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JOURNAL OF MATHEMATICAL PHYSICS 49, 125210 共2008兲 Mathematical foundation of quantum annealing Satoshi Morita1,a兲 and Hidetoshi Nishimori2 1 International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014 Trieste, Italy Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan 2 共Received 4 June 2008; accepted 10 September 2008; published online 15 December 2008兲 Quantum annealing is a generic name of quantum algorithms that use quantummechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic 共Monte Carlo兲 dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule 共time dependence of the control parameter兲 without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2995837兴 I. INTRODUCTION An optimization problem is a problem to minimize or maximize a real single-valued function of multivariables called the cost function.1,2 If the problem is to maximize the cost function f, it suffices to minimize f. It thus does not lose generality to consider minimization only. In the present paper we consider combinatorial optimization, in which variables take discrete values. Well-known examples are satisfiability problems, exact cover, maximum cut, Hamilton graph, and traveling salesman problem. In physics, the search of the ground state of spin systems is a typical example, in particular, systems with quenched randomness such as spin glasses. Optimization problems are classified roughly into two types, easy and hard ones. Loosely speaking, easy problems are those for which we have algorithms to solve in steps 共=time兲 polynomial in the system size 共polynomial complexity兲. In contrast, for hard problems, all known algorithms take exponentially many steps to reach the exact solution 共exponential complexity兲. For these latter problems it is virtually impossible to find the exact solution if the problem size exceeds a moderate value. Most of the interesting cases as exemplified above belong to the latter hard class. It is therefore important practically to devise algorithms which give approximate but accurate solutions efficiently, i.e., with polynomial complexity. Many instances of combinatorial optimization problems have such approximate algorithm. For example, the Lin–Kernighan algorithm is often used to solve the traveling salesman problem within a reasonable time.3 a兲 Electronic mail: [email protected]. 0022-2488/2008/49共12兲/125210/47/$23.00 49, 125210-1 © 2008 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-2 S. Morita and H. Nishimori J. Math. Phys. 49, 125210 共2008兲 In the present paper we will instead discuss generic algorithms, simulated annealing 共SA兲 and quantum annealing 共QA兲. The former was developed from the analogy between optimization problems and statistical physics.4,5 In SA, the cost function to be minimized is identified with the energy of a statistical-mechanical system. The system is then given a temperature, an artificially introduced control parameter, which, by reducing slowly from a high value to zero, we hope to drive the system to the state with the lowest value of the energy 共cost function兲, reaching the solution of the optimization problem. The idea is that the system is expected to stay close to thermal equilibrium during time evolution if the rate of decrease in temperature is sufficiently slow, and is thus led in the end to the zero-temperature equilibrium state, the lowest-energy state. In practical applications SA is immensely popular due to its general applicability, reasonable performance, and relatively easy implementation in most cases. SA is usually used as a method to obtain an approximate solution within a finite computation time since it needs an infinitely long time to reach the exact solution by keeping the system close to thermal equilibrium. Let us now turn our attention to QA 共see Refs. 6–11兲.1 In SA, we make use of thermal 共classical兲 fluctuations to let the system hop from state to state over intermediate energy barriers to search for the desired lowest-energy state. Why then not try quantum-mechanical fluctuations 共quantum tunneling兲 for state transitions if such may lead to better performance? In QA we introduce artificial degrees of freedom of quantum nature, noncommutative operators, which induce quantum fluctuations. We then ingeniously control the strength of these quantum fluctuations so that the system finally reaches the ground state, just like SA in which we slowly reduce the temperature. More precisely, the strength of quantum fluctuations is first set to a very large value for the system to search for the global structure of the phase space, corresponding to the hightemperature situation in SA. Then the strength is gradually decreased to finally vanish to recover the original system hopefully in the lowest-energy state. Quantum tunneling between different classical states replaces thermal hopping in SA. The physical idea behind such a procedure is to keep the system close to the instantaneous ground state of the quantum system, analogously to the quasiequilibrium state to be kept during the time evolution of SA. Similarly to SA, QA is a generic algorithm applicable, in principle, to any combinatorial optimization problem and is used as a method to reach an approximate solution within a given finite amount of time. The reader may wonder why one should invent yet another generic algorithm when we already have the powerful SA. A short answer is that QA outperforms SA in most cases, at least theoretically. Analytical and numerical results indicate that the computation time needed to achieve a given precision of the answer is shorter in QA than in SA. Also, the magnitude of error is smaller for QA than for SA if we run the algorithm for a fixed finite amount of time. We shall show some theoretical bases for these conclusions in this paper. Numerical evidence is found in Refs. 9–11 and 14–22. A drawback of QA is that a full practical implementation should rely on the quantum computer because we need to solve the time-dependent Schrödinger equation with a very large scale. Existing numerical studies have been carried out either for small prototype examples or for large problems by Monte Carlo simulations using the quantum-classical mapping by adding an extra 关Trotter or imaginary-time 共IT兲兴 dimension.23–25 The latter mapping involves approximations, which inevitably introduces additional errors as well as the overhead caused by the extra dimension. Nevertheless, it is worthwhile to clarify the usefulness and limitations of QA as a theoretical step toward a new paradigm of computation. This aspect is shared by quantum computation in general whose practical significance will be fully exploited on the quantum computer. The idea of QA is essentially the same as quantum adiabatic evolution 共QAE兲, which is now actively investigated as an alternative paradigm of quantum computation.26 It has been proved that QAE is equivalent to the conventional circuit model of quantum computation,27 but QAE is sometimes considered more useful than the circuit model for several reasons including robustness 1 The term quantum annealing first appeared in Refs. 12 and 13 in which the authors used quantum transitions for state search and the dynamical evolution of control parameters were set by hand as an algorithm. QA in the present sense using natural Schrödinger dynamics was proposed later independently in Refs. 6 and 7. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-3 Mathematical foundation of quantum annealing J. Math. Phys. 49, 125210 共2008兲 against external disturbance. In the literature of quantum computation, one is often interested in the computational complexity of the QAE-based algorithm for a given specific problem under a fixed value of acceptable error. QAE can also be used to find the final quantum state when the problem is not a classical optimization. In some contrast to these situations on QAE, studies of QA are often focused not on computational complexity but on the theoretical convergence conditions for infinite-time evolution and on the amount of errors in the final state within a fixed evolution time. Such a difference may have led some researchers to think that QA and QAE are to be distinguished from each other. We would emphasize that they are essentially the same and worth investigations by various communities of researchers. The structure of the present paper is as follows. Section II discusses the convergence condition of QA, in particular, the rate of decrease in the control parameter representing quantum fluctuations. It will be shown there that a qualitatively faster decrease in the control parameter is allowed in QA than in SA to reach the solution. This is one of the explicit statements of the claim more vaguely stated above that QA outperforms SA. In Sec. III we review the performance analysis of SA using quantum-mechanical tools. The well-known convergence condition for SA will be rederived from the perspective of quantum adiabaticity. The methods and results in this section help us strengthen the interrelation between QA, SA, and QAE. The error rate of QA after a finite-time dynamical evolution is analyzed in Sec. IV. There we explain how to reduce the final residual error after evolution of a given amount of time. This point of view is unique in the sense that most references of QAE study the time needed to reach a given amount of tolerable error, i.e., computational complexity. The results given in this section can be used to qualitatively reduce residual errors for a given algorithm without compromising computational complexity. Convergence conditions for stochastic implementation of QA are discussed in Sec. V. The results are surprising in that the rate of decrease in the control parameter for the system to reach the solution coincides with that found in Sec. II for the pure quantum-mechanical Schrödinger dynamics. The stochastic 共and therefore classical兲 dynamics shares the same convergence conditions as fully quantum dynamics. Summary and outlook are described in Sec. VI. The main parts of this paper 共Secs. II, IV, and V兲 are based on the Ph.D. thesis of Morita28 as well as several original papers of the present and other authors as will be referred to appropriately. The present paper is not a comprehensive review of QA since an emphasis is given almost exclusively to the theoretical and mathematical aspects. There exists an extensive body of numerical studies and the reader is referred to Refs. 9–11 for reviews. II. CONVERGENCE CONDITION OF QA: REAL-TIME SCHRÖDINGER EVOLUTION The convergence condition of QA with the real-time 共RT兲 Schrödinger dynamics is investigated in this section, following Ref. 29. We first review the proof of the adiabatic theorem30 to be used to derive the convergence condition. Then introduced is the Ising model with a transverse field as a simple but versatile implementation of QA. The convergence condition is derived by solving the condition for adiabatic transition with respect to the strength of the transverse field. A. Adiabatic theorem Let us consider the general Hamiltonian which depends on time t only through the dimensionless time s = t / , H共t兲 = H̃ 冉冊 t ⬅ H̃共s兲. 共1兲 The parameter is introduced to control the rate of change in the Hamiltonian. In natural quantum systems, the state vector 兩共t兲典 follows the RT Schrödinger equation, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-4 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori d 兩共t兲典 = H共t兲兩共t兲典, dt i 共2兲 or, in terms of the dimensionless time, i d ˜ 兩共s兲典 = H̃共s兲兩˜共s兲典, ds 共3兲 where we set ប = 1. We assume that the initial state is chosen to be the ground state of the initial Hamiltonian H共0兲 and that the ground state of H̃共s兲 is not degenerate for s ⱖ 0. We show in Sec. III that the transverse-field Ising model 共TFIM兲, to be used as H共t兲 in most parts of this paper, has no degeneracy in the ground state 共except possibly in the limit of t → ⬁兲. If is large, the Hamiltonian changes slowly and it is expected that the state vector keeps track of the instantaneous ground state. The adiabatic theorem provides the condition for adiabatic evolution. To see this, we derive the asymptotic form of the state vector with respect to the parameter . Since we wish to estimate how close the state vector is to the ground state, it is natural to expand the state vector by the instantaneous eigenstates of H̃共s兲. Before doing so, we derive useful formulas for the eigenstates. The kth instantaneous eigenstate of H̃共s兲 with the eigenvalue k共s兲 is denoted as 兩k共s兲典, H̃共s兲兩k共s兲典 = k共s兲兩k共s兲典. 共4兲 We assume that 兩0共s兲典 is the ground state of H̃共s兲 and that the eigenstates are orthonormal, 具j共s兲 兩 k共s兲典 = ␦ jk. From differentiation of 共4兲 with respect to s, we obtain 冓 冏冏 冔 j共s兲 −1 d k共s兲 = ds j共s兲 − k共s兲 冓 冏 冏 冔 j共s兲 dH̃共s兲 k共s兲 , ds 共5兲 where j ⫽ k. In the case of j = k, the same calculation does not provide any meaningful result. We can, however, impose the following condition: 冓 冏冏 冔 k共s兲 d k共s兲 = 0. ds 共6兲 This condition is achievable by the time-dependent phase shift: If 兩k̃共s兲典 = ei共s兲兩k共s兲典, we find 冓 冏冏 冔 k̃共s兲 冓 冏冏 冔 d d d k̃共s兲 = i + k共s兲 k共s兲 . ds ds ds 共7兲 The second term on the right-hand side is purely imaginary because 冋冓 冏 冏 冔册 冓 冏 冏 冔 k共s兲 d k共s兲 ds ⴱ + k共s兲 d d k共s兲 = 具k共s兲兩k共s兲典 = 0. ds ds 共8兲 Thus, condition 共6兲 can be satisfied by tuning the phase factor 共s兲 even if the original eigenstate does not satisfy it. Theorem 2.1: If the instantaneous ground state of the Hamiltonian H̃共s兲 is not degenerate for s ⱖ 0 and the initial state is the ground state at s = 0 , i.e., 兩˜共0兲典 = 兩0共0兲典 , the state vector 兩˜共s兲典 has the asymptotic form in the limit of large as 兩˜共s兲典 = 兺 c j共s兲e−i j共s兲兩j共s兲典, 共9兲 c0共s兲 ⬇ 1 + O共−2兲, 共10兲 j Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-5 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing i c j⫽0共s兲 ⬇ 关A j共0兲 − ei关 j共s兲−0共s兲兴A j共s兲兴 + O共−2兲, 共11兲 where j共s兲 ⬅ 兰s0ds⬘ j共s⬘兲 , ⌬ j共s兲 ⬅ j共s兲 − 0共s兲 , and A j共s兲 ⬅ 1 ⌬ j共s兲2 冓 冏 冏 冔 j共s兲 dH̃共s兲 0共s兲 . ds 共12兲 Proof: Substitution of 共9兲 into Schrödinger equation 共3兲 yields the equation for the coefficient c j共s兲 as dc j dH̃共s兲 ei关 j共s兲−k共s兲兴 = 兺 ck共s兲 具j共s兲兩 兩k共s兲典, ds k⫽j j共s兲 − k共s兲 ds 共13兲 where we used 共5兲 and 共6兲. Integration of this equation yields c j共s兲 = c j共0兲 + 兺 k⫽j 冕 s ds̃ck共s̃兲 0 ei关 j共s̃兲−k共s̃兲兴 j共s̃兲 − k共s̃兲 具j共s̃兲兩 dH̃共s̃兲 ds̃ 兩k共s̃兲典. 