A note on adiabatic theorem for Markov chains

Transcription

1 Yevgeniy Kovchegov Abstract We state and prove a version of an adiabatic theorem for Markov chains using well known facts about mixing times. We extend the result to the case of continuous time Markov chains with bounded generators. 1 Introduction he adiabatic theorem was first stated by Max Born and Vladimir Fock in It asserts that a physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum (see Wikipedia page on adiabatic theorem. In this work, we state and prove the corresponding theorem for Markov chains using some of the machinery of mixing times and relaxation times of Markov chains that was developed so successfully in the last thirty years (see Aldous (1983, Aldous and Fill, Burton and Kovchegov (2009, Levin et al (2008 and references therein. Stated in terms of Markov chains, the adiabatic theorem is intuitively simple and accessible. It is important to point out that despite clear similarities and relation between the two, the quantum adiabatic theorem and the adiabatic theorem of this paper are different. First we state a version of the adiabatic theorem. Given two Hamiltonians, H initial and H final, acting on a quantum system. Let H(s = (1 sh initial +sh final. Suppose the system evolves according to H(t/ from time t = 0 to time (the so called adiabatic evolution. he adiabatic theorem of quantum mechanics states that for large enough, the final state of the system will be close to the ground state of H final. hey are close in l 2 norm whenever C, where β is the least spectral gap of H(s β 3 over all s [0, 1], and C depends linearly on a square of the distance between H initial and H final. See Messiah (1958. here are many versions of the adiabatic theorem. Now, let us list the main concepts involved in the theory of mixing and relaxation times that we will need in the following sections. We refer the readers to Levin et al (2008 and Aldous and Fill for details. Department of Mathematics, Oregon State University, Corvallis, OR , USA kovchegy@math.oregonstate.edu 1

2 Definition 1. If µ and are two probability distributions over Ω, then the total variation distance is µ V = 1 2 x Ω µ(x (x = sup µ(a (A A Ω Observe that the total variation distance measures the coincidence between the distributions on a scale from zero to one. Definition 2. Suppose P is an irreducible and aperiodic finite Markov chain with stationary distribution π, i.e. πp = π. Given an > 0, the mixing time t mix ( is defined as t mix ( = inf { t : P t π V, for all probability distributions } Now, suppose P = condition ( p(x, y x,y is reversible, i.e. P satisfies the detailed balance π(xp(x, y = π(yp(y, x for all x, y in the sample space Ω It is easy to check that if P is reversible, then P is self-adjoint with respect to an inner product induced by π, and as such will have all real eigenvalues. he difference β = 1 λ (2 between the largest eigenvalue (i.e. λ 1 = 1 and the second largest (in absolute value eigenvalue λ (2 is called the spectral gap of P. he relaxation time is defined as τ rlx = 1 β One can show the following relationship between mixing and relaxation times. heorem 1. Suppose P is a reversible, irreducible and aperiodic Markov chain with state space Ω and stationary distribution π. hen (τ rlx 1 log(2 1 t mix ( τ rlx log( min x Ω π(x 1 2 Adiabatic theorem for Markov chains Given two transition probability operators, P initial and P final, with a finite state space Ω. Suppose P final is irreducible and aperiodic and π f is the unique stationary distribution for P final. Let P s = (1 sp initial + sp final Definition 3. Given > 0, a time is called the adiabatic time if it is the least such that max P 1 P 2 P 1 P 1 π f V, where the maximum is taken over all probability distributions over Ω. 2

3 heorem 2. Let t mix denote the mixing time for P final. hen the adiabatic time ( tmix (/2 2 = O Proof. Observe that P 1 P 2 where N = P 1 P 2 P N inequality, max P 1 P 2 P 1 where 0 S N 1! N! N. P 1 P 1 =! N! N NP N final + E, and E is the rest of the terms. P 1 π f V max P N final π f V Hence, by the triangle! N! N + S N, Let = Kt mix (/2 and N = (K 1t mix (/2, so that max P N final π f V /2. Observe that er N log xdx ( N log! N! N = ep j=n+1 log j ( N log 1 and therefore, expressing and N via t mix (/2, and simplifying, we obtain ( K 1 e K 1 t mix (/2 = e N log N ( N! N! N 1 So (! 0 S N 1 N! N K 1 e K 1 t mix (/2 and we need to find the least K such that ( K 1 1 e K 1 t mix (/2 /2 Now, since log(1 + x = x x2 2 + O(x3, the least such K is approximated as follows K hus for = Kt mix (/2 t mix(/2 2, max P 1 P 2 t mix (/2 2 log (1 /2 t mix(/2 P 1 P 1 π f V 3

