APPLYING QUANTUM COMPUTER FOR THE REALIZATION OF SPSA ALGORITHM Oleg Granichin, Alexey Wladimirovich

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1 APPLYING QUANTUM COMPUTER FOR THE REALIZATION OF SPSA ALGORITHM Oleg Granichin, Alexey Wladimirovich Department of Mathematics and Mechanics St. Petersburg State University Abstract The estimates of the algorithm of the simultaneously perturbation stochastic approximation (SPSA) are critical against the requirement of the simultaneous generation of a large amount of a random value realization that have to be independent from each other, that is a hard problem for the classical computer. The method of an arbitrary distribution simulation on the quantum computer and also the method of simultaneous simulation of different distributions or different realizations of some random value are considered, and applying the methods for the SPSA algorithm is discussed. The arbitrary distribution simulation algorithm with complexity close as desired to linear (but using more memory) is also given. Index Terms Multi dimensional optimization, parameter estimation, stochastic approximation, simultaneous perturbation, quantum Oleg granitchin@mail.ru mail@agv.spb.ru computing, SPSA. randomized algorithm, I. SPSA algorithm Recently electronics development has closely approached to creating intelligent control devices. Even now we have a real opportunity of effective use of new mathematical algorithms of theories of optimization, dynamic systems unknown parameters identification, mathematical experiment design and optimal and adaptive control for practice problems solving. Earlier we could only try to get the most probable set that contains a vector of unknown parameters of the functioning dynamic system (or plant) for some problems. But now the opportunity to get more precise solutions for them appears. The exact solution of any problem can be found if there is a precise formulation. In our real existing world all connections and relationships are so difficult and many-sided that it is impossible to give a strict mathematical description for many phenomena. Typically theoretic approach is to choose a mathematical model 1

2 close to a real process and to include different noises (disturbances) into it. Noises represent some kind of roughness of the mathematical model from one side and are characteristics of outside uncontrolled perturbations of a plant or a system from the other. It is enough strangely that for a long time it was not distinct understood that searching algorithms with sequential estimate changing {θ t } in an axis direction of some random centered vector t (mean value is zero) θ t = θ t 1 t Y t, can converge to a true parameters vector θ in the conditions when the observations Y t are done with an almost arbitrary noise in some points, which define by previous estimate θ t 1 and the vector t of simultaneous perturbation. We shall name algorithms of such type as randomized algorithms of an estimation since under almost arbitrary noise the substantiation of the consistency of their estimates essentially uses a stochastic nature of the simultaneous perturbation (see [1-2]). As an important partial case of the randomized estimation algorithms there are algorithms of the simultaneously perturbation stochastic approximation (SPSA). In [3] the two algorithms for constructing the design of an experiment {x t } and the sequence of estimates {θ t } were considered to the solve the problem of the finding a point of minimum of an unknown function f(x). The first one is x t = θ t 1 + β t t, y t = f(x t ) + v t, θ t = θ t 1 α t t y t, β t that uses one measurement per each iteration. Next algorithm, the smoothed version of the Kiefer Wolfowitz procedure, uses two measurements x 2t = θ t 1 + β t t, x 2t 1 = θ t 1 β t t, θ t = θ t 1 α t y 2t y 2t 1 t. β t 2 Here {α t }, {β t } are some numerical sequences tending to zero; { t } t=1,2,... is an observable sequence of independent random vectors in R d, which is called trial simultaneous perturbation, with distribution functions P t ( ); {v t } is a sequence of disturbances in the measurements channel. The main features of the SPSA algorithms are the following: at each algorithm iteration we need only one or two values of an unknown minimized function (noisy) measurements and the unknown function is measured not at a point of the previous estimate but at estimate s slightly excited position for all unknown vector components simultaneously. As it was shown by J. Spall (see [4]) in the multi dimensional case the essential reduction of the quantity of measurements at each iteration, in comparison with a classical Kiefer Wolfowitz procedure of stochastic approximation, does not increase amount of iterations, which are necessary for obtaining the same accuracy of the estimation. II. Quantum Computer The main problem of the substantiation of convergence of estimates that arises from practical use of SPSA algorithms is the way of generation of trial simultaneous perturbations. They are to be independent from noises in the observation channel as well as the trial simultaneous perturbation vector components should be independent between themselves. The algorithms efficiency reduces when we use a 2