共14兲 Since the initial state is chosen to be the ground state of H共0兲, c0共0兲 = 1 and c j⫽0共0兲 = 0. The second term on the right-hand side is of the order of −1 because its integrand rapidly oscillates for large . In fact, the integration by parts yields the −1 factor. Thus, c j⫽0共0兲 is of order −1 at most. Hence only the k = 0 term in the summation remains up to the order of −1, c j⫽0共s兲 ⬇ 冕 s ds̃ ei关 j共s̃兲−0共s̃兲兴 ⌬ j共s̃兲 0 具j共s̃兲兩 dH̃共s̃兲 ds̃ 兩0共s̃兲典 + O共−2兲, 共15兲 and the integration by parts yields 共11兲. 䊐 Remark: The condition for the adiabatic evolution is given by the smallness of the excitation probability. That is, the right-hand side of 共11兲 should be much smaller than unity. This condition is consistent with the criterion of the validity of the above asymptotic expansion. It is represented by Ⰷ 兩A j共s兲兩. 共16兲 Using the original time variable t, this adiabaticity condition is written as 冏 冏 1 dH共t兲 兩0共t兲典 = ␦ Ⰶ 1. 2 具j共t兲兩 ⌬ j共t兲 dt 共17兲 This is the usual expression of the adiabaticity condition. B. Convergence conditions of quantum annealing In this section, we derive the condition which guarantees the convergence of QA. The problem is what annealing schedule 共time dependence of the control parameter兲 would satisfy adiabaticity condition 共17兲. We solve this problem on the basis of the idea of Somma et al.31 developed for the analysis of SA in terms of quantum adiabaticity as reviewed in Sec. III. 1. Transverse-field Ising model Let us suppose that the optimization problem we wish to solve can be represented as the ground-state search of an Ising model of general form Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-6 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori N HIsing ⬅ − 兺 Jizi − 兺 Jijzi zj − 兺 Jijkzi zj zk − ¯ , i=1 ij 共18兲 ijk where the i␣ 共␣ = x , y , z兲 are the Pauli matrices, components of the spin 21 operator at site i. The eigenvalue of zi is +1 or −1, which corresponds the classical Ising spin. Most combinatorial optimization problems can be written in this form by, for example, mapping binary variables 共0 and 1兲 to spin variables 共⫾1兲. Another important assumption is that Hamiltonian 共18兲 is extensive, i.e., proportional to the number of spins N for large N. To realize QA, a fictitious kinetic energy is introduced typically by the time-dependent transverse field N HTF共t兲 ⬅ − ⌫共t兲 兺 xi , 共19兲 i=1 which induces spin flips, quantum fluctuations, or quantum tunneling between the two states zi = 1 and zi = −1, thus allowing a quantum search of the phase space. Initially the strength of the transverse field ⌫共t兲 is chosen to be very large, and the total Hamiltonian H共t兲 = HIsing + HTF共t兲 共20兲 is dominated by the second kinetic term. This corresponds to the high-temperature limit of SA. The coefficient ⌫共t兲 is then gradually and monotonically decreased toward 0, leaving eventually only the potential term HIsing. Accordingly the state vector 兩共t兲典, which follows the RT Schrödinger equation, is expected to evolve from the trivial initial ground state of transverse-field term 共19兲 to the nontrivial ground state of 共18兲, which is the solution of the optimization problem. An important issue is how slowly we should decrease ⌫共t兲 to keep the state vector arbitrarily close to the instantaneous ground state of total Hamiltonian 共20兲. The following theorem provides a solution to this problem as a sufficient condition. Theorem 2.2: Adiabaticity (17) for TFIM (20) yields the time dependence of ⌫共t兲 as ⌫共t兲 = a共␦t + c兲−1/共2N−1兲 共21兲 for t ⬎ t0 (for a given positive t0 ) as a sufficient condition of convergence of QA. Here a and c are constants of O共N0兲 and ␦ is a small parameter to control adiabaticity appearing in (17). The following theorem proved by Hopf32 will be useful in proving this theorem. See Appendix A for the proof. Theorem 2.3: If all the elements of a square matrix M are strictly positive, M ij ⬎ 0 , its maximum eigenvalue 0 and any other eigenvalues satisfy 兩兩 ⱕ −1 , +1 0 共22兲 M ik . M jk 共23兲 where is defined by ⬅ max i,j,k Proof of Theorem 2.2: We show that power decay 共21兲 satisfies adiabaticity condition 共17兲 which guarantees convergence to the ground state of HIsing as t → ⬁. For this purpose we estimate the energy gap and the time derivative of the Hamiltonian. As for the latter, it is straightforward to see Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-7 Mathematical foundation of quantum annealing 冏 具j共t兲兩 J. Math. Phys. 49, 125210 共2008兲 冏 d⌫共t兲 dH共t兲 兩0共t兲典 ⱕ − N dt dt 共24兲 since the time dependence of H共t兲 lies only in the kinetic term HTF共t兲, which has N terms. Note that d⌫ / dt is negative. To estimate a lower bound for the energy gap, we apply Theorem 2.3 to the operator M ⬅ 共E+ − H共t兲兲N. We assume that the constant E+ satisfies E+ ⬎ Emax + ⌫0, where ⌫0 ⬅ ⌫共t0兲 and Emax is the maximum eigenvalue of the potential term HIsing. All the elements of the matrix M are strictly positive in the representation that diagonalizes 兵zi 其 because E+ − H共t兲 is non-negative and irreducible, that is, any state can be reached from any other state within at most N steps. For t ⬎ t0, where ⌫共t兲 ⬍ ⌫0, all the diagonal elements of E+ − H共t兲 are larger than any nonzero off-diagonal element ⌫共t兲. Thus, the minimum element of M, which is between two states having all the spins in mutually opposite directions, is equal to N ! ⌫共t兲N, where N! comes from the ways of permutation to flip spins. Replacement of HTF共t兲 by −N⌫0 shows that the maximum matrix element of M has the upper bound 共E+ − Emin + N⌫0兲N, where Emin is the lowest eigenvalue of HIsing. Thus, we have ⱕ 共E+ − Emin + N⌫0兲N . N ! ⌫共t兲N 共25兲 If we denote the eigenvalue of H共t兲 by j共t兲, 共22兲 is rewritten as 关E+ − j共t兲兴N ⱕ −1 关E − 共t兲兴N . +1 + 0 共26兲 Substitution of 共25兲 into the above inequality yields ⌬ j共t兲 ⱖ 2关E+ − 0共t兲兴N! ⌫共t兲N ⬅ A⌫共t兲N , N共E+ − Emin + N⌫0兲N 共27兲 where we used 1 − 共共 − 1兲 / 共 + 1兲兲1/N ⱖ 2 / N共 + 1兲 for ⱖ 1 and N ⱖ 1. The coefficient A is estimated using the Stirling formula as A⬇ 冉 2冑2N关E+ − max N 0 兴 N Ne E+ − Emin + N⌫0 冊 N , 共28兲 is maxt⬎t0兵0共t兲其. This expression implies that A is exponentially small for large N. where max 0 Now, by combination of estimates 共24兲 and 共27兲, we find that the sufficient condition for convergence for t ⬎ t0 is − d⌫共t兲 N = ␦ Ⰶ 1, A2⌫共t兲2N dt 共29兲 where ␦ is an arbitrarily small constant. By integrating this differential equation, we obtain 共21兲. 䊐 Remark: The asymptotic power decay of the transverse field guarantees that the excitation probability is bounded by the arbitrarily small constant ␦2 at each instant. This annealing schedule is not valid when ⌫共t兲 is not sufficiently small because we evaluated the energy gap for ⌫共t兲 ⬍ ⌫0共t ⬎ t0兲. If we take the limit t0 → 0, ⌫0 increases indefinitely and the coefficient a in 共21兲 diverges. Then result 共21兲 does not make sense. This is the reason why a finite positive time t0 should be introduced in the statement of Theorem 2.2. 2. Transverse ferromagnetic interactions The same discussions as above apply to QA using the transverse ferromagnetic interactions in addition to a transverse field, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-8 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori 冉兺 N HTI共t兲 ⬅ − ⌫TI共t兲 i=1 冊 xi + 兺 xi xj . ij 共30兲 The second summation runs over appropriate pairs of sites that satisfy extensiveness of the Hamiltonian. A recent numerical study shows the effectiveness of this type of quantum kinetic energy.18 The additional transverse interaction widens the instantaneous energy gap between the ground state and the first excited state. Thus, it is expected that an annealing schedule faster than 共21兲 satisfies the adiabaticity condition. The following theorem supports this expectation. Theorem 2.4: The adiabaticity for the quantum system HIsing + HTI共t兲 yields the time dependence of ⌫共t兲 for t ⬎ t0 as ⌫TI共t兲 ⬀ t−1/共N−1兲 . 共31兲 Proof: The transverse interaction introduces nonzero off-diagonal elements to the Hamiltonian in the representation that diagonalizes zi . Consequently, any state can be reached from any other state within N / 2 steps at most. Thus, the strictly positive operator is modified to 共E+ − HIsing − HTI共t兲兲N/2, which leads to the lower bound for the energy gap as a quantity proportional to 䊐 ⌫TI共t兲N/2. The rest of the proof is the same as that of Theorem 2.2. The above result implies that additional nonzero off-diagonal elements of the Hamiltonian accelerate the convergence of QA. It is thus interesting to consider the many-body transverse interaction of the form N HMTI共t兲 = − ⌫MTI共t兲 兿 共1 + xi 兲. 共32兲 i=1 All the elements of HMTI are equal to −⌫MTI共t兲 in the representation that diagonalizes zi . In this system, the following theorem holds. Theorem 2.5: The adiabaticity for the quantum system HIsing + HTMI共t兲 yields the time dependence of ⌫共t兲 for t ⬎ t0 as ⌫MTI共t兲 ⬀ 2N−2 . ␦t 共33兲 Proof: We define the strictly positive operator as M = E+ − HIsing − HMTI共t兲. The maximum and minimum matrix elements of M are E+ − Emin + ⌫MTI共t兲 and ⌫MTI共t兲, respectively. Thus we have = E+ − Emin + ⌫MTI共t兲 , ⌫MTI共t兲 E+ − Emin 2⌫MTI共t兲 −1 , = ⱖ1− + 1 E+ − Emin + 2⌫MTI共t兲 E+ − Emin 共34兲 共35兲 The inequality for strictly positive operator 共22兲 yields ⌬ j共t兲 ⱖ 2⌫MTI共t兲共E+ − max 0 兲 ⬅ Ã⌫MTI共t兲, E+ − Emin 共36兲 where à is O共N0兲. Since the matrix element of the derivative of the Hamiltonian is bounded as 冏 具j共t兲兩 冏 d⌫MTI dH共t兲 , 兩0共t兲典 ⱕ − 2N dt dt 共37兲 we find that the sufficient condition for convergence with the many-body transverse interaction is Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-9 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing − 2N Ã2⌫MTI d⌫MTI = ␦ Ⰶ 1. 共t兲2 dt Integrating this differential equation yields annealing schedule 共33兲. 共38兲 䊐 3. Computational complexity The asymptotic power-law annealing schedules guarantee the adiabatic evolution during the annealing process. The power-law dependence on t is much faster than the log-inverse law for the control parameter in SA, T共t兲 = pN / log共␣t + 1兲, to be discussed in Sec. III, first proved by Geman and Geman.33 However, it does not mean that QA provides an algorithm to solve NP 共nondeterministic polynomial兲 problems in polynomial time. In the case with the transverse field only, the time for ⌫共t兲 to reach a sufficiently small value ⑀ 共which implies that the system is sufficiently close to the final ground state of HIsing whence HTF is a small perturbation兲 is estimated from 共21兲 as tTF ⬇ 冉冊 1 1 ␦ ⑀ 2N−1 . 共39兲 This relation clearly shows that the QA needs a time exponential in N to converge. For QA with many-body transverse interactions, the exponent of t in annealing schedule 共33兲 does not depend on the system size N. Nevertheless, it also does not mean that QA provides a polynomial-time algorithm because of the factor 2N. The characteristic time for ⌫MTI to reach a sufficiently small value ⑀ is estimated as tMTI ⬇ 2N−2 , ␦⑀ 共40兲 which again shows exponential dependence on N. These exponential computational complexities do not come as a surprise because Theorems 2.2, 2.4, and 2.5 all apply to any optimization problem written in generic form 共18兲, which includes the worst cases of most difficult problems. Similar arguments apply to SA.34 Another remark is on the comparison of ⌫共t兲共⬀t−1/共2N−1兲兲 in QA with T共t兲共⬀N / log共␣t + 1兲兲 in SA to conclude that the former schedule is faster than the latter. The transverse-field coefficient ⌫ in a quantum system plays the same role qualitatively and quantitatively as the temperature T does in a corresponding classical system at least in the Hopfield model in a transverse field.35 When the phase diagram is written in terms of ⌫ and ␣ 共the number of embedded patterns divided by the number of neurons兲 for the ground state of the model, the result has precisely the same structure as the T-␣ phase diagram of the finite-temperature version of the Hopfield model without a transverse field. This example serves as a justification of the direct comparison of ⌫ and T at least as long as the theoretical analyses of QA and SA are concerned. III. CONVERGENCE CONDITION OF SA AND QUANTUM ADIABATICITY We next study the convergence condition of SA to be compared with QA. This problem was originally solved by Geman and Geman33 using the theory of the inhomogeneous Markov chain as described in the quantum Monte Carlo context in Sec. V. It is quite surprising that their result is reproduced using the quantum adiabaticity condition applied after a classical-quantum mapping.31 This approach is reviewed in this section, following Ref. 31, to clarify the correspondence between the quasiequilibrium condition for SA in a classical system and the adiabaticity condition in the corresponding quantum system. The analysis will also reveal an aspect related to the equivalence of QA and QAE. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-10 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori A. Classical-quantum mapping The starting point is an expression of a classical thermal expectation value in terms of a quantum ground-state expectation value. A well-known mapping between quantum and classical systems is to rewrite the former in terms of the latter with an extra IT 共or Trotter兲 dimension.24 The mapping discussed in the present section is a different one, which allows us to express the thermal expectation value of a classical system in terms of the ground-state expectation value of a corresponding quantum system without an extra dimension. Suppose that the classical Hamiltonian, whose value we want to minimize, is written as an Ising spin system as in 共18兲: N H = − 兺 Jizi − 兺 Jijzi zj − 兺 Jijkzi zj zk − ¯ . i=1 ij 共41兲 ijk The thermal expectation value of a classical physical quantity Q共兵zi 其兲 is 具Q典T = 1 兺 e−HQ共兵i其兲, Z共T兲 兵其 共42兲 where the sum runs over all configurations of Ising spins, i.e., over the values taken by the z components of the Pauli matrices, zi = i共⫾1兲 共∀i兲. The symbol 兵i其 stands for the set 兵1 , 2 , . . . , N其. An important element is the following theorem. Theorem 3.1: Thermal expectation value (42) is equal to the expectation value of Q by the quantum wave function 兩共T兲典 = e−H/2 兺 兩兵i其典, 兵其 共43兲 where 兩兵i其典 is the basis state diagonalizing each zi as i . The sum runs over all such possible assignments. Assume T ⬎ 0 . Wave function (43) is the ground state of the quantum Hamiltonian Hq共T兲 = − 兺 Hqj共T兲 ⬅ − 兺 共xj − eH j兲, j 共44兲 j where H j is the sum of the terms of Hamiltonian (41) involving site j , H j = − J jzj − 兺 J jkzj zk − 兺 J jklzj zkzl − ¯ . k 共45兲 kl The coefficient is defined by = e−p with p = max j兩H j兩 . Proof: The first half is trivial: 具共T兲兩Q兩共T兲典 1 = 兺 e−H具兵i其兩Q兩兵i其典 = 具Q典T . 具共T兲兩共T兲典 Z共T兲 兵其 共46兲 To show the second half, we first note that xj 兺 兩兵i其典 = 兺 兩兵i其典 兵其 兵其 共47兲 since the operator xj just changes the order of the above summation. It is also easy to see that xje−H/2 = eH je−H/2xj 共48兲 because Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-11 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing xje−H/2xj = e−共H−H j兲/2xje−H j/2xj = e−共H−H j兲/2eH j/2 = eH je−H/2 共49兲 as both H and H j are diagonal in the present representation and H − H j does not include zj , so 关H − H j , xj 兴 = 0. We therefore have Hqj共T兲兩共T兲典 = 共xj − eH j兲e−H/2 兺 兩兵i其典 = 0. 兵其 共50兲 Thus 兩共T兲典 is an eigenstate of Hq共T兲 with eigenvalue 0. In the present representation, the nonvanishing off-diagonal elements of −Hq共T兲 are all positive and the coefficients of 兩共T兲典 are also all positive as one sees in 共43兲. Then 兩共T兲典 is the unique ground state of Hq共T兲 according to the 䊐 Perron–Frobenius theorem.36 A few remarks are in order. In the high-temperature limit, the quantum Hamiltonian is composed just of the transverse-field term, Hq共T → ⬁兲 = − 兺 共xj − 1兲. 共51兲 j Correspondingly the ground-state wave function 兩共T → ⬁兲典 is the simple summation over all possible states with equal weight. In this way the thermal fluctuations in the original classical system are mapped to the quantum fluctuations. The low-temperature limit has, in contrast, the purely classical Hamiltonian Hq共T ⬇ 0兲 → 兺 eH j 共52兲 j and the ground state of Hq共T ⬇ 0兲 is also the ground state of H as is apparent from definition 共43兲. Hence the decrease in thermal fluctuations in SA is mapped to the decrease in quantum fluctuations. As explained below, this correspondence allows us to analyze the condition for quasiequilibrium in the classical SA using the adiabaticity condition for the quantum system. B. Adiabaticity and convergence condition of SA The adiabaticity condition applied to the quantum system introduced above leads to the condition of convergence of SA. Suppose that we monotonically decrease the temperature as a function of time, T共t兲, to realize SA. Theorem 3.2: The adiabaticity condition for the quantum system of Hq共T兲 yields the time dependence of T共t兲 as T共t兲 = pN log共␣t + 1兲 共53兲 in the limit of large N . The coefficient ␣ is exponentially small in N . A few lemmas will be useful to prove this theorem. Lemma 3.3: The energy gap ⌬共T兲 of Hq共T兲 between the ground state and the first excited state is bounded below as ⌬共T兲 ⱖ a冑Ne−共p+c兲N , 共54兲 where a and c are N -independent positive constants in the asymptotic limit of large N . Proof: The analysis of Sec. II B 1 applies with the replacement of ⌫共t兲 by = e−p and 0共t兲 = 0. This latter condition comes from Hq共T兲兩共T兲典 = 0. The condition ⌫共t兲 ⬍ ⌫0共t ⬎ t0兲 is unnecessary here because the off-diagonal element can always be chosen smaller than the diagonal elements by adding a positive constant to the diagonal. Equation 共27兲 gives ⌬ j共t兲 ⱖ Ae−pN 共55兲 and A satisfies, according to 共28兲, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-12 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori A ⬇ b冑2Ne−cN , 共56兲 䊐 with b and c positive constants of O共N 兲. Lemma 3.4: The matrix element of the derivative of Hq共T兲 , relevant to the adiabaticity condition, satisfies 0 具1共T兲兩THq共T兲兩共T兲典 = − ⌬共T兲具1共T兲兩H兩共T兲典 , 2kBT2 共57兲 where 1共T兲 is the normalized first excited state of Hq共T兲 . Proof: By differentiating the identity Hq共T兲兩共T兲典 = 0, 共58兲 we find 冉 冊 冉 冊 1 H 兩共T兲典. Hq共T兲 兩共T兲典 = − Hq共T兲 兩共T兲典 = Hq共T兲 − T T 2kBT2 共59兲 This relation immediately proves the lemma if we notice that the ground-state energy of Hq共T兲 is 䊐 zero and therefore Hq共T兲兩1共T兲典 = ⌬共T兲兩1共T兲典. Lemma 3.5: The matrix element of H satisfies 兩具1共T兲兩H兩共T兲典兩 ⱕ pN冑Z共T兲. 共60兲 Proof: There are N terms in H = 兺 jH j, each of which is of norm of at most p. The factor 冑Z共T兲 䊐 appears from normalization of 兩共T兲典. Proof of Theorem 3.2: The condition of adiabaticity for the quantum system Hq共T兲 reads 1 ⌬共T兲 冑Z共T兲 2 冏 具1共T兲兩THq共T兲兩共T兲典 冏 dT = ␦, dt 共61兲 with sufficiently small ␦. If we rewrite the matrix element by Lemma 3.4, the left-hand side is 冏 冏 兩具1共T兲兩H兩共T兲典兩 dT . 2kBT2⌬共T兲冑Z共T兲 dt 共62兲 By replacing the numerator by its bound in Lemma 3.5 we have 冏 冏 pN dT = ˜␦ Ⰶ 1 2kBT2⌬共T兲 dt 共63兲 as a sufficient condition for adiabaticity. Using the bound of Lemma 3.3 and integrating the above differential equation for T共t兲 noticing dT / dt ⬍ 0, we reach the statement of Theorem 3.2. 䊐 C. Remarks Equation 共53兲 reproduces the Geman–Geman condition for convergence of SA.33 Their method of proof is to use the theory of the classical inhomogeneous 共i.e., time-dependent兲 Markov chain representing nonequilibrium processes. It may thus be naively expected that the classical system under consideration may not stay close to equilibrium during the process of SA since the temperature always changes. It therefore comes as a surprise that the adiabaticity condition, which is equivalent to the quasiequilibrium condition according to Theorem 3.1, leads to Theorem 3.2. The rate of temperature change in this latter theorem is slow enough to guarantee the quasiequilibrium condition even when the temperature keeps changing. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-13 Mathematical foundation of quantum annealing J. Math. Phys. 49, 125210 共2008兲 Also, Theorem 3.2 is quite general, covering the worst cases, as it applies to any system written as the Ising model of 共41兲. This fact means that one may apply a faster rate of temperature decrease to solve a given specific problem with small errors. The same comment applies to the QA situation in Sec. II. Another remark is on the relation of QA and QAE. Mathematical analyses of QA often focus their attention to the generic convergence conditions in the infinite-time limit as seen in Secs. II and V as well as in an earlier paper,7 although the residual energy after finite-time evolution has also been extensively investigated mainly in numerical studies. This aspect may have led some researchers to think that QA is different from QAE since the studies using the latter mostly concern the computational complexity of finite-time evolution for a given specific optimization problem using adiabaticity to construct an algorithm of QAE. As has been shown in the present and the previous sections, the adiabaticity condition also leads to the convergence condition in the infinite-time limit for QA and SA. In this sense QA, QAE, and even SA share essentially the same mathematical background. IV. REDUCTION IN ERRORS FOR FINITE-TIME EVOLUTION In Sec. II, we discussed the convergence condition of QA implemented for the TFIM. The power decrease in the transverse field guarantees the adiabatic evolution. This annealing schedule, however, does not provide practically useful algorithms because an infinitely long time is necessary to reach the exact solution. An approximate algorithm for finite annealing time should be used in practice. Since such a finite-time algorithm does not satisfy the generic convergence condition, the answer includes a certain amount of errors. An important question is how the error depends on the annealing time . Suzuki and Okada16 showed that the error after adiabatic evolution for time is generally proportional to −2 in the limit of large with the system size N kept finite. In this section, we analyze their results in detail and propose new annealing schedules which show smaller errors proportional to −2m 共m ⬎ 1兲.37 This method allows us to reduce errors by orders of magnitude without compromising the computational complexity apart from a possibly moderate numerical factor. A. Upper bound for excitation probability Let us consider general time-dependent Hamiltonian 共1兲. The goal of this section is to evaluate the excitation probability 共closely related with the error probability兲 at the final time s = 1 under adiabaticity condition 共16兲. This task is easy because we have already obtained the asymptotic form of excitation amplitude 共11兲. The upper bound for the excitation probability is derived as 兩具j共1兲兩˜共1兲典兩2 = 兩c j⫽0共1兲兩2 ⱗ 1 关兩A 共0兲兩 + 兩A j共1兲兩兴2 + O共−3兲. 2 j 共64兲 This formula indicates that the coefficient of the −2 term is determined only by the state of the system at s = 0 and 1 and vanishes if A j共s兲 is zero at s = 0 and 1. When the −2 term vanishes, a similar calculation yields the next order term of the excitation probability. If H̃⬘共0兲 = H̃⬘共1兲 = 0, the excitation amplitude c j⫽0共1兲 is at most of order −2 and then c0共1兲 ⬇ 1 + O共−3兲. Therefore we have c j⫽0共1兲 ⬇ 冕 1 ds 0 1 ei关 j共s兲−0共s兲兴 dH̃共s兲 具j共s兲兩 兩0共s兲典 + O共−3兲 ⬇ 2 关A共2兲 共0兲 − ei关 j共s兲−0共s兲兴A共2兲 j 共1兲兴 ⌬ j共s兲 ds j + O共−3兲, 共65兲 where we defined Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-14 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori FIG. 1. 共Color online兲 Examples of annealing schedules with reduced errors listed in 共70兲–共73兲. A共m兲 j 共s兲 ⬅ 1 dmH̃共s兲 兩0共s兲典. m+1 具j共s兲兩 ⌬ j共s兲 dsm 共66兲 To derive the second line of 共65兲, we used integration by parts twice and 共5兲 and 共6兲. The other −2 terms vanish because of the assumption H̃⬘共0兲 = H̃⬘共1兲 = 0. Thus the upper bound of the next order for the excitation probability under this assumption is obtained as 兩具j共1兲兩˜共1兲典兩2 ⱗ 1 共2兲 2 −5 关兩A 共0兲兩 + 兩A共2兲 j 共1兲兩兴 + O共 兲. 4 j 共67兲 It is easy to see that the −4 term also vanishes when H̃⬙共0兲 = H̃⬙共1兲 = 0. It is straightforward to generalize these results to prove the following theorem. Theorem 4.1: If the kth derivative of H̃共s兲 is equal to zero at s = 0 and 1 for all k = 1 , 2 , . . . , m − 1 , the excitation probability has the upper bound 兩具j共1兲兩˜共1兲典兩2 ⱗ 1 2 −2m−1 关兩A共m兲共0兲兩 + 兩A共m兲 兲. j 共1兲兩兴 + O共 2m j 共68兲 B. Annealing schedules with reduced errors Although we have so far considered the general time-dependent Hamiltonian, the ordinary Hamiltonian for QA with finite annealing time is composed of the potential term and the kinetic energy term, H̃共s兲 = f共s兲Hpot + 关1 − f共s兲兴Hkin , 共69兲 where Hpot and Hkin generalize HIsing and HTF in Sec. II, respectively. The function f共s兲, representing the annealing schedule, satisfies f共0兲 = 0 and f共1兲 = 1. Thus H̃共0兲 = Hkin and H̃共1兲 = Hpot. The ground state of Hpot corresponds to the solution of the optimization problem. The kinetic energy is chosen so that its ground state is trivial. The above Hamiltonian connects the trivial initial state and the nontrivial desired solution after evolution time . The condition for the −2m term to exist in the error is obtained straightforwardly from the results of Sec. V because Hamiltonian 共69兲 depends on time only through the annealing schedule f共s兲. It is sufficient that the kth derivative of f共s兲 is zero at s = 0 and 1 for k = 1 , 2 , . . . , m − 1. We note that f共s兲 should belong to Cm, that is, f共s兲 is an mth differentiable function whose mth derivative is continuous. Examples of the annealing schedules f m共s兲 with the −2m error rate are the following polynomials 共Fig. 1兲: Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-15 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing ½ ¼ ½ ¼ ½ ´ µ ´ µ ´ µ ´ µ ¼ ½ ½ ¼ ½ ¼ ¼ ½ ¼ ¼ ¼ ½ ¼ ¼ ¼ ¼ FIG. 2. 共Color online兲 The annealing-time dependence of the excitation probability for two-level system 共74兲 using schedules 共70兲–共73兲. The curved and straight lines show 共75兲 and 共76兲 for each annealing schedule, respectively. The parameters in 共74兲 are chosen to be h = 2 and ␣ = 0.2. f 1共s兲 = s, 共70兲 f 2共s兲 = s2共3 − 2s兲, 共71兲 f 3共s兲 = s3共10 − 15s + 6s2兲, 共72兲 f 4共s兲 = s4共35 − 84s + 70s2 − 20s3兲. 共73兲 The linear annealing schedule f 1共s兲, which shows the −2 error, has been used in the past studies. Although we here list only polynomials symmetrical with respect to the point s = 1 / 2, this is not essential. For example, f共s兲 = 共1 − cos共s2兲兲 / 2 also has the −4 error rate because f ⬘共0兲 = f ⬘共1兲 = f ⬙共0兲 = 0 but f ⬙共1兲 = −22. C. Numerical results 1. Two-level system To confirm the upper bound for the excitation probability discussed above, it is instructive to study the two-level system, the Landau–Zener problem, with the Hamiltonian HLZ共t兲 = − 冋 冉 冊册 1 t −f 2 hz − ␣x . 共74兲 The energy gap of HLZ共t兲 has the minimum 2␣ at f共s兲 = 1 / 2. If the annealing time is not large enough to satisfy 共16兲, nonadiabatic transitions occur. The Landau–Zener theorem38,39 provides the excitation probability Pex共兲 = 兩具1共1兲 兩 ˜共1兲典兩2 as 冋 Pex共兲 = exp − 册 ␣ 2 , f ⬘共sⴱ兲h 共75兲 where sⴱ denotes the solution of f共sⴱ兲 = 1 / 2. On the other hand, if is sufficiently large, the system evolves adiabatically. Then the excitation probability has upper bound 共68兲, which is estimated as Pex共兲 ⱗ 4h2␣2 2m共h2 + 4␣2兲m+2 冋冏 冏 冏 冏册 dm f dm f 共0兲 + 共1兲 dsm dsm 2 . 共76兲 We numerically solved Schrödinger equation 共2兲 for system 共74兲 with the Runge–Kutta method.40 Figure 2 shows the result for the excitation probability with annealing schedules 共70兲–共73兲. The initial state is the ground state of HLZ共0兲. The parameters are chosen to be h = 2 and ␣ = 0.2. The curved and straight lines show 共75兲 and 共76兲, respectively. In the small and large regions, the excitation probability perfectly fits to those two expressions. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori 0.29 −0.15 −0.54 0.08 0.77 0.66 −0.45 −0.73 0.31 0.29 125210-16 −0.44 −0.6 FIG. 3. 共Color online兲 Configuration of random interactions 兵Jij其 on the 3 ⫻ 3 square lattice which we investigated and spin configuration of the target state. The solid and dashed lines indicate ferromagnetic and antiferromagnetic interactions, respectively. 2. Spin-glass model We next carried out simulations of a rather large system, the Ising spin system with random interactions. The quantum fluctuations are introduced by the uniform transverse field. Thus, the potential and kinetic energy terms are defined by N Hpot = − 兺 Jijzi zj − h 兺 zi , 具ij典 共77兲 i=1 N Hkin = − ⌫ 兺 xi . 共78兲 i=1 The initial state, namely, the ground state of Hkin, is the all-up state along the x axis. The difference between the obtained approximate energy and the true ground-state energy 共exact solution兲 is the residual energy Eres. It is a useful measure of the error rate of QA. It has the same behavior as the excitation probability because it is rewritten as Eres ⬅ 具˜共1兲兩Hpot兩˜共1兲典 − 0共1兲 共79兲 = 兺 ⌬ j共1兲兩具j共1兲兩˜共1兲典兩2 . 共80兲 j⬎0 Therefore Eres is expected to be asymptotically proportional to −2m using the improved annealing schedules. We investigated the two-dimensional square lattice of size 3 ⫻ 3. The quenched random coupling constants 兵Jij其 are chosen from the uniform distribution between −1 and +1, as shown in Fig. 3. The parameters are h = 0.1 and ⌫ = 1. Figure 4 shows the dependence of the residual energy using annealing schedules 共70兲–共73兲. Straight lines representing −2m 共m = 1 , 2 , 3 , 4兲 are also shown for comparison. The data clearly indicate the −2m law for large . The irregular behavior around Eres ⬇ 10−25 comes from numerical rounding errors. 3. Database search problem As another example, we apply the improved annealing schedule to the database search problem of an item in an unsorted database. Consider N items, among which one is marked. The goal of this problem is to find the marked item in a minimum time. The pioneering quantum algorithm proposed by Grover41 solves this task in time of order 冑N, whereas the classical algorithm tests N / 2 items on average. Farhi et al.26 proposed a QAE algorithm and Roland and Cerf42 found a Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-17 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing FIG. 4. 共Color online兲 The annealing-time dependence of the residual energy for the two-dimensional spin-glass model with improved annealing schedules. The solid lines denote functions proportional to −2m 共m = 1 , 2 , 3 , 4兲. The parameter values are h = 0.1 and ⌫ = 1. QAE-based algorithm with the same computational complexity as Grover’s algorithm. Although their schedule is optimal in the sense that the excitation probability by the adiabatic transition is equal to a small constant at each time, it has the −2 error rate. We show that annealing schedules with the −2m error rate can be constructed by a slight modification in their optimal schedule. Let us consider the Hilbert space which has the basis states 兩i典 共i = 1 , 2 , . . . , N兲, and the marked state is denoted by 兩m典. Suppose that we can construct Hamiltonian 共69兲 with two terms, Hpot = 1 − 兩m典具m兩, N 共81兲 N 1 Hkin = 1 − 兺 兺 兩i典具j兩. N i=1 j=1 共82兲 The Hamiltonian Hpot can be applied without the explicit knowledge of 兩m典, the same assumption as in Grover’s algorithm. The initial state is a superposition of all basis states, 兩共0兲典 = 1 N 兩i典, 冑N 兺 i=1 共83兲 which does not depend on the marked state. The energy gap between the ground state and the first excited state, ⌬1共s兲 = 冑 1−4 N−1 f共s兲关1 − f共s兲兴, N 共84兲 has a minimum at f共s兲 = 1 / 2. The highest eigenvalue 2共s兲 = 1 is 共N − 2兲-fold degenerate. To derive the optimal annealing schedule, we briefly review the results reported by Roland and Cerf.42 When the energy gap is small 关i.e., for f共s兲 ⬇ 1 / 2兴, nonadiabatic transitions are likely to occur. Thus we need to change the Hamiltonian carefully. On the other hand, when the energy gap is not very small, too slow a change wastes time. Thus the speed in parameter change should be adjusted adaptively to the instantaneous energy gap. This is realized by tuning the annealing schedule to satisfy adiabaticity condition 共16兲 in each infinitesimal time interval, that is, 兩A1共s兲兩 = ␦, 共85兲 where ␦ is a small constant. In the database search problem, this condition is rewritten as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-18 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 FIG. 5. 共Color online兲 The optimal annealing schedules for the database search problem 共N = 64兲. The solid line denotes original optimal schedule 共87兲 and the dashed lines are for the modified schedules. 冑N − 1 df = ␦. N⌬1共s兲3 ds 共86兲 After integration under boundary conditions f共0兲 = 0 and f共1兲 = 1, we obtain f opt共s兲 = 1 2s − 1 + . 2 2冑N − 共N − 1兲共2s − 1兲2 共87兲 As plotted by a solid line in Fig. 5, this function changes most slowly when the energy gap takes the minimum value. It is noted that the annealing time is determined by the small constant ␦ as 冑N − 1 = ␦ 共88兲 , which means that the computation time is of order 冑N, similarly to Grover’s algorithm. Optimal annealing schedule 共87兲 shows the −2 error rate because its derivative is nonvanishing at s = 0 and 1. It is easy to see from 共87兲 that the simple replacement of s with f m共s兲 fulfills the condition for the −2m error rate. We carried out numerical simulations for N = 64 with such 共m兲 annealing schedules, f opt 共s兲 ⬅ f opt共f m共s兲兲, as plotted by dashed lines in Fig. 5. As shown in Fig. 6, 共m兲 共s兲 is proportional to −2m. The characteristic time c for the −2m error the residual energy with f opt FIG. 6. 共Color online兲 The annealing-time dependence of the residual energy for the database search problem 共N = 64兲 with the optimal annealing schedules described in Fig. 5. The solid lines represent functions proportional to −2m 共m = 1 , 2 , 3 , 4兲. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-19 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing 共m兲 rate to show up increases with m: Since the modified optimal schedule f opt 共s兲 has a steeper slope at s = 1 / 2 than f opt共s兲, a longer annealing time is necessary to satisfy adiabaticity condition 共86兲. 共m兲 共s兲 is only a factor of O共1兲, and therefore c is still Nevertheless, the difference in slopes of f opt scaled as 冑N. Significant qualitative reduction in errors has been achieved without compromising computational complexity apart from a numerical factor. D. Imaginary-time Schrödinger dynamics So far, we have concentrated on QA following the RT Schrödinger dynamics. From the point of view of physics, it is natural that the time evolution of a quantum system obeys the RT Schrödinger equation. Since our goal is to find the solution of optimization problems, however, we need not stick to physical reality. We therefore investigate QA following the IT Schrödinger dynamics here to further reduce errors. The IT evolution tends to filter out the excited states. Thus, it is expected that QA with the IT dynamics can find the optimal solution more efficiently than RT-QA. Stella et al.20 investigated numerically the performance of IT-QA and conjectured that 共i兲 the IT error rate is not larger than in the RT and that 共ii兲 the asymptotic behavior of the error rate for → ⬁ is identical for IT-QA and RT-QA. We prove their conjectures through the IT version of the adiabatic theorem. 1. Imaginary-time Schrödinger equation The IT Schrödinger equation is obtained by the transformation t → −it in the time derivative of the original RT Schrödinger equation: − d 兩⌿共t兲典 = H共t兲兩⌿共t兲典. dt 共89兲 The time dependence of the Hamiltonian does not change. If the Hamiltonian is time independent, we easily see that the excitation amplitude decreases exponentially relative to the ground state, 兩⌿共t兲典 = 兺 c je−it j兩j典 → 兺 c je−t j兩j典 = e−t0 兺 c je−t共 j−0兲兩j典. j j 共90兲 j However, it is not obvious that this feature survives in the time-dependent situation. An important aspect of the IT Schrödinger equation is nonunitarity. The norm of the wave function is not conserved. Thus, we consider the normalized state vector 兩共t兲典 ⬅ 1 冑具⌿共t兲兩⌿共t兲典 兩⌿共t兲典. 共91兲 The equation of motion for this normalized state vector is − d 兩共t兲典 = 关H共t兲 − 具H共t兲典兴兩共t兲典, dt 共92兲 where we defined the expectation value of the Hamiltonian, 具H共t兲典 ⬅ 具共t兲兩H共t兲兩共t兲典. 共93兲 The above equation is not linear but norm conserving, which makes the asymptotic expansion easy. In terms of the dimensionless time s = t / , the norm-conserving IT Schrödinger equation is written as − d ˜ 兩共s兲典 = 关H̃共s兲 − 具H̃共s兲典兴兩˜共s兲典. ds 共94兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-20 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori 2. Asymptotic expansion of the excitation probability To prove the conjecture by Stella et al.,20 we derive the asymptotic expansion of the excitation probability. The following theorem provides us with the IT version of the adiabatic theorem. Theorem 4.2: Under the same hypothesis as in Theorem 2.1, the state vector following the norm-conserving IT Schrödinger equation (94) has the asymptotic form in the limit of large as 兩˜共s兲典 = 兺 c j共s兲兩j共s兲典, 共95兲 c0共s兲 ⬇ 1 + O共−2兲, 共96兲 j c j⫽0共s兲 ⬇ A j共s兲 + O共−2兲. 共97兲 Proof: Norm-conserving IT Schrödinger equation 共94兲 is rewritten as the equation of motion for c j共s兲 as 冋 册 ck共s兲 dH̃共s兲 dc j 具j共s兲兩 =兺 兩k共s兲典 − c j共s兲 j共s兲 − 兺 l共s兲兩cl共s兲兩2 . ds k⫽j j共s兲 − k共s兲 ds l 共98兲 To remove the second term on the right-hand side, we define 冉冕 冋 s ds̃ j共s̃兲 − 兺 l共s̃兲兩cl共s̃兲兩2 c̃ j共s兲 ⬅ exp 0 l 册冊 c j共s兲 共99兲 and obtain the equation of motion for c̃ j共s兲 as e兵 j共s兲−k共s兲其 dH̃共s兲 dc̃ j = 兺 c̃k共s兲 具j共s兲兩 兩k共s兲典, ds k⫽j j共s兲 − k共s兲 ds 共100兲 where we defined j共s兲 ⬅ 兰s0ds⬘ j共s⬘兲 for convenience. Integration of this equation yields the integral equation for c̃ j共s兲. It is useful to introduce the following quantity: ␦共s兲 ⬅ 冕 s 0 ds̃ 兺 关l共s̃兲 − 0共s̃兲兴兩cl共s̃兲兩2 . 共101兲 l⫽0 Since the norm of the wave function is conserved, 兺l兩cl共s兲兩2 = 1 and therefore 兺l l共s兲兩cl共s兲2兩 = 0共s兲 + l⫽0 兺 关l共s兲 − 0共s兲兴兩cl共s兲2兩. 共102兲 Thus, the definition of c̃ j共s兲 is written as c̃ j共s兲 = e−␦共s兲e关 j共s兲−0共s兲兴c j共s兲. 共103兲 Finally we obtain the integral equation for c j共s兲: c0共s兲 = e ␦共s兲 +e ␦共s兲 冕 s 0 ds̃e −␦共s̃兲 兺 cl共s̃兲 l⫽0 具0共s̃兲兩 dH̃ ds̃ 兩l共s̃兲典 0共s̃兲 − l共s̃兲 , 共104兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-21 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing c j⫽0共s兲 = e ␦共s兲 −关 j共s兲−0共s兲兴 e 冕 s ds̃e −␦共s̃兲 关 j共s̃兲−0共s̃兲兴 e 兺 ck共s̃兲 具j共s̃兲兩 k⫽j 0 dH̃ ds̃ 兩k共s̃兲典 j共s̃兲 − k共s̃兲 , 共105兲 where we used the initial conditions c0共0兲 = 1 and c j⫽0 = 0. The next step is the asymptotic expansion of these integral equations for large . It is expected that c0共s兲 = 1 and c j⫽0共s兲 = 0 for → ⬁ because of the following argument: Since coefficient c0共s兲 is less than unity, ␦共s兲 should be O共−1兲 at most and e␦共s兲 = O共1兲. The second factor on the right-hand side of 共105兲 is small exponentially with because j共s兲 − 0共s兲 is positive and an increasing function of s. Thus, c j⫽0共s兲 → 0 and then c0共s兲 → 1 owing to the norm conservation law. Therefore we estimate the next term of order −1 under the assumption that c0共s兲 Ⰷ c j⫽0共s兲. Since ␦共s兲 is proportional to the square of c j⫽0共s兲, we have e␦共s兲 ⬇ 1. Thus, the e⫾␦共s兲 factors can be ignored in the −1 term estimation of 共105兲. Consequently, evaluation of the integral equations yields c j⫽0共s兲 ⬇ e−关 j共s兲−0共s兲兴 冕 s 具j共s̃兲兩 ds̃e关 j共s̃兲−0共s̃兲兴 0 dH̃ ds̃ 兩0共s̃兲典 ⌬ j共s̃兲 + O共−2兲. 共106兲 The excitation amplitude is estimated by integration by parts as 1 c j⫽0共s兲 ⬇ 关A j共s兲 − e−共 j共s兲−0共s兲兲A j共0兲兴 + O共−2兲, 共107兲 where A j共s兲 is defined by 共12兲. The second term in the square brackets is vanishingly small, which is a different point from the RT dynamics. From the above expression, we find ␦共s兲 = O共−2兲, that is, e␦共s兲 ⬇ 1 + O共−1兲. Therefore, we obtain 共96兲 and 共97兲, which is consistent with the assumption 䊐 c0共s兲 Ⰷ c j⫽0共s兲. Remark: The excitation probability at the end of a QA process is proportional to −2 in the large limit: 兩具j共1兲兩˜共1兲典兩2 ⬇ 1 兩A 共1兲兩2 + O共−3兲. 2 j 共108兲 Its difference from the upper bound for RT dynamics 共64兲 is only in the absence of A j共0兲. In the IT dynamics, this term decreases exponentially because of the factor e−共 j共s兲−0共s兲兲. This result proves the conjecture proposed by Stella et al.,20 that is, ⑀IT共兲 ⱕ ⑀RT共兲, ⑀IT共兲 ⬇ ⑀RT共兲 共 → ⬁兲. 共109兲 共110兲 Strictly speaking, the right-hand sides in the above equations denote the upper bound for the error rate for RT-QA, not the error rate itself. In some systems, for example, the two-level system, the error rate oscillates because A j共0兲 and A j共1兲 may cancel in 共11兲 and becomes smaller than that of IT-QA at some . However, QA for ordinary optimization problems has different energy levels at initial and final times, and thus such a cancellation seldom occurs. 3. Numerical verification We demonstrate a numerical verification of the above results by simulations of the IT and RT Schrödinger equations. For this purpose, we consider the following annealing schedules 共Fig. 7兲: Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-22 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori Õ ¾ × × Õ ½ FIG. 7. 共Color online兲 The annealing schedules defined in 共111兲. f sq1共s兲 and f sq2共s兲 have a vanishing slope at the initial time s = 0 and the final time s = 1, respectively. f sq1共s兲 = s2, f sq2共s兲 = s共2 − s兲. 