4 Observe that heorem 2 is independent of the distance. We can use l 2 norm in the definitions of adiabatic and mixing times, and arrive to the same result. A Markov chain P final over finite sample space Ω, if it is reversible, will have a spectral gap. Here we apply heorem 1 to the result in heorem 2. Corollary. Suppose P initial and P final are Markov chains with state space Ω. If P final is reversible, irreducible and aperiodic with its spectral gap β > 0, then [ ] = O log2 2 min x Ω π f (x β 2 3 Continuous time Markov processes In the case of continuous time Markov chains, an equivalent result is produced via the method of uniformization and order statistics. Suppose Q is a bounded Markov generator for a continuous time Markov chain P (t, and λ max i Ω j:j i q(i, j is the upper bound on the departure rates over all states. he method of uniformization provides an expression for the transition probabilities P (t as follows: (λt n P (t = e λt Pλ n n!, where P λ = I + 1 λ Q he expression is obtained via conditioning on the number of arrivals in a Poisson process with rate λ. he definition of a Mixing time is similar in the case of continuous time processes. Definition 4. Suppose P (t is an irreducible and finite continuous time Markov chain with stationary distribution π. Given an > 0, the mixing time t mix ( is defined as t mix ( = inf {t : P (t π V, for all probability distributions } Suppose Q initial and Q final are two bounded generators for continuous time Markov processes over a finite state space Ω, and π f is the only stationary distribution for Q final. Let Q[s] = (1 sq initial + sq final be a time non-homogeneous generator. Given > 0, let P (t 1, t 2 (0 t 1 t 2 denote a matrix of transition probabilities of a Markov process generated by Q [ ] t over [t 1, t 2 ] time interval. Observe that the continuous time Markov adiabatic evolution is governed by [ ] d t t dt = tq, t [0, ] while the quantum adiabatic evolution is described via the corresponding Schrödinger s equation dvt dt = iv t H ( t. 4

5 Definition 5. Given > 0, a time is called the adiabatic time if it is the least such that max P (0, π f V, where the maximum is taken over all probability distributions over Ω. We will state and prove the following adiabatic theorem. heorem 3. Let t mix denote the mixing time for Q final. ake λ such that λ max q initial (i, j and λ max q final (i, j, i Ω i Ω j:j i where q initial (i, j and q final (i, j are the rates in Q initial and Q final respectively. hen the adiabatic time λt mix(/2 2 + θ, where θ = t mix (/2 + /(4λ. ] Proof. Observe that λ max i Ω j:j i q t(i, j, where q t (i, j are the rates in Q [ t (0 t. ake = K j:j i ( ( t mix(/2, N = (K t mix(/2, 2K 2K and let P final (t = e tq final denote the transition probability matrix associated with the generator Q final. Now, we let P 0 = I + 1 λ Q initial and P 1 = I + 1 λ Q final. hen P 0 and P 1 are discrete Markov chains and ( (λ( N n P (0, = N P (N, = N e λ( N I n, n! where N = P (0, N and n! I n = ( N n... N<x 1 <x 2 < <x n< [( 1 x 1 P 0 + x ] [( 1 P x n P 0 + x ] n P 1 dx 1... dx n i.e. the order statistics of n arrivals within the [N, ] time interval. We used the fact that, when conditioned on the number of arrivals, the arrival times of a Poisson process are distributed as an order statistics of uniform random variables. Hence ( P (0, = N (λ( N n e λ( N ( N n n n! P 1 n = e λ(1 1 2K 1 t mix (/2 N ( N λ n (t mix (/2 n = e λt mix (/2 2K 1 N P final (t mix (/2 + E, n! P n 1... N + E x 1... x n dx 1... dx n + E 5

