Adiabatic times for Markov chains and applications

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1 Adiabatic times for Markov chains and applications Kyle Bradford and Yevgeniy Kovchegov Department of Mathematics Oregon State University Corvallis, OR , USA bradfork, Abstract We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of the Ising model over Z d /nz d. he theorems derived in this paper describe a type of adiabatic dynamics for l 1 R n + norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with l C n norm preserving ones, i.e. gradually changing unitary dynamics in C n. Keywords: time inhomogeneous Markov processes; ergodicity; mixing times; adiabatic; Glauber dynamics; Ising model AMS Subject Classification: 60J10, 60J7, 60J8 1 Introduction he long-term stability of time inhomogeneous Markov processes is an active area of research in the field of stochastic processes and their applications. See [9] and [10], and references therein. he adiabatic time, as introduced in [5], is a way to quantify the stability for a certain class of time inhomogeneous Markov processes. In order for us to introduce the reader to the type of adiabatic results that we will be working with in this paper, let us first mention earlier results that were published in [5], thus postponing a more elaborate discussion of the matter until subsection Preliminaries he mixing time quantifies the time it takes for a Markov chain to reach a state that is close enough to its stationary distribution. For the discrete-time finite state case we will look at the evolution of the Markov chain through its probability transition matrix. See [6] for a systematized account of mixing time theory and examples. Let V denote the total variation distance. 1

2 Adiabatic times and applications Definition 1. Suppose P is a discrete-time finite Markov chain with a unique stationary distribution π, i.e. πp = π. Given an > 0, the mixing time t mix is defined as t mix = inf { t : νp t π V, for all probability distributions ν }. o define the adiabatic time in its first and simplest form that we will expand and generalize a few pages down we have to consider a time inhomogeneous Markov chain whose probability transition matrix evolves linearly from an initial probability transition matrix P initial to a final probability transition matrix P final. Namely, we consider two transition probability operators, P initial and P final, on a finite state space Ω, and we suppose there is a unique stationary distribution π f of P final. We let P s = 1 sp initial + sp final 1 We use 1 to define a time inhomogeneous Markov chain P t over [0, ] time interval. he adiabatic time quantifies how gradual the transition from P initial to P final should be so that at time, the distribution is close to the stationary distribution π f of P final. Definition. Given > 0, a time is called the adiabatic time if it is the least such that max ν νp 1 P P 1 P 1 π f V where the maximum is taken over all probability distributions ν over Ω. With these definitions one would naturally ask how adiabatic and mixing times compare. his will be especially relevant given the emergence of quantum adiabatic computation and some instances of using adiabatic algorithms to solve certain classical computation problems. See [1] and [8]. It can be speculated that there may be scenarios in which the adiabatic time is more convenient to compute than mixing times. If we find the relationship between the two, it will give us an understanding of the adiabatic transition which is more prevalent in a context of physics in terms of mixing times and vice versa. he following adiabatic theorem was proved in [5]. heorem Kovchegov 009. Let t mix denote the mixing time for P final. hen the adiabatic time tmix / = O In subsection 1.4 we will give an example that shows the order of t mix / is the best bound for the adiabatic time in this setting. here Ω = {0, 1,,..., n} and P initial = and P. final =

