clusters
We now consider the weak-strong cluster problem whose interactions between the clusters are sparse. The problem Hamiltonian is defined aŝ where the longitudinal field in the strong cluster is h 1 = 1 and that in the weak cluster is  The quantum annealing Hamiltonian for this problem is given bŷ x 2rσ x 2r where s ∈ [0, 1] is the dimensionless time. After taking the thermodynamic limit N → ∞ and the zero-temperature limit, we calculate the magnetizations in the two clusters m z 1 and m z 2 using the imaginary-time path-integral formulation of the partition function and the saddlepoint method with the static ansatz as detailed in Appendix B.
We show the magnetizations m z 1 and m z 2 for γ 1 (s) = γ 2 (s) = s and (ξ 11 , ξ 22 , ξ 12 ) = (0, 0, ξ) in Fig. 8. We find that while the uniform transverse-field driver ξ = 0 causes a first-order transition, both of the non-stoquastic XX interaction between the clusters ξ < 0 and the stoquastic one ξ > 0 can remove the transition. In contrast to the case of dense interactions discussed in Section 2.1, there is no transition for large positive ξ and too large negative ξ.

Conclusion
We have studied the phase transitions of two weak-strong cluster problems with the ul- directions and different strengths, which causes a first-order phase transition in the absence of the catalysts. The difference between the two problems is the connectivity between the clusters: One has dense (all-to-all) interactions between the clusters and the other has sparse interactions.
We solved the problem by a semi-classical method and found that stoquastic or nonstoquastic catalysts can remove the first-order transition for the model with all-to-all interactions between the clusters. More precisely, we first showed that the transition disappears if a non-stoquastic catalyst is appended between the clusters with an appropriate strength while the transition persists if the catalyst is stoquastic, which is consistent with the already known result of numerical diagonalization of finite-size systems. 20) We also calculated the energy gap in the thermodynamic limit analytically and identified the optimal strength of the non-stoquastic XX interaction between the clusters that maximizes the minimum energy gap. The result again confirms the consequence of the numerical study. 20) In addition to the non-stoquastic catalyst between the clusters, we found other protocols to eliminate the firstorder transition, namely, a non-stoquastic catalyst in the strong cluster, a stoquastic catalyst in the weak cluster, and inhomogeneous transverse-field driving in which the transverse field is weaker in the strong cluster or stronger in the weak cluster. The latter result confirms general observations in previous studies on the usefulness of inhomogeneous field driving. [26][27][28][29] We next analyzed the problem with sparse interactions between the clusters by evaluating the partition function in the thermodynamic limit and the zero-temperature limit. Then, we found generally similar results as in the case with all-to-all interactions, except that a stoquastic catalyst between the clusters as well as a non-stoquastic one can remove the first-order transition.
It is noteworthy that our results are rare examples of two-body interacting systems for which the removal of first-order transitions with stoquastic or non-stoquastic catalysts has been shown analytically. Although it is generally difficult to predict for a given real-world optimization problem which type of catalyst (stoquastic or non-stoquastic) or inhomogeneous driving is effective to enhance the performance of quantum annealing, it is likely to be useful to introduce many-body drivers (XX interactions with adjustable sign and strength) and Governmental purposes notwithstanding any copyright annotation thereon.

Appendix A: Analysis of an Infinite-Range System Consisting of Several Subsystems
We analyze a mean-field spin system consisting of several subsystems by use of the semiclassical method. The system is supposed to have infinite-range (all-to-all) interactions in each subsystem and between subsystems. We first take the classical limit to calculate the magnetization and next include quantum fluctuations to evaluate the energy gap.