6 where E is the rest of the terms. hus, the total variation distance, max aking K λt mix(/2 P (0, π f V e λt mix (/2 2K 1 /2 + S N + 1/2, we bound the error term S N = E π f V 1 e λ(1 1 2K 1 t mix (/2 λ n (t mix (/2 n n! = 1 e λt mix (/2 2K 1 /2 as < 2 log ( ( 1 2. herefore λtmix (/2 + 1/2 (t mix (/2 + /(2λ. In the end, we would like to mention a possible application. he Glauber dynamics of a finite Ising model is a continuous time Markov process. Its mixing time can be estimated using path coupling. he adiabatic transformation of the Markov generator corresponds to an adiabatic transformation of the Hamiltonian. he above result can be applied to obtain the adiabatic time for the transformation. he result can be adjusted when the adiabatic evolution of the generator is nonlinear. 3.1 Application: adiabatic Glauber dynamics of a one dimensional Ising model In the original, non-adiabatic case, the Glauber dynamics is used to generate distribution π(x = 1 Z(β e H(x over all spin configurations x { 1, +1} S, where S denotes all the sites of a graph, H(x = β u v x(ux(v, β is the reciprocal of the temperature, and Z(β is the normalization constant. Consider a non-linear adiabatic Glauber dynamics of an Ising model on S = Z/nZ. here, for time s [0, 1], the Hamiltonian H s = (1 ψ(sh initial + ϕ(sh final, where ψ(s and ϕ(s are continuous functions on [0, 1] such that ψ(0 = ϕ(0 = 0 and ψ(1 = ϕ(1 = 1. Each site j Z/nZ has an independent exponential clock with parameter one associated with it. When it rings, the spin x(j is reselected using the following probability P (x(j = +1 = e Hs(x + e Hs(x + e Hs(x +, where x + (i = x (i = x(i for i j, x + (j = +1 and x (j = 1. Here H s is the Hamitonian at the transition time s. he time non-homogeneous generator associated with the above dynamics is denoted by Q[s]. 6

7 We will compare the above adiabatic Glauber dynamics with the ordinary Glauber dynamics used for generating Ising models, where at each site j S, its spin is reset after waiting for an exponential interarrival time with parameter one, and the probabilities are determined by the local Hamiltonians. Let Q initial and Q final be the time homogenious generators of ordinary Glauber dynamics associated with the respective Hamiltonians, H initial and H final. Let us consider the case when H initial (x = 0, H final (x = β j Z/nZ x(jx(j+1 and ϕ(s = 1 4β log p(s 1 p(s, e 2β where p(s = (1 s s. hen e 2β +e 2β e 2βϕ(s e 2βϕ(s +e 2βϕ(s Q[s] = (1 sq initial + sq final So the question of how slow the nonlinear adiabatic transition H t/ = (1 ψ(t/ H initial + ϕ(t/ H final = p(s, and therefore has to happen in order to obtain the distribution that is close to that of Ising model with Hamiltonian H final at the end of the adiabatic evolution is addressed in heorem 3. In particular, for β < 1 2, the adiabatic time ( [ ] n log n + log(2/ 2 O 1 tanh(2β as t mix (/2 log n+log(2/ 1 tanh(2β by heorem 15.1 in Levin et al (2008. he above example connects an adiabatic evolution of a physical system to that of Markov operators, and invites to further study the latter. Acknowledgment he author would like to thank the anonymous referee for pointing at the errors in an earlier draft, and suggesting an improvement. References Aldous, D., Random Walks on Finite Groups and Rapidly Mixing Markov Chains. Seminaire de Probabilites XVII, Lecture Notes in Math., 986, Aldous,D., Fill, J.A. Reversible Markov Chains and Random Walks on Graphs. On the web: Burton, R.M., Kovchegov, Y., Mixing times via super-fast coupling. In press. 7

8 Levin, D.A., Peres, Y., Wilmer, E.L., Markov Chains and Mixing imes. Amer. Math. Soc., Providance, RI. Messiah, A., Quantum mechanics. John Wiley and Sons, NY 8