3 Adiabatic times and applications 3 Similar adiabatic results hold in the case of continuous-time Markov chains. here, the concept of an adiabatic time is defined within the same setting and a relationship with the mixing time is shown. Let us state a continuous adiabatic result from [5], and then prove a more general statement of the theorem in the next section. Once again we define the mixing time as a measurement of the time it takes for a Markov chain to reach a state that is close enough to its stationary distribution. For the continuous-time, finite-state case we look at the evolution of the Markov chain through its probability transition matrix as a function over time. Definition 3. Suppose P t is a finite continuous-time Markov chain with a unique stationary distribution π. Given an > 0, the mixing time t mix is defined as t mix = inf {t : νp t π V, for all probability distributions ν}. o define an adiabatic time we have to look at the linear evolution of a generator for the initial probability transition matrix to a generator for the final probability transition matrix. Suppose Q initial and Q final are two bounded generators for continuous-time Markov processes on a finite state space Ω, and π f is the unique stationary distribution for Q final. Let us define a time inhomogeneous generator Q[s] = 1 sq initial + sq final Given > 0 and 0 t 1 t, let P t 1, t denote a matrix of transition probabilities of a Markov process generated by Q[ t ] over the time interval [t 1, t ]. With this new generator we define the adiabatic time to be the smallest transition time such that regardless of our starting distribution, the continuous-time Markov chain generated by Q[ t ] arrives at a state close enough to our stationary distribution π f. Definition 4. Given > 0, a time is called the adiabatic time if it is the least such that max νp 0, π f V ν where the maximum is taken over all probability distributions ν over Ω. he above definition for continuous-time Markov chains is similar to the one in the discrete time setting. he corresponding adiabatic theorem for the continuous times case was proved in [5]. heorem Kovchegov 009. Let t mix denote the mixing time for Q final. ake λ such that λ max i Ω j:j i qinitial and λ max i Ω j:j i qfinal, where q initial and q final are the rates in Q initial and Q final respectively. hen the adiabatic time where θ = t mix / + /4λ. λt mix/ his is once again the best bound as can be shown through the corresponding example. In the next section we will state the adiabatic results for Markov chains that generalize the above mentioned theorems in [5] and provide examples of applications in statistical mechanics. Section is dedicated to proofs. + θ

4 Adiabatic times and applications 4 1. Results and discussion Here we extend the results from [5], and thus expand the range of problems that can be analyzed with these types of adiabatic theorems. One such problem that we will discuss in subsection 1.3 deals with adiabatic Glauber dynamics for the Ising model. Now, in order to solve a larger class of problems, we redefine the adiabatic transition for both the discrete and continuous cases. We consider an{ adiabatic } dynamics { where } transition probabilities change gradually from P initial = to P final = so that, for each pair of states i and j, p initial p final the corresponding mutation of p from p initial to p final is implemented differently and not always linearly. In the case of discrete time steps, this means defining p [s] = 1 φ sp initial + φ sp final, 3 where φ : [0, 1] [0, 1] are continuous functions such that φ 0 = 0 and φ 1 = 1 for all locations. We require functions φ to be such that the new operators P s = {p [s]} are Markov chains for all s [0, 1], e.g. φ φ i,k j, k Ω. he above definition generalizes 1. If we suppose there is a unique stationary distribution π f for P final, then the Definition of adiabatic time given in the previous section will hold for the adiabatic dynamics defined in 3. he new is related to mixing time via the following adiabatic theorem, that we will prove in section. ]} be an inhomogeneous heorem 1 Discrete Adiabatic heorem. Let P t = { [ p t discrete-time Markov chain over [0, ]. Let φs = min φ s be the pointwise minimum function of all of the φ functions. If m 1 is an integer such that φ is m + 1 times continuously differentiable in a neighborhood of 1, φ k 1 = 0 for all integers k such that 1 k < m and φ m 1 0, then = O t m+1 m mix / 1 m he above is, in fact, the best bound in the new setting as shown through the example given later. See subsection 1.4. Now we extend the notion of adiabatic dynamics for the continuous-time Markov generators as follows. We let q [s] = 1 φ sq initial + φ sq final for all pairs i j, 4 where once again φ : [0, 1] [0, 1] are continuous functions such that φ 0 = 0 and φ 1 = 1 for all locations. Also, we let Q[s] denote the corresponding Markov operator.