共111兲 The former has a zero slope at the initial time s = 0 and the latter at s = 1. Thus, the A j共0兲 and A j共1兲 terms vanish with f sq1共s兲 and f sq2共s兲, respectively. Since the error rate for IT-QA depends only on A j共1兲, IT-QA with f sq2共s兲 should show the −4 error rate, while RT-QA with f sq2共s兲 exhibits the −2 law. On the other hand, RT-QA and IT-QA with f sq1共s兲 should have the same error rate for large . Figure 8 shows the residual energy with two annealing schedules for the spin-glass model presented in Sec. IV C 2, which explicitly supports our results. V. CONVERGENCE CONDITION OF QA: QUANTUM MONTE CARLO EVOLUTION So far, we have discussed QA with the Schrödinger dynamics. When we solve the Schrödinger equation on the classical computer, the computation time and memory increase exponentially with the system size. Therefore, some approximations are necessary to simulate QA processes for large-size problems. In most numerical studies, stochastic methods are used. In this section, we investigate two types of quantum Monte Carlo methods and prove their convergence theorems, following Ref. 43. A. Inhomogeneous Markov chain Since we prove the convergence of stochastic processes, it is useful to recall various definitions and theorems for inhomogeneous Markov processes.5 We denote the space of discrete states by S and assume that the size of S is finite. A Monte Carlo step is characterized by the transition probability from state x 共苸S兲 to state y 共苸S兲 at time step t: Residual Energy ½ ½ ¼ IT, sq1 ½ ¼ IT, sq2 RT, sq1 ½ RT, sq2 ¼ 10 ½ ¼ ¼ ½ ¼ ½ ¼ ½ ¼ FIG. 8. 共Color online兲 The annealing-time dependence of the residual energy for IT- and RT-QA with annealing schedules f sq1共s兲 and f sq2共s兲. The system is the spin-glass model presented in Sec. IV C 2. The solid lines stand for functions proportional to −2 and −4. The parameters are h = 0.1 and ⌫ = 1. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-23 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing G共y,x;t兲 = 再 共x ⫽ y兲 P共y,x兲A共y,x;t兲 1 − 兺 z苸S P共z,x兲A共z,x;t兲 共x = y兲, 冎 共112兲 where P共y , x兲 and A共y , x ; t兲 are called the generation probability and the acceptance probability, respectively. The former is the probability to generate the next candidate state y from the present state x. We assume that this probability does not depend on time and satisfies the following conditions: ∀x,y 苸 S:P共y,x兲 = P共x,y兲 ⱖ 0, 共113兲 ∀x 苸 S:P共x,x兲 = 0, 共114兲 ∀x 苸 S: 兺 P共y,x兲 = 1, 共115兲 y苸S n−1 ∀x,y 苸 S, ∃ n ⬎ 0, ∃ z1, . . . ,zn−1 苸 S: 兿 P共zk+1,zk兲 ⬎ 0, z0 = x, zn = y. 共116兲 k=0 The last condition represents irreducibility of S, that is, any state in S can be reached from any other state in S. We define Sx as the neighborhood of x, i.e., the set of states that can be reached by a single step from x: Sx = 兵y兩y 苸 S, P共y,x兲 ⬎ 0其. 共117兲 The acceptance probability A共y , x ; t兲 is the probability to accept the candidate y generated from state x. The matrix G共t兲, whose 共y , x兲 component is given by 共112兲, 关G共t兲兴y,x = G共y , x ; t兲, is called the transition matrix. Let P denote the set of probability distributions on S. We regard a probability distribution p 共苸P兲 as the column vector with the component 关p兴x = p共x兲. The probability distribution at time t, started from an initial distribution p0 共苸P兲 at time t0, is written as p共t,t0兲 = Gt,t0 p0 ⬅ G共t − 1兲G共t − 2兲 ¯ G共t0兲p0 . 共118兲 A Markov chain is called inhomogeneous when the transition probability depends on time. In the following sections, we will prove that inhomogeneous Markov chains associated with QA are ergodic under appropriate conditions. There are two kinds of ergodicity, weak and strong. Weak ergodicity means that the probability distribution becomes independent of the initial conditions after a sufficiently long time: ∀t0 ⱖ 0:lim sup兵储p共t,t0兲 − p⬘共t,t0兲储兩p0,p0⬘ 苸 P其 = 0, t→⬁ 共119兲 where p共t , t0兲 and p⬘共t , t0兲 are the probability distributions with different initial distributions p0 and p0⬘. The norm is defined by 储p储 = 兺 兩p共x兲兩. 共120兲 x苸S Strong ergodicity is the property that the probability distribution converges to a unique distribution irrespective of the initial state: ∃r 苸 P, ∀ t0 ⱖ 0:lim sup兵储p共t,t0兲 − r储兩p0 苸 P其 = 0. t→⬁ 共121兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-24 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori The following two theorems provide conditions for weak and strong ergodicities of an inhomogeneous Markov chain.5 For proofs see Appendix B. Theorem 5.1 (condition for weak ergodicity): An inhomogeneous Markov chain is weakly ergodic if and only if there exists a strictly increasing sequence of positive numbers 兵ti其 , 共i = 0 , 1 , 2 , . . .兲 , such that ⬁ 关1 − ␣共Gt 兺 i=0 i+1,ti 兲兴 → ⬁, 共122兲 where ␣共Gti+1,ti兲 is the coefficient of ergodicity defined by 再 冎 ␣共Gti+1,ti兲 = 1 − min 兩 兺 min兵G共z,x兲,G共z,y兲其兩x,y 苸 S , z苸S 共123兲 with the notation G共z , x兲 = 关Gti+1,ti兴z,x . The coefficient of ergodicity measures the variety of the transition probability. If G共z , x兲 is independent of a state x, ␣共G兲 is equal to zero. Theorem 5.2 (condition for strong ergodicity): An inhomogeneous Markov chain is strongly ergodic if the following three conditions hold: 共1兲 共2兲 共3兲 The Markov chain is weakly ergodic. For all t there exists a stationary state pt 苸 P such that pt = G共t兲pt . pt satisfies ⬁ 储pt − pt+1储 ⬍ ⬁. 兺 t=0 共124兲 Moreover, if p = lim pt , then p is equal to the probability distribution r in (121). t→⬁ We note that the existence of the limit is guaranteed by 共124兲. This inequality implies that the probability distribution pt共x兲 is a Cauchy sequence: ∀ ⬎ 0, ∃ t0 ⬎ 0, ∀ t,t⬘ ⬎ t0:兩pt共x兲 − pt⬘共x兲兩 ⬍ . 共125兲 B. Path-integral Monte Carlo method Let us first discuss convergence conditions for the implementation of QA by the path-integral Monte Carlo 共PIMC兲 method.24,25 The basic idea of PIMC is to apply the Monte Carlo method to the classical system obtained from the original quantum system by the path-integral formula. We first consider the example of ground-state search of the Ising spin system whose quantum fluctuations are introduced by adding a transverse field. The total Hamiltonian is defined in 共20兲. Although we only treat the two-body interaction for simplicity in this section, the existence of arbitrarily many-body interactions between the z components of the Pauli matrix and longitudinal random magnetic field −兺Jizi in addition to the above Hamiltonian would not change the following argument. In the path-integral method, the d-dimensional TFIM is mapped to a 共d + 1兲-dimensional classical Ising system so that the quantum system can be simulated on the classical computer. In numerical simulations, the Suzuki–Trotter formula23,24 is usually employed to express the partition function of the resulting classical system, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-25 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing 冉 M N M 冊  共k兲 共k兲 共k+1兲 , Z共t兲 ⬇ 兺 exp 兺 兺 Jij共k兲 i j + ␥共t兲 兺 兺 i i M k=1 具ij典 共k兲 i=0 k=1 兵S 其 i 共126兲 共= ⫾ 1兲 denotes a where M is the length along the extra dimension 共Trotter number兲 and 共k兲 i classical Ising spin at site i on the kth Trotter slice. The nearest-neighbor interaction between adjacent Trotter slices, 冉 冊 1 ⌫共t兲 ␥共t兲 = log coth , 2 M 共127兲 is ferromagnetic. Approximation 共126兲 becomes exact in the limit M → ⬁ for a fixed  = 1 / kBT. The magnitude of interaction 共127兲 increases with time t and tends to infinity as t → ⬁, reflecting the decrease in ⌫共t兲. We fix M and  to arbitrarily large values, which corresponds to the actual situation in numerical simulations. Therefore the theorem presented below does not directly guarantee the convergence of the system to the true ground state, which is realized only after taking the limits M → ⬁ and  → ⬁. We will rather show that the system converges to the thermal equilibrium represented by the right-hand side of 共126兲, which can be chosen arbitrarily close to the true ground state by taking M and  large enough. With the above example of TFIM in mind, it will be convenient to treat a more general expression than 共126兲, Z共t兲 = 冉 兺 exp x苸S − 冊 F0共x兲 F1共x兲 − . T0 T1共t兲 共128兲 Here F0共x兲 is the cost function whose global minimum is the desired solution of the combinatorial optimization problem. The temperature T0 is chosen to be sufficiently small. The term F1共x兲 derives from the kinetic energy, which is the transverse field in the TFIM. Quantum fluctuations are tuned by the extra temperature factor T1共t兲, which decreases with time. The first term −F0共x兲 / T0 corresponds to the interaction term in the exponent of 共126兲, and the second term −F1共x兲 / T1共t兲 generalizes the transverse-field term in 共126兲. For partition function 共128兲, we define the acceptance probability of PIMC as A共y,x;t兲 = g q共x;t兲 = 冉 冊 q共y;t兲 , q共x;t兲 冉 共129兲 冊 1 F0共x兲 F1共x兲 exp − − . Z共t兲 T0 T1共t兲 共130兲 This q共x ; t兲 is the equilibrium Boltzmann factor at a given fixed T1共t兲. Function g共u兲 is the acceptance function, a monotone increasing function satisfying 0 ⱕ g共u兲 ⱕ 1 and g共1 / u兲 = g共u兲 / u for u ⱖ 0. For instance, for the heat bath and the Metropolis methods, we have g共u兲 = u , 1+u g共u兲 = min兵1,u其, 共131兲 共132兲 respectively. The conditions mentioned above for g共u兲 guarantee that q共x ; t兲 satisfies the detailed balance condition G共y , x ; t兲q共x ; t兲 = G共x , y ; t兲q共y ; t兲. Thus, q共x ; t兲 is the stationary distribution of the homogeneous Markov chain defined by the transition matrix G共t兲 with a fixed t. In other words, q共x ; t兲 is the right eigenvector of G共t兲 with eigenvalue 1. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-26 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori 1. Convergence theorem for PIMC-QA We first define a few quantities. The set of local maximum states of F1 is written as Sm, Sm = 兵x兩x 苸 S, ∀ y 苸 Sx, F1共y兲 ⱕ F1共x兲其. 共133兲 We denote by d共y , x兲 the minimum number of steps necessary to make a transition from x to y. Using this notation we define the minimum number of maximum steps needed to reach any other state from an arbitrary state in the set S \ Sm, R = min兵max兵d共y,x兲兩y 苸 S其兩x 苸 S \ Sm其. 共134兲 Also, L0 and L1 stand for the maximum changes in F0共x兲 and F1共x兲, respectively, in a single step, L0 = max兵兩F0共x兲 − F0共y兲兩兩P共y,x兲 ⬎ 0, x,y 苸 S其, 共135兲 L1 = max兵兩F1共x兲 − F1共y兲兩兩P共y,x兲 ⬎ 0, x,y 苸 S其. 共136兲 Our main results are summarized in the following theorem and corollary. Theorem 5.3 [strong ergodicity of system (128)]: The inhomogeneous Markov chain generated by (129) and (130) is strongly ergodic and converges to the equilibrium state corresponding to the first term of the right-hand side of (130), exp共−F0共x兲 / T0兲 , if T1共t兲 ⱖ RL1 . log共t + 2兲 共137兲 Application of this theorem to the PIMC implementation of QA represented by 共126兲 immediately yields the following corollary. Corollary 5.4 (strong ergodicity of QA-PIMC for TFIM): The inhomogeneous Markov chain generated by the Boltzmann factor on the right-hand side of (126) is strongly ergodic and converges to the equilibrium state corresponding to the first term on the right-hand side of (126) if ⌫共t兲 ⱖ 1 M tanh−1 .  共t + 2兲2/RL1 共138兲 Remark: For sufficiently large t, the above inequality reduces to ⌫共t兲 ⱖ M 共t + 2兲−2/RL1 .  共139兲 This result implies that a power decay of the transverse field is sufficient to guarantee the convergence of QA of TFIM by the PIMC. Notice that RL1 is of O共N兲. Thus 共139兲 is qualitatively similar to 共21兲. To prove strong ergodicity it is necessary to prove weak ergodicity first. The following lemma is useful for this purpose. Lemma 5.5 (lower bound on the transition probability): The elements of the transition matrix defined by (112), (129), and (130) have the following lower bound: 冉 P共y,x兲 ⬎ 0 ⇒ ∀ t ⬎ 0:G共y,x;t兲 ⱖ wg共1兲exp − L0 L1 − T0 T1共t兲 冊 共140兲 and ∃t1 ⬎ 0, ∀ x 苸 S \ S m, 冉 ∀ t ⱖ t1:G共x,x;t兲 ⱖ wg共1兲exp − 冊 L0 L1 − . T0 T1共t兲 共141兲 Here, w stands for the minimum nonvanishing value of P共y , x兲, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-27 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing w = min兵P共y,x兲兩P共y,x兲 ⬎ 0, x,y 苸 S其. 共142兲 Proof of Lemma 5.5: The first part of Lemma 5.5 is proved straightforwardly. Equation 共140兲 follows directly from the definition of the transition probability and the property of the acceptance function g. When q共y ; t兲 / q共x ; t兲 ⬍ 1, we have G共y,x;t兲 ⱖ wg 冉 冊 冉 冊 q共x;t兲 q共y;t兲 L0 L1 ⱖ wg共1兲exp − − . q共y;t兲 q共x;t兲 T0 T1共t兲 共143兲 On the other hand, if q共y ; t兲 / q共x ; t兲 ⱖ 1, 冉 G共y,x;t兲 ⱖ wg共1兲 ⱖ wg共1兲exp − 冊 L0 L1 − , T0 T1共t兲 共144兲 where we used the fact that both L0 and L1 are positive. Next, we prove 共141兲. Since x is not a member of Sm, there exists a state y 苸 Sx such that F1共y兲 − F1共x兲 ⬎ 0. For such a state y, 冉 冉 lim g exp − t→⬁ F0共y兲 − F0共x兲 F1共y兲 − F1共x兲 − T0 T1共t兲 冊冊 共145兲 =0 because T1共t兲 tends to zero as t → ⬁ and 0 ⱕ g共u兲 ⱕ u. Thus, for all ⬎ 0, there exists t1 ⬎ 0 such that 冉 冉 ∀t ⬎ t1:g exp − F0共y兲 − F0共x兲 F1共y兲 − F1共x兲 − T0 T1共t兲 冊冊 ⬍ . 共146兲 We therefore have 兺 P共z,x兲A共z,x;t兲 = P共y,x兲A共y,x;t兲 + z苸S\兵y其 兺 P共z,x兲A共z,x;t兲 ⬍ P共y,x兲 + z苸S\兵y其 兺 P共z,x兲 z苸S = 1 − 共1 − 兲P共y,x兲, 共147兲 and consequently, G共x,x;t兲 ⬎ 共1 − 兲P共y,x兲 ⬎ 0. 共148兲 Since the right-hand side of 共141兲 can be arbitrarily small for sufficiently large t, we obtain the second part of Lemma 5.5. 䊐 Proof of weak ergodicity implied in Theorem 5.3: Let us introduce the following quantity: xⴱ = arg min兵max兵d共y,x兲兩y 苸 S其兩x 苸 S \ Sm其. 共149兲 Comparison with the definition of R in 共134兲 shows that the state xⴱ is reachable by at most R transitions from any state. Now, consider the transition probability from an arbitrary state x to xⴱ. From the definitions of R and xⴱ, there exists at least one transition route within R steps: x ⬅ x0 ⫽ x1 ⫽ x2 ⫽ ¯ ⫽ xl = xl+1 = ¯ = xR ⬅ xⴱ . Then Lemma 5.5 yields that, for sufficiently large t, the transition probability at each time step has the following lower bound: 冉 G共xi+1,xi ;t − R + i兲 ⱖ wg共1兲exp − 冊 L0 L1 − . T0 T1共t − R + i兲 共150兲 Thus, by taking the product of 共150兲 from i = 0 to i = R − 1, we have Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-28 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori Gt,t−R共xⴱ,x兲 ⱖ G共xⴱ,xR−1 ;t − 1兲G共xR−1,xR−2 ;t − 2兲 ¯ G共x1,x;t − R兲 冉 R−1 ⱖ wg共1兲exp 兿 i=0 − 冊 冉 冊 L0 L1 RL0 RL1 − ⱖ wRg共1兲R exp − − , T0 T1共t − R + i兲 T0 T1共t − 1兲 共151兲 where we have used monotonicity of T1共t兲. Consequently, it is possible to find an integer k0 ⱖ 0 such that, for all k ⬎ k0, the coefficient of ergodicity satisfies 冉 1 − ␣共GkR,kR−R兲 ⱖ wRg共1兲R exp − 冊 RL0 RL1 − , T0 T1共kR − 1兲 共152兲 where we eliminate the sum over z in 共123兲 by replacing it with a single term for z = xⴱ. We now substitute annealing schedule 共137兲. Then weak ergodicity is immediately proved from Theorem 5.1 because we obtain ⬁ 共1 − ␣共G 兺 k=1 kR,kR−R 冉 冊兺 RL0 兲兲 ⱖ w g共1兲 exp − T0 R R ⬁ k=k0 1 → ⬁. kR + 1 共153兲 䊐 Proof of Theorem 5.3: To prove strong ergodicity, we refer to Theorem 5.2. Condition 共1兲 has already been proved. As has been mentioned, Boltzmann factor 共130兲 satisfies q共t兲 = G共t兲q共t兲, which is condition 共2兲. Thus the proof will be complete if we prove condition 共3兲 by setting pt = q共t兲. For this purpose, we first prove that q共x ; t兲 is monotonic for large t: ∀t ⱖ 0, ∃t1 ⬎ 0, ∀ x 苸 Smin 1 :q共x;t + 1兲 ⱖ q共x;t兲, ∀ t ⱖ t 1, ∀ x 苸 S \ Smin 1 :q共x;t + 1兲 ⱕ q共x;t兲, 共154兲 共155兲 denotes the set of global minimum states of F1. where Smin 1 To prove this monotonicity, we use the following notations for simplicity: 冉 A共x兲 = exp − 冊 F0共x兲 , T0 B= 兺 A共x兲, 共156兲 x苸Smin 1 ⌬共x兲 = F1共x兲 − Fmin 1 . 