we can write the HamiltonianĤ asĤ = Nh({m a }). We assume that h({m a }) is a polynomial of degree P = O(N 0 ), i.e. a linear combination ofm α 1 a 1 · · ·m α p a p (p = 0, 1, . . . , P), and the coefficients are of O(N 0 ). Now we take the thermodynamic limit N → ∞ ⇐⇒ S → ∞. In this limit, the non-commutativity of the components ofm a is negligible andm 2 a approaches unity:

A.2 Quantum fluctuation
In order to derive the energy gap, we extend the method used in Refs. 17 and 25 to the system of several large spins. First we wish to expand the spin operatorsŜ a around the classical limit. We introduce rotated spin operatorsŜ a whose z-components are the spin operators in the directions of m a =: e z a . We choose unit vectors e x a and e y a such that {e α a } α=x,y,z is an orthonormal set and e x a × e y a = e z a . Defining the components ofŜ a asŜ α a = e α a ·Ŝ a , we obtain where T a = (T αβ a ) α,β=x,y,z = (e x a , e y a , e z a ) ∈ SO(3) is a special orthogonal matrix. Then,Ŝ a satisfy the commutation relations [Ŝ α a ,Ŝ β b ] = iδ ab γ αβγŜ γ a . Letm a =Ŝ a /S andm ± a =m x a ±im y a . We perform the Holstein-Primakoff transformation 30) whereb a are bosonic operators satisfying [b a ,b † b ] = δ ab and [b a ,b b ] = 0. The fact thatm z a = m a ·m a approaches unity in the classical limit S → ∞ implies that the number operatorsn a =b † aba take values sufficiently smaller than S in the low-energy states for large S . Expanding the operators in S −1 results in We thus obtainm where we defined the Hessian matrix as However, the value of c is not needed for determining the energy gap.
Since the magnetizations m a in the ground state satisfy Eq. (A·4) and e x a and e y a are orthogonal to m a , we find the expansion of the Hamiltonian density operator and For α = β, h αα ab is symmetric under the exchange of the lower indices: h αα ab = h αα ba . Let us diagonalize the operatorε. We perform the Bogoliubov transformation and assume that the new bosonic operatorsb a diagonalizeε: Here, ε a and c should be real numbers forε to be Hermitian andb a should satisfy the commutation relations Combining these equations with Eq. (A·21), we derive where and Z ± * ab := (Z ± ab ) * . On the other hand, Eq. (A·23) yields the commutation relation Comparing the coefficients ofb c andb † c in Eqs. (A·27) and (A·29) results in Notice that we can derive the equivalent equations by calculating [b † a ,ε] with Eqs. (A·18), (A·22), and (A·23).
Let us define the A-dimensional matrices M = (µ a δ ab ), Z ± = (Z ± ab ) (A·32) and the A-dimensional vectors Then, the set of Eqs. (A·30) and (A·31) is written as the eigenvalue equation where is a 2A-dimensional matrix and ψ a : shows that ε a are the eigenvalues of E and ψ T a are the corresponding left eigenvectors. We can show that ε a ∈ R holds if |u a | 2 |v a | 2 . In the following, we assume that all of the eigenvalues of E are real numbers. Notice that the eigenvalue equation (A·34) yields another eigenvalue equation where This means that the 2A-dimensional matrix E has 2A eigenvalues ε ±1 , . . . , ε ±A . Let ε 1 , . . . , ε A be non-negative eigenvalues of E, which become the frequencies of the quasi-particles created byb † a . The commutation relations (A·24) are equivalent to the following constraints on u a and v a : These constraints are rewritten as the pseudo orthonormality of ψ a , We can show that Eq. (A·34) automatically yields Eq. (A·39) if ε a ε b . For each degenerate eigenvalue, we impose the constraint (A·39) on the corresponding eigenvectors. The constraint for a = b, |u a | 2 − |v a | 2 = sgn(a), gives the normalization condition of the eigenvectors When ε a are real numbers for all a and the pseudo orthonormality (A·39) holds, the energy gaps between the ground state and the low-energy excited states of the original Hamiltonian H are given by with n a ∈ {0, 1, 2, . . . }. Here, are the energy gaps created by the quasi-particle excitationsb † a . We can assume that without loss of generality. Then, the energy gap between the ground and first excited states is Notice that our