5 Adiabatic times and applications 5 If there is a unique stationary distribution π f for Q final, then the Definition 4 of adiabatic time will apply for the extended adiabatic dynamics in 4, and the new can be again related to mixing time. heorem Continuous Adiabatic heorem. Let Q [ ] t t [0, ] generate the inhomogeneous discrete-time Markov chain. Let φs = min φ s be the pointwise minimum function of all of the φ functions. Suppose m 1 is an integer such that φ is m + 1 times continuously differentiable in a neighborhood of 1, φ k 1 = 0 for all integers k such that 1 k < m and φ m 1 0. If we take λ such that λ max q initial and λ max i Ω i Ω j:j i j:j i q final, where q initial and q final are the rates in Q initial and Q final respectively. hen = O [λ ] 1 m m+1 t m mix / he reader can reference the proof of this theorem in section. Again this is the best bound in the new setting as can be shown through the same example. See subsection 1.4. Observe that we could take φ : [0, 1] [0, 1] such that φs min φ s in both adiabatic theorems thus guaranteeing the function is nice enough. Now we check that the above continuous adiabatic theorem is scale invariant. For a positive M, we scale the initial and final generators to be 1 M Q initial and 1 M Q final respectively. hen the adiabatic evolution is slowed down M times, and the new ] 1 m t m+1 adiabatic time should be of order M [ λ m mix / with the old t mix and λ taken before scaling. On the other hand the new mixing time will be Mt mix, and the new λ is λ M as the rates are M times lower. Plugging the new parameters into the expression in the theorem, we obtain [ ] 1 λ m Mtmix m+1 m M = M [ λ confirming the theorem is invariant under time scaling. ] 1 m t m+1 m mix Let us revisit adiabatic theorems in physics and quantum mechanics. he reader can find a version of the quantum adiabatic theorem in [7] and multiple other sources. he adiabatic results in physics consider a system that transitions from one state to another, while the energy function changes from an initial H initial to H final. If the change in the energy function happens slowly enough, for the system that is initially at one of the equilibrium states i.e. an eigenstate of the initial energy function H initial, the resulting state will end up -close to the corresponding eigenvector of the final energy function H final. hat is, provided the change in the external conditions is

6 Adiabatic times and applications 6 gradual enough, the jth eigenstate of H initial is carried to an -proximity of the jth eigenstate of H final. Often the adiabatic results concern with one eigenstate, the ground state. hinking of Schrödinger equation as an l C n norm preserving linear dynamics, and a finite Markov process as a natural description of an l 1 R n + norm preserving linear dynamics, the ground state of one would correspond to the stationary state of the other. It is important to mention that in addition to all of the above properties, the quantum adiabatic theorems often require the transition to be gradual enough for the state to be within an -proximity of the corresponding ground state at each time during the transition. aking this into account, the complete analogue of the quantum adiabatic theorem for l 1 R n + would be the one in which the initial distribution is µ 0 = π initial and µ t π t < t [0, ], where µ t = µ 0 P 1 P P t is the distribution of the time inhomogeneous Markov chain at time t [0, ], π initial is the stationary distribution of P initial, and π t is the stationary distribution P t. See [8] for a related result. While we are currently working on proving the above mentioned complete analogue in both discrete and continuous cases, the adiabatic results of this section are sufficiently strong for answering our questions concerning adiabatic Glauber dynamics as stated in the following subsection. Observe that the results of the next subsection could not be obtained using the adiabatic theorems of [5]. Finally, we would like to point out that the models of the adiabatic Markov evolution considered in this paper are similar to simulated annealing. See [3], [4], and []. We expect some of the adiabatic Markov chain results to be used for a class of optimization problems by introducing Monte Carlo Markov Chains with the self-adjusting rates. 1.3 Applications to Ising models with adiabatic Glauber dynamics Let us first state a version of the quantum adiabatic theorem. Given two Hamiltonians, H initial and H final, acting on a quantum system. Let Hs = 1 sh initial + sh final 5 Suppose the system evolves according to Ht/ from time t = 0 to time. hen if is large enough, the final state of the system will be close to the ground state of H final. hey are close in the l norm whenever C, where is the least spectral gap 3 of Hs over all s [0, 1], and C depends linearly on a square of the distance between H initial and H final. Now, switching to canonical ensembles of statistical mechanics will land us in a Gibbs measure space with familiar probabilistic properties, i.e. the Markov property of statistical independence. We consider a nearest-neighbor Ising model. here the spins can be of two types, -1 and +1. he spins interact only with nearest neighbors. A