共157兲 If x 苸 Smin 1 , the Boltzmann distribution can be rewritten as A共x兲 q共x;t兲 = B+ 冉 冊 ⌬共y兲 兺min exp − T1共t兲 A共y兲 y苸S\S1 . 共158兲 Since ⌬共y兲 ⬎ 0 by definition, the denominator decreases with time. Thus, we obtain 共154兲. To prove 共155兲, we consider the derivative of q共x ; t兲 with respect to T1共t兲, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-29 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing 冋 兺 A共x兲 B⌬共x兲 + q共x;t兲 = T1共t兲 冉 冊冋 y苸S\Smin 1 ⌬共x兲 T1共t兲 exp T1共t兲 2 冉 共F1共x兲 − F1共y兲兲exp − B+ 冉 冊 ⌬共y兲 A共y兲 T1共t兲 冊 ⌬共y兲 兺min exp − T1共t兲 A共y兲 y苸S\S1 册 册 2 . 共159兲 Only F1共x兲 − F1共y兲 in the numerator has the possibility of being negative. However, the first term B⌬共x兲 is larger than the second one for sufficiently large t because exp共−⌬共y兲 / T1共t兲兲 tends to zero as T1共t兲 → ⬁. Thus there exists t1 ⬎ 0 such that q共x ; t兲 / T共t兲 ⬎ 0 for all t ⬎ t1. Since T1共t兲 is a decreasing function of t, we have 共155兲. Consequently, for all t ⬎ t1, we have 储q共t + 1兲 − q共t兲储 = 兺 关q共x;t + 1兲 − q共x;t兲兴 − x苸Smin 1 =2 兺 兺 关q共x;t + 1兲 − q共x;t兲兴 x苸Smin 1 关q共x;t + 1兲 − q共x;t兲兴, 共160兲 x苸Smin 1 where we used 储q共t兲储 = 兺x苸Sminq共x ; t兲 + 兺x苸Sminq共x ; t兲 = 1. We then obtain 1 1 ⬁ 兺 储q共t + 1兲 − q共t兲储 = 2 兺 关q共x;⬁兲 − q共x;t1兲兴 ⱕ 2储q共x;⬁兲储 = 2. 共161兲 x苸Smin 1 t=t1 Therefore q共t兲 satisfies condition 共3兲: ⬁ 兺 储q共t + 1兲 − q共t兲储 = t=0 t1−1 兺 t=0 ⬁ 储q共t + 1兲 − q共t兲储 + 兺 储q共t + 1兲 − q共t兲储 ⱕ t=t1 t1−1 关储q共t + 1兲储 + 储q共t兲储兴 + 2 兺 t=0 = 2t1 + 2 ⬍ ⬁, 共162兲 which completes the proof of strong ergodicity. 2. Generalized transition probability In Theorem 5.3, the acceptance probability is defined by the conventional Boltzmann form, 共129兲 and 共130兲. However, we have the freedom to choose any transition 共acceptance兲 probability as long as it is useful to achieve our objective since our goal is not to find finite-temperature equilibrium states but to identify the optimal state. There have been attempts to accelerate the annealing schedule in SA by modifying the transition probability. In particular, Nishimori and Inoue34 proved weak ergodicity of the inhomogeneous Markov chain for classical SA using the probability of Tsallis and Stariolo.44 There the property of weak ergodicity was shown to hold under the annealing schedule of temperature inversely proportional to a power of time steps. This annealing rate is much faster than the log-inverse law for the conventional Boltzmann factor. A similar generalization is possible for QA-PIMC by using the following modified acceptance probability: 共163兲 A共y,x;t兲 = g共u共y,x;t兲兲, 冋 u共y,x;t兲 = e−关F0共y兲−F0共x兲兴/T0 1 + 共q − 1兲 F1共y兲 − F1共x兲 T1共t兲 册 1/共1−q兲 , 共164兲 where q is a real number. In the limit q → 1, this acceptance probability reduces to the Boltzmann form. Similarly to the discussions leading to Theorem 5.3, we can prove that the inhomogeneous Markov chain with this acceptance probability is weakly ergodic if Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-30 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori T1共t兲 ⱖ b , 共t + 2兲c 0⬍cⱕ q−1 , R 共165兲 where b is a positive constant. We have to restrict ourselves to the case q ⬎ 1 for a technical reason as was the case previously.34 We do not reproduce the proof here because it is quite straightforward to generalize the discussions for Theorem 5.3 in combination with the argument of Ref. 34. Result 共165兲 applied to the TFIM is that, if the annealing schedule asymptotically satisfies ⌫共t兲 ⱖ 冉 冊 2共t + 2兲c M exp − ,  b 共166兲 the inhomogeneous Markov chain is weakly ergodic. Notice that this annealing schedule is faster than the power law of 共139兲. We have been unable to prove strong ergodicity because we could not identify the stationary distribution for a fixed T1共t兲 in the present case. 3. Continuous systems In the above analyses we treated systems with discrete degrees of freedom. Theorem 5.3 does not apply directly to a continuous system. Nevertheless, by discretization of the continuous space we obtain the following result. Let us consider a system of N distinguishable particles in a continuous space of finite volume with the Hamiltonian N H= 1 兺 p2 + V共兵ri其兲. 2m共t兲 i=1 i 共167兲 The mass m共t兲 controls the magnitude of quantum fluctuations. The goal is to find the minimum of the potential term, which is achieved by a gradual increase in m共t兲 to infinity according to the prescription of QA. After discretization of the continuous space 共which is necessary anyway in any computer simulation with finite precision兲 and an application of the Suzuki–Trotter formula, the equilibrium partition function acquires the following expression in the representation to diagonalize spatial coordinates: 冉 Z共t兲 ⬇ Tr exp − M N M 冊  Mm共t兲 2 V共兵r共k兲 , 兺 兺 兺 兩r共k+1兲 − r共k兲 i 其兲 − i 兩 M k=1 2 i=1 k=1 i 共168兲 with the unit ប = 1. Theorem 5.3 is applicable to this system under the identification of T1共t兲 with m共t兲−1. We therefore conclude that a logarithmic increase in the mass suffices to guarantee strong ergodicity of the potential-minimization problem under spatial discretization. The coefficient corresponding to the numerator of the right-hand side of 共137兲 is estimated as RL1 ⬇ M 2NL2/ , 共169兲 − r共k兲 兩r共k+1兲 i i 兩. To obtain this coefficient, let us consider two where L denotes the maximum value of extremes. One is that any state is reachable at one step. By definition, R = 1 and L1 ⬇ M 2NL2 / , which yields 共169兲. The other case is that only one particle can move to the nearest-neighbor point at one time step. With a 共ⰆL兲 denoting the lattice spacing, we have L1 ⬇ M 2 MLa 关L − 共L − a兲2兴 ⬇ . 2  共170兲 Since the number of steps to reach any configuration is estimated as R ⬇ NML / a, we again obtain 共169兲. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-31 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing C. Green’s function Monte Carlo method The PIMC simulates only the equilibrium behavior at finite temperature because its starting point is the equilibrium partition function. Moreover, it follows an artificial time evolution of Monte Carlo dynamics, not the natural Schrödinger dynamics. An alternative approach to improve these points is Green’s function Monte Carlo 共GFMC兲 method.25,45–47 The basic idea is to solve the IT Schrödinger equation by stochastic processes. In the present section we derive sufficient conditions for strong ergodicity in GFMC. The evolution of states by the IT Schrödinger equation starting from an initial state 兩0典 is expressed as 冉冕 t 兩共t兲典 = T exp − 0 冊 dt⬘H共t⬘兲 兩0典, 共171兲 where T is the time-ordering operator. The right-hand side can be decomposed into a product of small-time evolutions, 兩共t兲典 = lim Ĝ0共tn−1兲Ĝ0共tn−2兲 ¯ Ĝ0共t1兲Ĝ0共t0兲兩0典, n→⬁ 共172兲 where tk = k⌬t, ⌬t = t / n, and Ĝ0共t兲 = 1 − ⌬tH共t兲. In the GFMC, one approximates the right-hand side of this equation by a product with large but finite n and replaces Ĝ0共t兲 with Ĝ1共t兲 = 1 − ⌬t共H共t兲 − ET兲, where ET is called the reference energy to be taken approximately close to the final groundstate energy. This subtraction of the reference energy simply adjusts the standard of energy and changes nothing physically. However, practically, this term is important to keep the matrix elements positive and to accelerate convergence to the ground state as will be explained shortly. To realize the process of 共172兲 by a stochastic method, we rewrite this equation in a recursive form, k+1共y兲 = 兺 Ĝ1共y,x;tk兲k共x兲, 共173兲 x where k共x兲 = 具x 兩 k典 and 兩x典 denotes a basis state. The matrix element of Green’s function is given by Ĝ1共y,x;t兲 = 具y兩1 − ⌬t关H共t兲 − ET兴兩x典. 共174兲 Equation 共173兲 looks similar to a Markov process but is significantly different in several ways. An important difference is that Green’s function is not normalized, 兺yĜ1共y , x ; t兲 ⫽ 1. In order to avoid this problem, one decomposes Green’s function into a normalized probability G1 and a weight w: Ĝ1共y,x;t兲 = G1共y,x;t兲w共x;t兲, 共175兲 where G1共y,x;t兲 ⬅ Ĝ1共y,x;t兲 兺y Ĝ1共y,x;t兲 , w共x;t兲 ⬅ Ĝ1共y,x;t兲 . G1共y,x;t兲 共176兲 Thus, using 共173兲, the wave function at time t is written as n共y兲 = 兺 ␦y,xnw共xn−1 ;tn−1兲w共xn−2 ;tn−2兲 ¯ w共x0 ;t0兲G1共xn,xn−1 ;tn−1兲G1共xn−1,xn−2 ;tn−2兲 兵xk其 ¯ G1共x1,x0 ;t0兲0共x0兲. 共177兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-32 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori The algorithm of GFMC is based on this formula and is defined by a weighted random walk in the following sense. One first prepares an arbitrary initial wave function 0共x0兲, all elements of which are non-negative. A random walker is generated, which sits initially 共t = t0兲 at the position x0 with a probability proportional to 0共x0兲. Then the walker moves to a new position x1 following the transition probability G1共x1 , x0 ; t0兲. Thus this probability should be chosen non-negative by choosing parameters appropriately as described later. Simultaneously, the weight of this walker is updated by the rule W1 = w共x0 ; t0兲W0 with W0 = 1. This stochastic process is repeated to t = tn−1. One actually prepares M independent walkers and let those walkers follow the above process. Then, according to 共177兲, the wave function n共y兲 is approximated by the distribution of walkers at the final step weighted by Wn, M 1 兺 W共i兲n ␦y,x共i兲n , M→⬁ M i=1 n共y兲 = lim 共178兲 where i is the index of a walker. As noted above, G1共y , x ; t兲 should be non-negative, which is achieved by choosing sufficiently small ⌬t 共i.e., sufficiently large n兲 and selecting ET within the instantaneous spectrum of the Hamiltonian H共t兲. In particular, when ET is close to the instantaneous ground-state energy of H共t兲 for large t 共i.e., the final target energy兲, Ĝ1共x , x ; t兲 is close to unity whereas other matrix components of Ĝ1共t兲 are small. Thus, by choosing ET this way, one can accelerate convergence of GFMC to the optimal state in the last steps of the process. If we apply this general framework to the TFIM with the z-diagonal basis, the matrix elements of Green’s function are immediately calculated as 冦 1 − ⌬t关E0共x兲 − ET兴 共x = y兲 共x and y differ by a single-spin flip兲 Ĝ1共y,x;t兲 = ⌬t⌫共t兲 0 共otherwise兲, 冧 共179兲 E0共x兲 = 具x兩共−兺ijJijzi zj 兲兩x典. One should choose ⌬t and ET such that 1 − ⌬t共E0共x兲 − ET兲 ⱖ 0 for where all x. Since w共x , t兲 = 兺yĜ1共y , x ; t兲, the weight is given by w共x;t兲 = 1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲. 共180兲 One can decompose this transition probability into the generation probability and the acceptance probability as in 共112兲: 冦 1 共single-spin flip兲 P共y,x兲 = N 0 共otherwise兲, A共y,x;t兲 = 冧 N⌬t⌫共t兲 . 1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲 共181兲 共182兲 We shall analyze the convergence properties of stochastic processes under these probabilities for TFIM. 1. Convergence theorem for GFMC-QA Similarly to the QA by PIMC, it is necessary to reduce the strength of quantum fluctuations slowly enough in order to find the ground state in the GFMC. The following theorem provides a sufficient condition in this regard. Theorem 5.6 (strong ergodicity of QA-GFMC): The inhomogeneous Markov process of the random walker for the QA-GFMC of TFIM, (112), (181), and (182), is strongly ergodic if Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-33 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing ⌫共t兲 ⱖ b , 共t + 1兲c 0⬍cⱕ 1 . N 共183兲 The lower bound of the transition probability given in the following lemma will be used in the proof of Theorem 5.6. Lemma 5.7: The transition probability of random walk in the GFMC defined by (112), (181), and (182) has the lower bound P共y,x兲 ⬎ 0 ⇒ ∀ t ⬎ 0:G1共y,x;t兲 ⱖ ∃t1 ⬎ 0, ∀ t ⬎ t1:G1共x,x;t兲 ⱖ ⌬t⌫共t兲 , 1 − ⌬t共Emin − ET兲 + N⌬t⌫共t兲 ⌬t⌫共t兲 , 1 − ⌬t共Emin − ET兲 + N⌬t⌫共t兲 共184兲 共185兲 where Emin is the minimum value of E0共x兲 , Emin = min兵E0共x兲兩x 苸 S其. 共186兲 Proof of Lemma 5.7: The first part of Lemma 5.7 is trivial because the transition probability is an increasing function with respect to E0共x兲 when P共y , x兲 ⬎ 0 as seen in 共182兲. Next, we prove the second part of Lemma 5.7. According to 共179兲 and 共180兲, G1共x , x ; t兲 is written as G1共x,x;t兲 = 1 − N⌬t⌫共t兲 . 1 − ⌬t关E0共x兲 − ET兴 + N⌬t⌫共t兲 共187兲 Since the transverse field ⌫共t兲 decreases to zero with time, the second term on the right-hand side tends to zero as t → ⬁. Thus, there exists t1 ⬎ 0 such that G1共x , x ; t兲 ⬎ 1 − for ∀ ⬎ 0 and ∀t ⬎ t1. On the other hand, the right-hand side of 共185兲 converges to zero as t → ⬁. We therefore have 共185兲. 䊐 Proof of Theorem 5.6: We show that condition 共183兲 is sufficient to satisfy the three conditions of Theorem 5.2. 共1兲 From Lemma 5.7, we obtain a bound on the coefficient of ergodicity for sufficiently large k as 兲ⱖ 1 − ␣共GkN,kN−N 1 冋 ⌬t⌫共kN − 1兲 1 − ⌬t共Emin − ET兲 + N⌬t⌫共kN − 1兲 册 N 共188兲 in the same manner as we derived 共152兲, where we used R = N. Substituting annealing schedule 共183兲, we can prove weak ergodicity from Theorem 5.1 because ⬁ 关1 − 兺 k=1 共2兲 ⬁ ␣共GkN,kN−N 兲兴 1 ⱖ b ⬘N 兺 cN , k=k 共kN兲 共189兲 0 which diverges when 0 ⬍ c ⱕ 1 / N. The stationary distribution of the instantaneous transition probability G1共y , x ; t兲 is q共x;t兲 ⬅ w共x;t兲 兺x苸S w共x;t兲 = 1 ⌬tE0共x兲 , − 2N 2N关1 + ⌬tET + N⌬t⌫共t兲兴 共190兲 which is derived as follows. The transition probability defined by 共112兲, 共181兲, and 共182兲 is rewritten in terms of weight 共180兲 as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-34 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori G1共y,x;t兲 = 冦 1− N⌬t⌫共t兲 共x = y兲 w共x;t兲 ⌬t⌫共t兲 w共x;t兲 共x 苸 Sy ; single-spin flip兲 0 共otherwise兲. 