method of calculating the energy gap is not applicable to a first-order transition point because the quasi-particle excitationb † 1 is a fluctuation around the single global minimum of the classical potential h({m a }). Typically, the classical potential h({m a }) is a double-well potential at a first-order transition and the ground and first excited states are superpositions of the states localized at the two minima. To calculate the gap between these states analytically, the discrete WKB or instantonic method would be needed. [31][32][33][34] Appendix B: Analysis of a System with Sparse Interactions between Subsystems where the first term h m ({m a }) is the mean-field part written as a function ofm a = A N rσar and the second term is the sum of sparse interactions between the subsystems which are determined by the matrices K ab = (K αβ ab ) α,β=x,y,z . We assume that h m ({m a }) is a polynomial of degree P = O(N 0 ) with coefficients of O(N 0 ). We can set K αβ aa = 0 and K αβ ab = K βα ba ∈ R without loss of generality, because the term with the coefficient matrix K aa can be included in the mean-field part and [σ α a ,σ β b ] = 0 for a b. Now we introduce the path-integral representation of the partition function Z = Tr e −βĤ with inverse temperature β: Here, |{n ar } = ar |n ar is the product state of the spin coherent states |n ar determined by unit vectors n ar . The spin coherent state |n ar at each site is the normalized eigenstate of σ ar · n ar with the eigenvalue one. In addition, Dn ar is the functional measure which is the product of the measures on the 2-sphere over imaginary time τ ∈ [0, β].
The energy expectation value in the spin coherent state is given by In the third and fifth lines of this equation, we used the fact that the number of (r 1 , . . . , r p ) including equal indices is of O(N p−1 ), while the number of (r 1 , . . . , r p ) whose elements are different from each other is N A N A − 1 · · · N A − p + 1 = O(N p ). Hence, we obtain the simplified expression We ignore the term of O(N −1 ), which yields a non-extensive correction to {n ar }|Ĥ|{n ar } .
Combining Eqs. (B·2), (B·3), and (B·5) yields We insert the path integral of the delta functional for a = 1, . . . , A into Eq. (B·6). Then, we obtain − am a (τ) · n ar (τ) − 1 2 ab n ar (τ) · K ab n br (τ) where we simplified the product over r = 1, . . . , N/A as the (N/A)th power, because the factor in the expression of Z depends on r only through n ar .
Let us evaluate the asymptotic form of the partition function in the thermodynamic limit N → ∞ by the stationary-phase approximation. We assume that the stationary path (i.e., the set of the functions m a (τ) andm a (τ) for which the functional derivatives of the integrand of Then, we can write the partition function as an ordinary integral with respect to m a andm a : Replacing the path integral over n a (τ) with the trace of an exponentiated operator results in We defined the effective Hamiltonian of an A-spin system aŝ Notice that Eq. (B·15) still holds in the zero-temperature limit β → ∞.

Appendix C: Results for the Total XX Catalyst
We consider the weak-strong cluster problem with the total XX catalyst, which has both of intercluster and intracluster XX interactions. The Hamiltonian is given by Eq. (2) for dense intercluster interactions and Eq. (4) for sparse ones. We set γ 1 (s) = γ 2 (s) = s and (ξ 11 , ξ 22 , ξ 12 ) = (ξ/2, ξ/2, ξ) in both cases. Notice that in the case of the dense intercluster interactions, the XX catalyst is proportional to the total x-magnetization operator squared ((m x 1 +m x 2 )/2) 2 . Figure C·1 shows the magnetization in the weak cluster m z 2 for the dense interactions between the clusters and for sparse ones. We find that the total XX catalyst cannot eliminate the first-order transition, whether the intercluster interactions are dense or sparse and whether the 22/25 catalyst is stoquastic or non-stoquastic. The result in the case of the dense intercluster interactions with the non-stoquastic catalyst ξ < 0 is consistent with the numerical consequence. 20) 23/25