7 Adiabatic times and applications 7 Hamiltonian determines the energy-value of the interactions of the configuration of spins. Here, for a microstate, we multiply its energy by the thermodynamic beta and call it the Hamiltonian of the microstate. In other words, letting x be a configuration of spins, the Hamiltonian we use in this paper will be defined as Hx = β M xixj i j where β is the thermodynamic beta, i.e. its inverse is the temperature times Boltzmann s constant, M = {M } is a symmetric matrix and for locations i and j, M = 0 if i is not a nearest neighbor to j and M = 1 if i is a nearest neighbor to j. he Markov property of statistical independence is reflected through the local Hamiltonian defined at every location j as follows H loc xj = β i:i j xixj, where i j means i and j are nearest neighbors on the graph. In the original, non-adiabatic case, the Glauber dynamics is used to generate the following Gibbs distribution πx = 1 Zβ e Hx over all spin configurations x { 1, +1} S, where S denotes all the sites of a graph, and Zβ is the normalization constant. Let us describe how the Glauber dynamics works in the case when each vertex of the connected graph is of the same degree. here, for each location j, we have an independent exponential clock with parameter one associated with it. When the clock rings, the spin xj of configuration x at the site j on the graph is reselected using the following probability P xj = +1 = e Hloc x + j { } e Hloc x j + e = tanh H loc x Hloc x + j + j where x + i = x i = xi for i j, x + j = +1 and x j = 1. Here P xj = 1 = 1 P xj = +1 Also H loc x j = H loc x + j. Now we have a continuous-time Markov process, where the state space is the collection of the configurations of spins. Now, consider an adiabatic evolution of Hamiltonians as in 5. here at each time t, Hs = 1 sh initial + sh final, where s = t. he local Hamiltonians must therefore evolve accordingly, H loc s = 1 sh loc initial + shloc final

8 Adiabatic times and applications 8 and the adiabatic Glauber dynamics is the one where when the clock rings, the spin xj is reselected with probabilities P s xj = +1 = e Hloc s e Hloc s x + j x j + e Hloc s x + j Here too, H loc s x j = H loc x + j. s he stationary distribution of the Q final -generated Markov process, i.e. Glauber dynamics with H initial energy, is, for a configuration x, πx = e H finalx all config. x e H finalx Let β 0 and β 1 denote the values of thermodynamic beta for H initial and H final respectively Adiabatic Glauber dynamics on Z /nz Consider nonlinear adiabatic Glauber dynamics of an Ising model on a two-dimensional torus Z /nz. here any two neighboring spin configurations x and { y in { 1, +1} n differ at only one site on the graph, say v Z /nz xu if u v. hat is yu = xu if u = v. he transition rates evolve according to the adiabatic Glauber dynamics rules, and the transition rates can be represented as q x,y [s] = 1 φ x,y sq initial x,y + φ x,y sq final x,y as in 4. Here the functions φ x,y s for two neighbors x and y depend entirely on the spins around the discrepancy site v. Namely if all four neighbors of v are of the same spin +1 or 1, then φ x,y s = cosh 4β 1 sinhs4β 0 4β 1 sinh4β 0 4β 1 cosh 4β 0 + s4β 0 4β 1 If it is three of one kind, and one of the other i.e three +1 and one 1, or three 1 and one +1 as illustrated below 1 +1 v then φ x,y s = cosh β 1 sinhsβ 0 β 1 sinhβ 0 β 1 cosh β 0 + sβ 0 β 1 If there are two of each kind, any function works, as both, the initial and the final, local Hamiltonians produce the same transition rates qx,y initial = qx,y final = 1/.

9 Adiabatic times and applications 9 Suppose tanhβ 1 < 1. Observe that cosh 4β 1 sinhs4β 0 4β 1 sinh4β 0 4β 1 cosh 4β 0 + s4β 0 4β 1 cosh β 1 sinhsβ 0 β 1 sinhβ 0 β 1 cosh β 0 + sβ 0 β 1 for s [0, 1]. herefore by heorem 15.1 in [6] and heorem of this paper, the adiabatic time [ ] = O C n logn + log, where C = β 0 β 1 [cothβ 0 β 1 tanh β 1 ] [1 tanhβ 1 ]. Here, at every vertex on the torus we attached a Poisson clock with rate one, and therefore we can take λ = n. Also m = 1 in the theorem, and one can find the expression for t mix in [6] Adiabatic Glauber dynamics on Z d /nz d he adiabatic Glauber dynamics of an Ising model on a d-dimensional torus Z d /nz d solves similarly. here the minimum function φs of the heorem is same as in the case of d = cosh β 1 sinhsβ 0 β 1 φs = sinhβ 0 β 1 cosh β 0 + sβ 0 β 1 and the adiabatic time = O C nd [ ] logn + log where again C = β 0 β 1 [cothβ 0 β 1 tanh β 1 ] [1 tanhβ 1 ]. if tanhβ 1 < 1 d, Notice that the time scaling argument that followed the statement of heorem works here as well. hat is, if we use one Poisson clock of rate one for all vertices, or equivalently place Poisson clocks of rate n d at every individual vertex, the new adiabatic time will be as λ = 1 here. = O C nd 1.4 Optimality of the bound [ ] logn + log In this subsection we give examples that show that the t mix order of adiabatic time given in Kovchegov [5], and t m+1 m mix order for more general settings of this current paper are in fact optimal. We consider discrete probability transition matrices P initial = and P. final =