3 classical computer that sequentially executes elementary operations one by one. The title simultaneous perturbation itself has an insistent demand for practical use to be parallel. At the same time the problem of parallel calculations organization on computers of classical type is difficult when dealing with the large function arguments vector dimensionality (five, tens,..., thousands). As an alternative, model of hypothetical quantum computer will be considered below. The word hypothetical is consciously quoted since after the P.Shor report [5] on the Berlin mathematical congress in 1998 many serious authors began to write about quantum computers as an engineering of the nearest future. Unlike the classical computer treats bits, receiving values from the set {0, 1}, the quantum computer treats the quantum bits or qubits, representing typically a two-state microscopic system, possibly an atom or nuclear spin or polarized photon, the behavior of which (e.g., entanglement, interference, superposition, stochasticity,... ) can be accurately explained using the rules of quantum theory only. The typical mathematical model of a calculation on the quantum computer with n qubits [6] is a rotation, i.e. unitary transformation of a normalized vector in the N-dimensional Hilbert space, where N = 2 n. Let denote by q i the states which correspond to the basis vectors. They are referred as pure states, all another as mixed states or pure state superpositions. Before the calculation the vector is in the state 0 and as the result it changes its state to (1) W = N 1 i=0 ψ i q i, where the summation by i is done for all binary lines of length n, ψ i are compex numbers: i ψ i 2 = 1. Sizes a i = ψ i 2 are named as probabilistic amplitudes, and W is named as a superposition of the basis vectors q i. The last stage of calculation is measurement, i.e. vector projection on some axis, and its result is some pure state (more exactly, its number). We can get the state q i with the probability a i. III. Simulation of an arbitrary random distribution on a quantum computer Efforts for searching algorithms for solving deterministic problems on the quantum computer more effectively than on the classical computer were undertaken in some recent years [7-10]. But applying a principal stochastic mode of the quantum computer is also very interesting, in part, for solving probabilistic problems including a simulation of random processes. So far applying quantum computer have been proposed only for a simulation of physical processes [11-12], but there is a more wider set of problems where a simulation may be used [13-14]. The problem of the arbitrary distribution simulation on the quantum computer and as particular case, the problem of randomized simultaneous perturbation simulation in the SPSA algorithm will be considered below. Let us have a random value ξ defined on an interval [0, 1] with the distribution function F (x) = P{ξ < x}. Let us also have the quantum computer with n 3

4 qubits. There and below the value n is supposed to be sufficiently large ( sufficiently in van Vijngarden sence [15], i.e. none of the considered problems should be solved inadequately solely on account of an insufficiency of this value). Let us approximate the distribution function F (x) with the discrete distribution defined at 2 n points with equal distances from each other. Split the interval [0, 1] to 2 n equal intervals with the points x i = i/2 n, i = 1,..., 2 n, and consider p i, i = 1,..., 2 n the probability of falling in the i-th interval: (2) p 1 = F (x 1 ), p i = F (x i ) F (x i 1 ). Then drive our quantum computer in the state superposition (1) where a i = p i and make the measurement. Since the probability of getting the state q i after the measurement is a i as mentioned above than we can to interpret getting the state q i as falling out the value x i. As a partial case, the uniform distribution is simulated with the uniform superposition N 1 i=0 1 2 n q i, that may be obtained at n steps by applying the transformation 1 2 ( )to each qubit, and for a Bernoulli distribution simulation there is sufficient one qubit transformed as same. Notice that the quantum computer will produce true random values as opposite as pseudo-random ones made by the classical computer, and unlike the classical computer, for which there exist simulation algorithms only for a small amount of distributions, with another ones simulating only by an indirect way, usually through the uniform distribution, the quantum computer will simulate an arbitrary distribution with practically the same effectiveness. The critical requirement for a simulation, partly for the SPSA algorithm, is that obtained values to be independent. Unlike applying the classical computer, when a substantiation of an independency is not a simple problem, in our case it directly follows from the mathematical model of quantum mechanics. Besides, we obtain a simple method of testing a real quantum computer, particularly ascertaining its corresponding to the adopted mathematical model (that drives us closer to experimental testing the quantum mechanics, see [11]). For that there is sufficient to simulate on it some simple distribution, for example, uniform or Bernoulli one, and after test by methods developed for verifying pseudorandom engines [13] if the distribution simulated is really the required one and if the values obtained are really independent. By splitting the set {0,..., n 1} to M subsets with the powers H i, i = 1,..., M we may simultaneously simulate discrete distributions, each of that is defined in 2 H i points. These may be both different distributions and different realisations of the same random value (in the last case all H i should be equal). As a partial case we may simultaneously simulate n Bernoulli test realizations with n cubits. It is very important that values obtained so will also be independent. Preceding explains why applying the quantum computer may be useful for the SPSA algorithm. When we use the classical computer the different realizations of the random value may be obtained only 4