冧 共191兲 Thus, we have 兺 G1共y,x;t兲q共x;t兲 = x苸S 冋 1− 册 ⌬t⌫共t兲 w共x;t兲 N⌬t⌫共t兲 w共y;t兲 + 兺 w共y;t兲 A A x苸Sy w共x;t兲 = q共y;t兲 − N⌬t⌫共t兲 ⌬t⌫共t兲 + 兺 1, A A x苸Sy 共192兲 where A denotes the normalization factor, 兺 w共x;t兲 = Tr x苸S 冋 冉 册 冊 1 − ⌬t − 兺 Jijzi zj − ET + N⌬t⌫共t兲 = 2N关1 + ⌬tET + N⌬t⌫共t兲兴, 具ij典 共193兲 共3兲 where we used Tr 兺 Jijzi zj = 0. Since the volume of Sy is N, 共192兲 indicates that q共x ; t兲 is the stationary distribution of G1共y , x ; t兲. The right-hand side of 共190兲 is easily derived from the above equation. Since the transverse field ⌫共t兲 decreases monotonically with t, the above stationary distribution q共x ; t兲 is an increasing function of t if E0共x兲 ⬍ 0 and is decreasing if E0 ⱖ 0. Consequently, using the same procedure as in 共160兲, we have 储q共t + 1兲 − q共t兲储 = 2 兺 E0共x兲⬍0 关q共x;t + 1兲 − q共x;t兲兴, 共194兲 and thus ⬁ 储q共t + 1兲 − q共t兲储 = 2 兺 关q共x;⬁兲 − q共x;0兲兴 ⱕ 2. 兺 t=0 E 共x兲⬍0 共195兲 0 ⬁ 储q共t + 1兲 − q共t兲储 is finite, which completes the proof of condition 共3兲. 䊐 Therefore the sum 兺t=0 Remark: Theorem 5.6 asserts convergence of the distribution of random walkers to equilibrium distribution 共190兲 with ⌫共t兲 → 0. This implies that the final distribution is not delta peaked at the ground state with minimum E0共x兲 but is a relatively mild function of this energy. The optimality of the solution is achieved after one takes the weight factor w共x ; t兲 into account: The repeated multiplication of weight factors as in 共177兲, in conjunction with the relatively mild distribution coming from the product of G1 as mentioned above, leads to the asymptotically delta-peaked wave function n共y兲 because w共x ; t兲 is larger for smaller E0共x兲 as seen in 共180兲. 2. Alternative choice of Green’s function So far we have used Green’s function defined in 共174兲, which is linear in the transverse field, allowing single-spin flips only. It may be useful to consider another type of Green’s function which accommodates multispin flips. Let us try the following form of Green’s function: 冉 冊 冉 冊 Ĝ2共t兲 = exp ⌬t⌫共t兲 兺 xi exp ⌬t 兺 Jijzi zj , i ij 共196兲 which is equal to Ĝ0共t兲 to the order ⌬t. The matrix element of Ĝ2共t兲 in the z-diagonal basis is Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-35 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing Ĝ2共y,x;t兲 = coshN共⌬t⌫共t兲兲tanh␦共⌬t⌫共t兲兲e−⌬tE0共x兲 , 共197兲 where ␦ is the number of spins in different states in x and y. According to the scheme of GFMC, we decompose Ĝ2共y , x ; t兲 into the normalized transition probability and the weight: G2共y,x;t兲 = 冋 cosh共⌬t⌫共t兲兲 e⌬t⌫共t兲 册 N tanh␦共⌬t⌫共t兲兲, 共198兲 w2共x;t兲 = e⌬tN⌫共t兲e−⌬tE0共x兲 . 共199兲 It is remarkable that the transition probability G2 is independent of E0共x兲, although it depends on x through ␦. Thus, the stationary distribution of random walk is uniform. This property is lost if one interchanges the order of the two factors in 共196兲. The property of strong ergodicity can be shown to hold in this case as well. Theorem 5.8 (strong ergodicity of QA-GFMC 2): The inhomogeneous Markov chain generated by (198) is strongly ergodic if ⌫共t兲 ⱖ − 1 log共1 − 2b共t + 1兲−1/N兲. 2⌬t 共200兲 Remark: For sufficiently large t, the above annealing schedule is reduced to ⌫共t兲 ⱖ b . ⌬t共t + 1兲1/N 共201兲 Since the proof is quite similar to the previous cases, we just outline the idea of the proof. The transition probability G2共y , x ; t兲 becomes smallest when ␦ = N. Consequently, the coefficient of ergodicity is estimated as 1 − ␣共Gt+1,t 2 兲ⱖ 冋 1 − e−2⌬t⌫共t兲 2 册 N . We note that R is equal to 1 in the present case because any state is reachable from an arbitrary state in a single step. From Theorem 5.1, the condition 冋 1 − e−2⌬t⌫共t兲 2 册 N ⱖ b⬘ t+1 共202兲 is sufficient for weak ergodicity. From this, one obtains 共200兲. Since the stationary distribution of G2共y , x ; t兲 is uniform as mentioned above, strong ergodicity readily follows from Theorem 5.2. Similarly to the case of PIMC, we can discuss the convergence condition of QA-GFMC in systems with continuous degrees of freedom. The resulting sufficient condition is a logarithmic increase in the mass as will be shown now. The operator Ĝ2 generated by Hamiltonian 共167兲 is written as 冉 N Ĝ2共t兲 = exp − 冊 ⌬t 兺 p2 e−⌬tV共兵ri其兲 . 2m共t兲 i=1 i Thus, Green’s function is calculated in a discretized space as 冉 Ĝ2共y,x;t兲 ⬀ exp − N 共203兲 冊 m共t兲 兺 兩r⬘ − ri兩2 − ⌬tV共兵ri其兲 , 2⌬t i=1 i 共204兲 where x and y represent 兵ri其 and 兵ri⬘其, respectively. Summation over y, i.e., integration over 兵ri⬘其, yields the weight w共x ; t兲, from which the transition probability is obtained: Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-36 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori w共x;t兲 ⬀ e−⌬tV共兵ri其兲 , 冉 N 共205兲 冊 m共t兲 G2共y,x;t兲 ⬀ exp − 兺 兩r⬘ − ri兩2 . 2⌬t i=1 i 共206兲 The lower bound for the transition probability depends exponentially on the mass: G2共y , x ; t兲 ⱖ e−Cm共t兲. Since 1 − ␣共Gt+1,t 2 兲 has the same lower bound, the sufficient condition for weak ergodicity is e−Cm共t兲 ⱖ 共t + 1兲−1, which is rewritten as m共t兲 ⱕ C−1 log共t + 1兲. 共207兲 Constant C is proportional to NL2 / ⌬t, where L denotes the maximum value of 兩r⬘ − r兩. The derivation of C is similar to 共169兲 because G2共t兲 allows any transition to arbitrary states at one time step. VI. SUMMARY AND PERSPECTIVE In this paper we have studied the mathematical foundation of QA, in particular, the convergence conditions and the reduction in residual errors. In Sec. II, we have seen that the adiabaticity condition of the quantum system representing QA leads to the convergence condition, i.e., the condition for the system to reach the solution of the classical optimization problem as t → ⬁ following the RT Schrödinger equation. The result shows the asymptotic power decrease in the transverse field as the condition for convergence. This rate in decrease of the control parameter is faster than the logarithmic rate of temperature decrease for convergence of SA. It nevertheless does not mean the qualitative reduction in computational complexity from classical SA to QA. Our method deals with a very generic system that represents most of the interesting problems including worst instances of difficult problems, for which drastic reduction in computational complexity is hard to expect. Section III reviews the quantum-mechanical derivation of the convergence condition of SA using the classical-quantum mapping without an extra dimension in the quantum system. The adiabaticity condition for the quantum system has been shown to be equivalent to the quasiequilibrium condition for the classical system at finite temperature, reproducing the well-known convergence condition of SA. The adiabaticity condition thus leads to the convergence condition of both QA and SA. Since the studies of QAE often exploit the adiabaticity condition to derive the computational complexity of a given problem, the adiabaticity may be seen as a versatile tool traversing QA, SA, and QAE. Section IV is for the reduction in residual errors after finite-time quantum evolution of RT and IT Schrödinger equations. This is a different point of view from the usual context of QAE, where the issue is to reduce the evolution time 共computational complexity兲 with the residual error fixed to a given small value. It has been shown that the residual error can become significantly smaller by the ingenious choice of the time dependence of coefficients in the quantum Hamiltonian. This idea allows us to reduce the residual error for any given QAE-based algorithm without compromising the computational complexity apart from a possibly moderate numerical factor. In Sec. V we have derived the convergence condition of QA implemented by quantum Monte Carlo simulations of path-integral and Green’s function methods. These approaches bear important practical significance because only stochastic methods allow us to treat practical large-size problems on the classical computer. A highly nontrivial result in this section is that the convergence condition for the stochastic methods is essentially the same power-law decrease in the transversefield term as in the Schrödinger dynamics of Sec. II. This is surprising since the Monte Carlo 共stochastic兲 dynamics is completely different from the Schrödinger dynamics. Something deep may lie behind this coincidence and it should be an interesting target of future studies. The results presented and/or reviewed in this paper serve as the mathematical foundation of QA. We have also stressed the similarity/equivalence of QA and QAE. Even the classical SA can be viewed from the same framework of quantum adiabaticity as long as the convergence condi- Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-37 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing tions are concerned. Since the studies of very generic properties of QA seem to have been almost completed, fruitful future developments would lie in the investigations of problems specific to each case of optimization task by analytical and numerical methods. ACKNOWLEDGMENTS We thank G. E. Santoro, E. Tosatti, and S. Suzuki for discussions and G. Ortiz for useful comments and correspondence on the details of the proofs of the theorems in Sec. III. Financial supports by CREST 共JST兲, DEX-SMI, and JSPS are gratefully acknowledged. APPENDIX A: HOPF’S INEQUALITY In this appendix, we prove inequality 共22兲. Although Hopf32 originally proved this inequality for positive linear integral operators, we concentrate on a square matrix for simplicity. Let M be a strictly positive m ⫻ m matrix. The strict positivity means that all the elements of M are positive, namely, M ij ⬎ 0 for all i, j, which will be denoted by M ⬎ 0. Similarly, M ⱖ 0 means that M ij ⱖ 0 for all i, j. We use the same notation for a vector, that is, v ⬎ 0 means that all the elements vi are positive. The product of the matrix M and an m element column vector v is denoted as usual by M v and its ith element is m 共M v兲i = 兺 M ijv j . 共A1兲 j=1 The strict positivity for M is equivalent to Mv ⬎ 0 if v ⱖ 0, v ⫽ 0, 共A2兲 where 0 denotes the zero vector. Of course, if v = 0, then M v = 0. Any real-valued vector v and any strictly positive vector p ⬎ 0 satisfy min i 共M v兲i 共M v兲i vi vi ⱕ min ⱕ max ⱕ max , pi i 共Mp兲i i 共Mp兲i i pi 共A3兲 冋 冉 冊册 共A4兲 冋冉 冊 册 共A5兲 because 冉 冊 共M v兲i − min i vi 共Mp兲i = pi 冉 冊 vi max 共Mp兲i − 共M v兲i = i pi m M ij 兺 j=1 m M ij 兺 j=1 v j − min i max i vi p j ⱖ 0, pi vi p j − v j ⱖ 0. pi The above inequality implies that the difference between the maximum and minimum of 共M v兲i / 共Mp兲i is smaller than that of vi / pi. Following Ref. 32 we use the notation osc i vi vi vi ⬅ max − min , pi i pi i pi 共A6兲 which is called the oscillation. For a complex-valued vector, we define oscvi = sup osc Re共vi兲. i 兩兩=1 i 共A7兲 It is easy to derive, for any complex c, Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-38 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori osc共cvi兲 = 兩c兩oscvi . i 共A8兲 i We can also easily prove that, if oscivi = 0, vi does not depend on i. We suppose that the simple ratio of matrix elements is bounded, M ik ⱕ M jk 共A9兲 for all i, j,k. This assumption is rewritten by the product form as 共M v兲i ⱕ , 共M v兲 j v ⱖ 0, v ⫽ 0, 共A10兲 for all i, j and such v. The following theorem states that inequality 共A3兲 is sharpened under additional assumption 共A10兲. Theorem A.1: If M satisfies conditions (A2) and (A10) for any p ⬎ 0 and any complex-valued v, osc i 共Mv兲i − 1 vi ⱕ osc . 共Mp兲i + 1 i pi 共A11兲 Proof. We consider a real-valued vector v at first. For fixed i, j and fixed p ⬎ 0, we define Xk by m 共M v兲i 共M v兲 j − = 兺 X kv k . 共Mp兲i 共Mp兲 j k=1 共A12兲 We do not have to know the exact form of Xk = Xk共i , j , p兲. When v = ap, the left-hand side of the above equation vanishes, which implies 兺kXk pk = 0. Thus, we have m 共M v兲i 共M v兲 j − = 兺 Xk共vk − apk兲. 共Mp兲i 共Mp兲 j k=1 共A13兲 Now we choose vi a = min , i pi vi b = max . i pi 共A14兲 Since vk − apk = 共b − a兲pk − 共bpk − vk兲, vk − apk takes its minimum 0 at vk = apk and its maximum 共b − a兲pk at vk = bpk. Therefore, the right-hand side of 共A13兲 with p given attains its maximum for v = ap− − bp+ = ap + 共b − a兲p+ , 共A15兲 where we defined p−i = 再 pi 共Xi ⱕ 0兲 0 共Xi ⬎ 0兲, 冎 再 p+i = 0 共Xi ⱕ 0兲 pi 共Xi ⬎ 0兲. 冎 共A16兲 Consequently, we have 冋 册 共Mp+兲i 共Mp+兲 j 共M v兲i 共M v兲 j − ⱕ − 共b − a兲. 共Mp兲i 共Mp兲 j 共Mp兲i 共Mp兲 j 共A17兲 Since, by assumptions, M ⬎ 0 and p ⬎ 0, we have Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-39 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing Mp− ⱖ 0, Mp+ ⱖ 0, Mp = Mp− + Mp+ ⬎ 0. 共A18兲 Moreover, Mp− ⬎ 0 if p− ⫽ 0 and Mp+ ⬎ 0 if p+ ⫽ 0. In either case, namely, p− = 0 or p+ = 0, the expression in the square brackets of 共A17兲 vanishes because p+ is equal to either p or 0. Thus, we may assume that both Mp− ⬎ 0 and Mp+ ⬎ 0. Therefore the expression in inequality 共A17兲 is rewritten as 1 1 共Mp+兲i 共Mp+兲 j − ⱕ , − 共Mp兲i 共Mp兲 j 1 + t 1 + t⬘ t⬅ 共Mp−兲i , 共Mp+兲i t⬘ ⬅ 共Mp−兲 j . 共Mp+兲 j 共A19兲 Since, from assumption 共A10兲, t and t⬘ are bounded from −1 to , we find t⬘ ⱕ 2t, which yields 1 1 共Mp+兲i 共Mp+兲 j − . − ⱕ 共Mp兲i 共Mp兲 j 1 + t 1 + 2t 共A20兲 For t ⬎ 0, the right-hand side of the above inequality takes its maximum value 共 − 1兲 / 共 + 1兲 at t = −1. Finally, we obtain 共M v兲i 共M v兲 j − 1 vi osc − ⱕ 共Mp兲i 共Mp兲 j + 1 i pi 共A21兲 for any i, j. Hence it holds for the sup of the left-hand side, which yields 共A11兲. For a complex-valued vector v, we replace vi by Re共vi兲. Since M Re共v兲 = Re共 M v兲, the same argument for the real vector case yields 冉 osc Re i 冊 冉 冊 −1 共M v兲i vi osc Re ⱕ . 共Mp兲i pi +1 i 共A22兲 Taking the sup with respect to , 兩兩 = 1, on both sides, we obtain 共A11兲. We apply this theorem to the eigenvalue problem, M v = v . 