10 Adiabatic times and applications 10 over n + 1 states, {0, 1,,..., n}, and let the discrete-time adiabatic probability transition matrix to be P s = 1 sp initial + sp final as in [5]. Let π f again denote the stationary distribution for P final. Here π f = 0,..., 0, 1 and the mixing time t mix = n for any 0, 1. Now, since µp initial inequality = ρ for any probability distribution µ, we have the following νp 1 P Observe that ρp j fin νp 1 P Now, because n+1 π f V ρ j=0 1 j! j! j = π f for any 0 j n. herefore π f V 1 = j= n+1 j= n+1 = 1 1 j! j! j P j fin π f V! j! j! j 1! j 1! n! n = 1 n n n for n, we see that νp 1 P π f V 1 hus νp 1 P π f V 1 e n 4 proving that the order of adiabatic time = O n n n 1 e 4 n implies 4 log1 n 4 in [5] is optimal. tmix / = t mix 4, Optimal bound for general settings he same example works in the more general setting considered in this paper. For the same P initial and P final, let p [s] = 1 φ sp initial + φ sp final as in 4. Suppose φ s = φs for all pairs of states i, j, and suppose m 1 is an integer such that φ is m + 1 times continuously differentiable in a neighborhood of 1, φ k 1 = 0 for all integers k such that 1 k < m

11 Adiabatic times and applications 11 and φ m 1 0. hen νp 1 P π f V ρ 1 1 φl/ and therefore νp 1 P l=0 π f V = 1 l= n+1 1 l= n+1 = 1 j=l+1 1 φl/ j= n+1 j=l+1 φj/ P l final π f j=l+1 φj/ φj/. φj/ φj/ he minimum function φx = 1 + φm 1x 1 m m! + O x 1 m+1 and νp 1 P as log1 + x x. π f V 1 It is a well known fact that j= n+1 j=l 1 + 1m φ m 1 j m m m! P j= n+1 1+ = 1 e log 1m φ m 1 j m m m! +O1 j/ m+1 1 e 1m φ m 1 Pn 1 m j=1 jm +On/ m+1 n 1 j k = j=1 k j=0 where B j is the jth Bernoulli number. Suppose νp 1 P π f V, then log1 1m+1 φ m 1 m V + O 1 j/ m+1 B j k n k+1 j, 6 k + 1 j j m j=0 hus confirming that the order of adiabatic time = O optimal. B j m n m+1 j + On/ m+1 m + 1 j j m+1 t m mix m 1 «in heorem 1 is

12 Adiabatic times and applications 1 Naturally, there is a similar example in the continuous case. here Q initial = and Q final = Proofs In this section we give formal proofs to both adiabatic theorems, heorem 1 and heorem..1 Proof of heorem 1 Proof. We write p [s] = 1 φ sp initial + φ s φsp final and define transition probability matrix ˆP = {ˆp } to be such that We will thus have that Observe that 1 φsˆp = 1 φ sp initial νp 1 P where ν N = νp 1 P P s = 1 φs ˆP + φsp final P 1 P 1 = natural numbers with N <. By the triangle inequality, we have max νp 1 P ν P 1 j=n+1 + φsp final + φ s φsp final φj/ ν N P N final + E P N, and E is the rest of the terms, and both and N are P 1 π f V max νp N ν [ ] where 0 S N 1 j=n+1 φ j. final π f V j=n+1 Let set N = t mix /, where > 0 is small. hen we have that max νp N ν final π f V φj/ / j=n+1 φj/ + S N