5 sequentially that may appear to be inacceptable in real time problems with a large amount of dimensions. Unlike that the quantum computer is able to construct the needed random value set for one evaluation cycle. IV. Simulation of an arbitrary random distribution with a rate, close to linear as desired Unfortunately, the algorithm discussed have a sufficient lack in a common case. A method of a physical realization of a transformation directly producing an arbitrary state superposition is unknown at the present time. With existing methods it may be obtained only by sequential applying of one-, two- or three-qubit transformations, the amount of that is arising exponential by n in general (see [16-17]). The problem may be slightly facilitated with the fact that we have some degree of freedom because of interesting in not amplitudes ψ i but their module squares, but in many cases the lack may be serious. There is known [16] that a superposition with each amplitude differing from the required one not far from ɛ may be built for a time that is linear by 1/ɛ. It also may appear to be unacceptable in some cases critical against approximation accuracy and/or time. Following will be offered the arbitrary distribution simulation algorithm with the efficiency close to linear by n as desired, but now for the discrete distribution simulation in N points there is necessary not log N but N cubits. Split the interval [0, 1] to n intervals and define p i as in (2) with the difference that the parameter i now has limits from 1 not to 2 n but to n. Define r i as Cp i, where the constant C < 1 is some arbitrary factor. Now consider the state superposition (1) obtained after applying to each from n qubits transformation (3) b j b j 1, where j = 0,..., n 1 and b j are some meanwhile unknown parameters characterizing rotation angles with 0 < b j < 1 for each j. In the case the state amplitudes will be ψ i = n 1 j=0 d j, where d j is equal b j, if j-th digit in the binary representation of i is equal 0 and 1 bj contrary wise. In part, if i = e k = 2 k, i.e. the number which binary representation contain only one 1 at the k-th place then we have ψ ek = and besides we get 1 b k n 1 ψ 0 = n 1 j=0 Consequently we derive (4) ψ 0 ψ ek = j=0 j k b j. bk 1 bk. b j, Now let us find such b j that corresponding ψ ek will be equal to r k 1. Choosing an arbitrary ψ 0 such as n 0 < ψ 2 0 < 1 r k = 1 C, k=1 we may find required parameters from (4). So after applying the transformation (3) with obtained values b j and making the 5

6 measurement we will get each of states e j with the probability r j. Now we again can to interpret the getting state e j as falling out the value x j, with ignoring all another states. Choosing the factor C as close to 1 as necessary, we may make the probability of obtaining one of states e j as large as desired, that providing algorithm efficiency as close to linear as desired. Practically, we are limited only with necessarily of choosing ψ 0 < 1 C as large to provide the algorithm computational stability. References 1. O. N. Granichin A stochastic recursive procedure with dependent noises in the observation that uses sample perturbations in the input // Vestnik Leningrad Univ. Math., 1989, 22: 1(4), p B.T. Polyak, A.B. Tsybakov On stochastic approximation with arbitrary noise (the KW case) / In: Topics in Nonparametric Estimation. Khasminskii R.Z. eds.. // Advances in Soviet Mathematics, Amer. Math. Soc.. Providence, 1992, 12, p O. N. Granichin Randomized algorithms for stochastic approximation under arbitrary disturbances // Automation and Remote Control, 2002, 63: 2, p J.C. Spall Multivariate stochastic approximation using a simultaneous perturbation gradient approximation// IEEE Trans. Automat. Contr., 37, 1992, P.W. Shor Quantum computing. Proc. of the 9th Int. Math. Congress. Berlin, 1998, 6. D. Deutsch Quantum theory, the Church-Turing principle and the universal quantum computer // Proc. Roy. Soc. London, Ser. A, 400, 1985, p D. Deutsch, R. Jozsa Rapid solutions of problems by quantum computer // Proc. Roy. Soc. London, Ser. A, 449, 1992, p L.K. Grover Quantum mechanics helps in searching for a needle in a haystack // Phys. Rev. Lett., 79, 1997, p A. Kitaev Quantum computations: algorithms and error correction // Russian Math. Surveys, 52:6, 1997, p P.W. Shor Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer // SIAM J. Comput, 26:5, 1997, p R. Feynmann Quantum mechanical computers // Found. Phys., 16, 1986, p S. Lloyd Universal quantum simulators. Science, 273, 1996, 1073 p. 13. S.M. Ermakov Monte Carlo method and adjacent issues. Moscow, Nauka, M. Vidyasagar Statistical learning theory and randomized algorithms for control // IEEE Control Systems, 1998, 12, p Revised report on the algorithmic language Algol 68. Ed. by A. van Vijngaarten et al. Springer Verlag, A. Barenco, C. H. Bennett, R. Cleve, D. P. Di Vincenco, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter Elementary gates for quantum computation // Phys. Rev. A., 54, 1995, p B. Tsirelson Quantum information processing. Lecture notes. Tel Aviv University,