共A23兲 The Perron–Frobenius theorem states that a non-negative square matrix, M ⱖ 0, has a real eigenvalue 0 satisfying 兩兩 ⱕ 0 for any other eigenvalue . This result is sharpened for a strictly positive matrix, M ⬎ 0, as the following theorems. Theorem A.2: Under hypotheses (A2) and (A10), eigenvalue equation (A23) has a positive solution = 0 ⬎ 0 and v = q ⬎ 0. Moreover, for any vector p 共p ⱖ 0 , p ⫽ 0兲, the sequence qn = M np 共M np兲k 共A24兲 with k fixed converges toward such q . Theorem A.3: Under the same hypotheses, (A23) has no other non-negative solutions than = 0 and v = cq . For = 0 , (A23) has no other solutions than v = cq . Theorem A.4: Under the same hypotheses, any (complex) eigenvalue ⫽ 0 of (A23) satisfies 兩兩 ⱕ −1 . +1 0 共A25兲 Remark: We note that the factor 共 − 1兲 / 共 + 1兲 is the best possible if there is no further condition. For example, M= 冉 冊 1 , 1 ⬎ 1, 共A26兲 has eigenvalues 0 = + 1 and = − 1. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-40 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori Proof of Theorem A.2: Let us consider two vectors p and p which are non-negative and unequal to 0 and define pn+1 = Mpn, pn+1 = Mpn, p0 = p, p0 = p. 共A27兲 From hypothesis 共A2兲, both pn and pn are strictly positive for n ⬎ 0. We find by repeated applications of Theorem A.1 that, for n ⬎ 1, osc i 冉 冊 −1 p̄n,i ⱕ pn,i +1 n−1 osc i p̄1,i , p1,i 共A28兲 where we used the notation pn,i = 共pn兲i. Consequently, there exists a finite constant ⬎ 0, such that p̄n,i → pn,i 共n → ⬁兲 共A29兲 for every i. We normalize the vectors pn and pn as qn = pn , pn,k qn = pn p̄n,k , 共A30兲 with k fixed. Hypothesis 共A10兲 implies that −1 ⱕ qn,i ⱕ , −1 ⱕ q̄n,i ⱕ . 共A31兲 Thus, we find that 兩q̄n,i − qn,i兩 = qn,i 冏 冏 pn,k p̄n,i p̄n,k pn,k p̄n,i − ⱕ osc . p p p p̄n,k n,i p̄n,k i n,i n,k 共A32兲 Now we specialize to the case that p = Mp = p1, namely, pn = Mpn = pn+1, qn = qn+1 . 共A33兲 Using 共A28兲 and 共A29兲, we estimate 共A32兲 for qn+1,i − qn,i, which implies that the sequence qn converges to a limit vector q. Because of 共A31兲, we have q ⬎ 0. Now 共A29兲 reads pn+1,i 共Mpn兲i 共Mqn兲i = = → 0 . pn,i 共pn兲i 共qn兲i 共A34兲 Consequently, Mq = 0q. For any other initial vector p, the sequence qn converges to the same limit 䊐 as qn because of 共A28兲, 共A29兲, and 共A32兲. Theorem A.2 is thereby proved. Proof of Theorem A.3: We assume that v ⱖ 0 and v ⫽ 0 is a solution of eigenvalue equation 共A23兲. Since hypothesis 共A2兲 implies M v ⬎ 0, we have ⬎ 0 and v ⬎ 0. We use this v as an initial vector p in Theorem A.2 and apply the last part of this theorem to M nv nv v = = . 共M nv兲k nvk vk 共A35兲 Hence, the limit q is equal to v / vk, that is, v = cq and = 0. Therefore the first part of Theorem A.3 is proved. Next, we take 0 ⬎ 0 and q ⬎ 0 from Theorem A.2 and consider a solution of M v = v. The application of Theorem A.1 to q and v yields Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-41 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing 兩兩 vi 共M v兲i − 1 vi vi osc , osc = osc = osc ⱕ 0 i qi + 1 i qi i 0q i i 共Mq兲i 共A36兲 where we used 共A8兲. If = 0, the above inequality implies that osci vi / qi = 0 or v = cq, which provides the second part of Theorem A.3. Proof of Theorem A.4: We consider 共A36兲. If ⫽ 0 and v ⫽ 0, v cannot be equal to cq. 䊐 Therefore osci vi / qi ⬎ 0, and then 共A36兲 yields 共A25兲. APPENDIX B: CONDITIONS FOR ERGODICITY In this appendix, we prove Theorems 5.1 and 5.2 which provide conditions for weak and strong ergodicities of an inhomogeneous Markov chain.5 1. Coefficient of ergodicity Let us recall the definition of the coefficient of ergodicity, ␣共G兲 = 1 − min 再兺 冎 共B1兲 min兵G共z,x兲,G共z,y兲其 . x,y苸S z苸S First, we prove that this coefficient is rewritten as 1 ␣共G兲 = max 2 x,y苸S 再兺 冎 兩G共z,x兲 − G共z,y兲兩 . z苸S 共B2兲 Proof of 共B2兲: For fixed x , y 苸 S, we define two subsets of S by + = 兵z 苸 S兩G共z,x兲 − G共z,y兲 ⬎ 0其, SG − = 兵z 苸 S兩G共z,x兲 − G共z,y兲 ⱕ 0其. SG Since the transition matrix satisfies 兺y苸SG共y , x兲 = 1, we have 兺 冋 关G共z,x兲 − G共z,y兲兴 = 1 − + z苸SG 兺 − z苸SG 册冋 G共z,x兲 − 1 − 兺 册 G共z,y兲 = − − z苸SG 共B3兲 兺 关G共z,x兲 − G共z,y兲兴. − z苸SG 共B4兲 Thus, we find 1 兺 兩G共z,x兲 − G共z,y兲兩 = 2 z苸S = 兺 + z苸SG 关G共z,x兲 − G共z,y兲兴 = 兺 max兵0,G共z,x兲 − G共z,y兲其 z苸S 兺 关G共z,x兲 − min兵G共z,x兲,G共z,y兲其兴 = 1 − z苸S 兺 min兵G共z,x兲,G共z,y兲其 z苸S 共B5兲 for any x , y. Hence taking the maximum with respect to x , y on both sides, we obtain 共B2兲. 䊐 To derive the conditions for weak and strong ergodicities, the following lemmas are useful. Lemma B.1: Let G be a transition matrix. Then the coefficient of ergodicity satisfies 0 ⱕ ␣共G兲 ⱕ 1. 共B6兲 Lemma B.2: Let G and H be transition matrices on S . Then the coefficient of ergodicity satisfies ␣共GH兲 ⱕ ␣共G兲␣共H兲. 共B7兲 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-42 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori Lemma B.3: Let G be a transition matrix and H be a square matrix on S such that 兺 H共z,x兲 = 0 共B8兲 储GH储 ⱕ ␣共G兲储H储, 共B9兲 z苸S for any x 苸 S . Then we have where the norm of a square matrix defined by 储A储 ⬅ max x苸S 再兺 冎 兩A共z,x兲兩 . z苸S 共B10兲 Proof of Lemma B.1: The definition of ␣共G兲 implies ␣共G兲 ⱕ 1 because G共y , x兲 ⱖ 0. From 共B2兲, ␣共G兲 ⱖ 0 is straightforward. Proof of Lemma B.2: Let us consider a transition matrix G, a column vector a such that 兺z苸Sa共z兲 = 0, and their product b = Ga. We note that the vector b satisfies 兺z苸Sb共z兲 = 0 because 冋 兺 b共z兲 = z苸S 兺 兺 G共z,y兲a共y兲 = y苸S 兺 a共y兲 z苸S 兺 G共z,y兲 z苸S y苸S 册 兺 a共y兲 = 0. = 共B11兲 y苸S We define subsets of S by S+a = 兵z 苸 S兩a共z兲 ⬎ 0其, S−a = 兵z 苸 S兩a共z兲 ⱕ 0其, S+b = 兵z 苸 S兩b共z兲 ⬎ 0其, S−b = 兵z 苸 S兩b共z兲 ⱕ 0其. 共B12兲 Since 兺z苸Sa共z兲 = 兺z苸Sb共z兲 = 0, we find 兺 兩a共z兲兩 = 兺 a共z兲 − z苸Sz+ z苸S 兺 a共z兲 = 2 z苸Sz− 兺 兺 兩b共z兲兩 = 2 兺 b共z兲 = − 2 z苸Sz+ z苸S a共z兲 = − 2 z苸Sz+ 兺 共B13兲 a共z兲, z苸Sz− 兺 共B14兲 b共z兲. z苸Sz− Therefore, we obtain 冋 兺 兩b共z兲兩 = 2 兺 u苸S 兺 G共z,u兲a共u兲 = 2 兺 兺 z苸S+b z苸S u苸S+a ⱕ 2 max v苸S G共z, v兲 z苸S+b 再兺 z苸S a共u兲 + 2 min w苸S u苸S+a 兩a共u兲兩 ⱕ max u苸S 1 max 2 v,w苸S 冋 兺 兺 u苸S−a 册 G共z,u兲 a共u兲 z苸S+b 再兺 冎兺 再兺 冎兺 再兺 再兺 冎兺 冎兺 − G共z,w兲兴 = 册 G共z,u兲 a共u兲 + 2 z苸S+b v,w苸S z苸S 兩G共z, v兲 − G共z,w兲兩 冎兺 u苸S G共z,w兲 z苸S+b a共u兲 = max v,w苸S u苸S−a max兵0,G共z, v兲 − G共z,w兲其 关G共z, v兲 z苸S+b 兩a共u兲兩 u苸S 兩a共u兲兩 = ␣共G兲 兺 兩a共u兲兩, 共B15兲 u苸S where we used 共B5兲 and 共B2兲. Next, we consider transition matrices G, H, and F = GH. We take a共z兲 = H共z , x兲 − H共z , y兲, and then 共B15兲 is rewritten as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-43 J. Math. Phys. 49, 125210 共2008兲 Mathematical foundation of quantum annealing 兺 兩F共z,x兲 − F共z,y兲兩 ⱕ ␣共G兲 u苸S 兺 兩H共u,x兲 − H共u,y兲兩 共B16兲 z苸S for any x , y. Hence this inequality holds for the maximum of both sides with respect to x , y, which yields Lemma B.2. Proof of Lemma B.3: Let us consider F = GH. We can take a共z兲 = H共z , x兲 in 共B15兲 because of the assumption 兺y苸SH共y , x兲 = 0. Thus we have 兺 兩F共z,x兲兩 ⱕ ␣共G兲 u苸S 兺 兩H共u,x兲兩 共B17兲 z苸S for any x. Hence this inequality holds for the maximum of both sides with respect to x, which provides Lemma B.3. 䊐 2. Conditions for weak ergodicity The following theorem provides the reason why ␣共G兲 is called the coefficient of ergodicity. Theorem B.4: An inhomogeneous Markov chain is weakly ergodic if and only if the transition matrix satisfies lim ␣共Gt,s兲 = 0 共B18兲 t→⬁ for any s ⬎ 0. Proof: We assume that the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic. For fixed x , y 苸 S, we define probability distributions by px共z兲 = 再 1 共z = x兲 0 共otherwise兲, 冎 py共z兲 = 再 1 共z = y兲 0 共otherwise兲. 冎 共B19兲 Since px共t , s ; z兲 = 兺u苸SGt,s共z , u兲px共u兲 = Gt,s共z , x兲 and py共t , s ; z兲 = Gt,s共z , y兲, we have 兺 兩Gt,s共z,x兲 − Gt,s共z,y兲兩 = z苸S 兺 兩px共t,s;z兲 − py共t,s;z兲兩 ⱕ sup兵储p共t,s兲 − p⬘共t,s兲储兩p0,p0⬘ 苸 P其. z苸S 共B20兲 Taking the maximum with respect to x , y on the left-hand side, we obtain 2␣共Gt,s兲 ⱕ sup兵储p共t,s兲 − p⬘共t,s兲储兩p0,p0⬘ 苸 P其. 共B21兲 Therefore the definition of weak ergodicity 共119兲 yields 共B18兲. We assume 共B18兲. For fixed p0 , q0 苸 P, we define the transition probabilities by H = 共p0,q0, . . . ,q0兲, 共B22兲 F = Gt,sH = 共p共t,s兲,q共t,s兲, . . . ,q共t,s兲兲, 共B23兲 where p共t , s兲 = Gt,s p0 and q共t , s兲 = Gt,sq0. From 共B2兲, the coefficient of ergodicity for F is rewritten as ␣共F兲 = 1 1 兩p共t,s;z兲 − q共t,s;z兲兩 = 储p共t,s兲 − q共t,s兲储. 兺 2 z苸S 2 共B24兲 Thus Lemmas B.1 and B.2 yield 储p共t,s兲 − q共t,s兲储 ⱕ 2␣共Gt,s兲␣共H兲 ⱕ 2␣共Gt,s兲. 共B25兲 Taking the sup with respect to p0 , q0 苸 S and the limit t → ⬁, we obtain Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-44 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori lim sup兵储p共t,s兲 − q共t,s兲储兩p0,q0 苸 P其 ⱕ 2 lim ␣共Gt,s兲 = 0 t→⬁ 共B26兲 t→⬁ for any s ⬎ 0. Therefore the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic. Next, we prove Theorem 5.1. For this purpose, the following lemma is useful. Lemma B.5: Let a0 , a1 , . . . , an , . . . be a sequence such that 0 ⱕ ai ⬍ 1 for any i . ⬁ ⬁ ai = ⬁ ⇒ 兿 共1 − ai兲 = 0. 兺 i=0 i=n Proof. Since 0 ⱕ 1 − ai ⱕ e−ai, we have m m 0 ⱕ 兿 共1 − ai兲 ⱕ 兿 e i=n i=n 共B27兲 冉 冊 m −ai ⱕ exp − 兺 ai . 共B28兲 i=n ⬁ In the limit m → ⬁, the right-hand side converges to zero because of the assumption 兺i=0 ai = ⬁. Therefore we obtain 共B27兲. Proof of Theorem 5.1: We assume that the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic. Theorem B.4 yields lim 关1 − ␣共Gt,s兲兴 = 1 共B29兲 t→⬁ for any s ⬎ 0. Thus, there exists t1 such that 1 − ␣共Gt1,t0兲 ⬎ 1 / 2 with t0 = s. Similarly, there exists tn+1 such that 1 − ␣共Gtn+1,tn兲 ⬎ 1 / 2 for any tn ⬎ 0. Therefore, n 关1 − ␣共Gt 兺 i=0 i+1,ti 1 兲兴 ⬎ 共n + 1兲. 2 共B30兲 Taking the limit n → ⬁, we obtain 共122兲. We assume 共122兲. Lemma B.5 yields ⬁ ⬁ i=n i=n 兿 兵1 − 关1 − ␣共Gti+1,ti兲兴其 = 兿 ␣共Gti+1,ti兲 = 0. 共B31兲 For fixed s and t such that t ⬎ s ⱖ 0, we define n and m by tn−1 ⱕ s ⬍ tn and tm ⬍ t ⱕ tm+1. Thus, from Lemma B.2, we obtain 冋兿 m ␣共Gt,s兲 ⱕ ␣共Gt,tm兲␣共Gtm,tm−1兲 ¯ ␣共Gtn+1,tn兲␣共Gtn,s兲 = ␣共Gt,tm兲 i=n 册 ␣共Gti+1,ti兲 ␣共Gtn,s兲. 共B32兲 In the limit t → ⬁, m goes to infinity and then the right-hand side converges to zero because of 共B31兲. Thus we have lim ␣共Gt,s兲 = 0 t→⬁ 共B33兲 for any s. Therefore, from Theorem B.4, the inhomogeneous Markov chain generated by G共t兲 is weakly ergodic. 䊐 3. Conditions for strong ergodicity The goal of this section is to give the proof of Theorem 5.2. Before that, we prove the following theorem, which also provides the sufficient condition for strong ergodicity. Theorem B.6: An inhomogeneous Markov chain generated by G共t兲 is strongly ergodic if there Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-45 Mathematical foundation of quantum annealing J. Math. Phys. 49, 125210 共2008兲 exists the transition matrix H on S such that H共z , x兲 = H共z , y兲 for any x , y , z 苸 S and lim 储Gt,s − H储 = 0 t→⬁ 共B34兲 for any s ⬎ 0 . Proof: We consider p0 苸 P and p共t , s兲 = Gt,s p0. For fixed u 苸 S, we define a probability distribution r by r共z兲 = H共z , u兲. We find 储p共t,s兲 − r储 = 兺 兩 兺 Gt,s共z,x兲p0共x兲 − H共z,u兲兩 = z苸S 兺 兩 x苸S 兺 关Gt,s共z,x兲 − H共z,u兲兴p0共x兲兩 z苸S x苸S ⱕ 兺 兺 兩Gt,s共z,x兲 − H共z,u兲兩 = z苸S 兺 兺 兩Gt,s共z,x兲 − H共z,x兲兩 ⱕ x苸S 兺 储Gt,s − H储 x苸S z苸S x苸S 共B35兲 = 兩S兩储Gt,s − H储. Taking the sup with respect to p0 苸 P and using assumption 共B34兲, we obtain lim sup兵储p共t,s兲 − r储兩p0 苸 P其 = 0. t→⬁ 共B36兲 Therefore, the inhomogeneous Markov chain generated by G共t兲 is strongly ergodic. 䊐 Proof of Theorem 5.2: We assume that the three conditions in Theorem 5.2 hold. Since condition 共3兲 is rewritten as 兺 ⬁ ⬁ 兺 兩pt共x兲 − pt+1共x兲兩 = 兺 储pt − pt+1储 ⬍ ⬁, x苸S t=0 共B37兲 t=0 we have ⬁ 兩pt共x兲 − pt+1共x兲兩 ⬍ ⬁ 兺 t=0 共B38兲 for any x 苸 S. Thus, the stationary state pt converges to p = limt→⬁ pt. Now, let us define transition matrices H and H共t兲 by H共z , x兲 = p共z兲 and H共z , x ; t兲 = pt共z兲, respectively. For t ⬎ u ⬎ s ⱖ 0, 储Gt,s − H储 ⱕ 储Gt,uGu,s − Gt,uH共u兲储 + 储Gt,uH共u兲 − H共t − 1兲储 + 储H共t − 1兲 − H储. 共B39兲 Thus, we evaluate each term on the right-hand side and show that 共B34兲 holds. First term. Lemma B.3 yields that 储Gt,uGu,s − Gt,uH共u兲储 ⱕ ␣共Gt,u兲储Gu,s − H共u兲储 ⱕ 2␣共Gt,u兲, 共B40兲 where we used 储Gu,s − H共u兲储 ⱕ 2. Since the Markov chain is weakly ergodic 关condition 共1兲兴, Theorem B.4 implies that ∀ ⬎ 0, ∃ t1 ⬎ 0, ∀ t ⬎ t1:储Gt,uGu,s − Gt,uH共u兲储 ⬍ . 3 共B41兲 Second term. Since pt = G共t兲pt 关condition 共2兲兴, we find H共u兲 = G共u兲H共u兲 = Gu+1,uH共u兲 共B42兲 Gt,uH共u兲 = Gt,u+1H共u兲 = Gt,u+1关H共u兲 − H共u + 1兲兴 + Gt,u+1H共u + 1兲. 共B43兲 and then The last term on the right-hand side of the above equation is similarly rewritten as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-46 J. Math. Phys. 49, 125210 共2008兲 S. Morita and H. Nishimori Gt,u+1H共u + 1兲 = Gt,u+2关H共u + 1兲 − H共u + 2兲兴 + Gt,u+2H共u + 2兲. 共B44兲 We recursively apply these relations and obtain t−2 G H共u兲 = 兺 G t,u t−2 t,v+1 关H共v兲 − H共v + 1兲兴 + G t,t−1 v=u H共t − 1兲 = 兺 Gt,v+1关H共v兲 − H共v + 1兲兴 + H共t − 1兲. v=u 共B45兲 Thus the second term in 共B39兲 is rewritten as t−2 储G H共u兲 − H共t − 1兲储 = 储 兺 G t,u t−2 t,v+1 v=u 关H共v兲 − H共v + 1兲兴储 ⱕ 兺 储Gt,v+1关H共v兲 − H共v + 1兲兴储. v=u 共B46兲 Lemmas B.1 and B.3 yield that 储Gt,v+1关H共v兲 − H共v + 1兲兴储 ⱕ 储H共v兲 − H共v + 1兲储 = 储pv − pv+1储, 共B47兲 where we used the definition of H共t兲. Thus we obtain t−2 储G H共u兲 − H共t − 1兲储 ⱕ 兺 储pv − pv+1储. t,u 共B48兲 v=u ⬁ Since 兺t=0 储pt − pt+1储 ⬍ ⬁ 关condition 共3兲兴, for all ⬎ 0, there exists t2 ⬎ 0 such that t−2 ∀t ⬎ ∀ u ⱖ t2: 兺 储pv − pv+1储 ⬍ . 3 v=u 共B49兲 Therefore ∀ ⬎ 0, ∃ t2 ⬎ 0, ∀ t ⬎ ∀ u ⱖ t2:储Gt,uH共u兲 − H共t − 1兲储 ⬍ . 3 共B50兲 Third term. From the definitions of H and H共t兲, they clearly satisfy lim 储H共t兲 − H储 = 0, t→⬁ 共B51兲 which implies that ∀ ⬎ 0, ∃ t3 ⬎ 0, ∀ t ⬎ t3:储H共t − 1兲 − H储 ⬍ . 3 共B52兲 Consequently, substitution of 共B41兲, 共B50兲, and 共B52兲 into 共B39兲 yields that 储Gt,s − H储 ⬍ + + ⬍ 3 3 3 共B53兲 for all t ⬎ max兵t1 , t2 , t3其. Since is arbitrarily small, 共B34兲 holds for any s ⬎ 0 and then the given Markov chain is strongly ergodic from Theorem B.6, which completes the proof of the first part of Theorem 5.2. Next, we assume p = limt→⬁ pt. For any distribution q0, we have Hq0 = p because 兺 H共z,x兲q0共x兲 = p共z兲 x苸S 兺 q0共x兲 = p共z兲. 共B54兲 x苸S Thus, we obtain Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 125210-47 Mathematical foundation of quantum annealing J. Math. Phys. 49, 125210 共2008兲 储q共t,t0兲 − p储 = 储共Gt,t0 − H兲q0储 ⱕ 储Gt,t0 − H储. 共B55兲 Hence it holds for the sup with respect to q0 苸 P, which yields 共121兲 in the limit of t → ⬁. 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