13 Adiabatic times and applications 13 [ ] Setting 1 j=n+1 φ j / we obtain log 1 / j=n+1 log φj/ We plug in the approximation of the minimum function φ around x = 1 obtaining log 1 / herefore φx = 1 + φm 1x 1 m m! j=n+1 log log 1 / 1m+1 φ m 1 m m! + O x 1 m m φ m 1 j m m m! N 1 j=1 + O 1 j/ m+1 N j m m+ + O m+1 Observe that 1 m+1 φ m 1 0 as φ : [0, 1] [0, 1] and φ1 = 1. tmix / 1 By 6, j=1 j m = m B k m k=0 m+1 k k tmix / m+1 k, where B k is the kth Bernoulli number, and therefore > log 1 / 1m+1 φ m 1 m m! m B k m N t mix / m+1 k m+ +O m + 1 k k m+1 k=0 In order for the right hand side of the above equation to be log 1 / close to zero, it is sufficient for to be of order of O. Proof of heorem m+1 t m mix / m 1 Proof. Define ˆQ to be a Markov generate with off-diagonal entries hen writing ˆq = 1 φ s 1 φs qinitial q [s] = 1 φ sq initial + φ s φs q final 1 φs + φ s φsq final. + φsq final would imply Observe that λ max i Ω Q[s] = 1 φs ˆQ + φsq final j:j i ˆq and λ max i Ω [ ] t q j:j i

14 Adiabatic times and applications 14 as λ max i Ω j:j i q initial and λ max i Ω j:j i q final Let P final t = e tq final denote the transition probability matrix associated with the generator Q final, and let P 0 = I + 1 ˆQ λ and P 1 = I + 1 λ Q final. he P 0 and P 1 are discrete Markov chains. Conditioning on the number of arrivals within the [N, ] time interval λ N n νp 0, = ν N P N, = ν N e λ N I n n! where ν N = νp 0, N and n! I n = N n N<s 1 < <s n< [ 1 φ n=0 s1 P 0 + φ [ 1 φ s1 ] P 1 sn P 0 + φ i.e. the order statistics of the n. herefore, combining the terms with P final, we obtain νp 0, = ν N n=0 λ n P n final n! = e λ N λ n n ν N n=0 e λ N n! where E denotes the rest of the terms. ake K > 0 and define = 1 K 1 K N P n final N φ 1 φxdx N φxdx 1 t mix / sn ] P 1 ds 1 ds n s1 sn φ ds 1 ds n + E n + E, and N = K 1 K 1 K 1 K φxdx 1 t mix / Recall the approximation of the minimum function φ around x = 1 φx = 1 + φm 1x 1 m m! + O x 1 m+1 and therefore 1 K 1 K φxdx = 1 K 1 + γk K m,

15 Adiabatic times and applications 15 where γk = 1 m φ m 1 m+1! + OK 1. hus we can write νp 0, = e λ N λ n N n ν N n=0 n! P n final We see, using standard uniformization argument, that νp 0, = e λ 1+ γk K m 1tmix / νn = e λ γk K m +γk n=0 [ N 1 + γk λ n t mix / n n! P n final t mix / νn exp {Q final t mix /} + E m ] n Now, since 1 m φ m 1 0, we have that, for any probability distribution ν, νp 0, π f V = ν exp {Q final t mix /} π f V e λ where, by the triangle inequality, and, by definition of t mix, 0 S N 1 e λ 1+ γk K m 1 tmix / ν exp {Q final t mix /} π f V e λ γk K m +γk λ n t mix / n n=0 γk K m +γk n! t mix / < / + E + E t mix / + SN, aking K = c λ/ 1 m tmix / 1 m with constant c >> 1 m+1 φ m 1 m+1!, we obtain γk > log1 / λ K m t mix / + γk and therefore hus 0 S N 1 e λ = Kt mix/ 1 + γk K m γk K m +γk = O [λ t mix / < / ] 1 m m+1 t m mix / References [1] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd and O. Regev, Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation SIAM Review, Vol.50, No. 4., 008, [] V. Cerny, A thermodynamical approach to the travelling salesman problem: an efficient simulation algorithm Journal of Optimization heory and Applications, 45, 1985, 41-51

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