On the Speed of Quantum Computers with Finite Size Clocks

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1 On the Speed of Quantum Computers with Finite Size Clocks Tino Gramss SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 On the speed of quantum computers with nite size clocks Tino Gramss Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM October 3, 1995 Abstract In previous work, the speed of locally coupled quantum computers with innite size clocks has been investigated in detail. One of the results is that the Feynman quantum computer and the Margolus cellular automaton compute at a constant rate. We show that this is no longer true for computers with nite size clocks and we will argue that nite size clocks are more appropriate for the computers under consideration. Furthermore, we give alternative denitions for computational speed that seem to be appropriate for nite size computers.

3 1 Introduction In 1985, Feynman introduced a model for a serial computer that is able to model the computation of a deterministic computation in a closed, locally interacting quantum system [Fe85]. Margolus was able to generalize Feynman's ideas to a quantum model of a cellular automaton [Ma8, Ma9]. The result of a Feynman or Margolus computer is obtained by performing a quantum measurement. The outcome of such a measurement is not certain. A nal result will only be obtained with a probability that is smaller than one. However, in case a nal result is measured, this result can be identied with certainty. Locally coupled quantum computers with certain outcomes have been proposed in [Pe85]. As for classical computers, computational clocks can be dened for such quantum computers in order to analyze their dynamic behavior. This clock can either be nite (i.e. it runs in a cycle) or innite. Innite size clock computers have been analyzed in detail in [Gr9] for the Feynman computer and in [Bi93] for the Margolus computer. Of particular interest is the average number of computational steps per time unit that a quantum computer performs. Quantum computational speed has rst been dened in [Ma8]. In [Ma9] it has been shown that the Margolus automaton computes at a constant rate, in other words, that its computational speed is constant. This also applies to the Feynman computer with an innite clock [Gr9]. In [Bi93] it is shown that the maximal speed is proportional to the number of sites of a quantum cellular automaton, just as for a classical cellular automaton 1. In a previous work [Gr95a], the Schrodinger equation for the Feynman computer with a nite size clock was solved. Based on this work, we will here analyze the computational speed of the nite size clock Feynman computer. Why nite size clocks? A reversible, nite size computer unavoidably has a cyclic time evolution. The state sequence repeats after say N computational steps. Coupling an innite size clock to a nite size computer means that two identical states of the computer which occur every N steps belong to dierent computational times. The clock's state sequence does not repeat. In this case, the denition of velocity, based on such clocks, does not take into account that two states of the computer at dierent times might not be distinguishable. This is why, itmightmake more sense to couple a suitably chosen nite size clock to the computer. In this work, it will be shown that there are essential dierences between a nite and an innite size clock computer. For example, it is no longer true that the computational velocity is constant. We now proceed by analyzing the velocity for nite size clock Feynman computers. p 1 In [Gr9] it is claimed that jhvij k, but this result turned out to be wrong.

4 3 Computational velocity Say computational time is dened by an hermitian operator N. That is, if the computer has done n steps of computation and is then in computational state j ni, h n j N j ni = n. A nite size, cyclic computational clock C that transforms a computational state j ni E to the following, (n+1)modn E (n+1)modn = C j ni is dened in [Gr95a]. With the computational states as a basis we get for the matrix representations of N and C: 3 and N = C = 1... N ; (1) : () Based on the computational time operator N, the operator for the computational velocity has been dened in [Ma8] by hvi = d dt hni = ;i h[n H]i : (3) As outlined in [Gr9], H = C + C y represents the Hamiltonian of the computer's clock that is coupled to the computation. This is true for nite and innite size clock computers. For innite size clock computers, the analysis in [Ma9] and [Gr9] yields V 1 = ;i (C ; C y ). This is no longer true for a nite size clock like (). The matrix representation of the velocity operator can be easily found. As for the innite size clock 3

5 computers, we dene hvi as in (3). But now, H = C + C y, (1), (), and (3) yield V = 1 ;N N ; : () In [Gr95a] the cyclic Feynman computer with an even number of gates and nite size clock has been analyzed. One result is the solution to the corresponding Schrodinger equation. If we start in a single computational state with entries j() = j the wave function for the computer is j (t)i, vector with entries n(t) = 1 P m= (;it) m m! mp k= ; m k n (;m+k)modn n = ::: N ; 1 : (5) Based on that result, it is not dicult to show that the velocity is not constant for most of the states. We setj ()i = j i. Evaluating (5) for small t yields n(t) = P l n;l ( l ; it ( l ;1 + l 1 ))+O ; t = n ; it ( n+1 + n;1)+o ; t : For the mean value of computational time we get with (1) h (t) j N j (t)i = P n = P n ; ; ; y y n n + it n+1 + y n;1 ( n ; it ( n+1 + n;1)) + 3 O t ; ; n j nj +Im y it n n+1 + y n;1 + t j n+1 + ; n;1j + 3 O t : The derivative of computational time is the computational velocity. Thus, for small t hvi = d dt hni = lim t! = P n n Im ; i n ; y n+1 + y n;1 h (t+t) jnj (t+t)i;h (t) jnj (t)i t + t j n+1 + n;1j + O(t ) : The \acceleration" A, dened as the time derivative of the velocity, is hai = d dt hvi = X n n j n+1 + n;1j + O (t) :

6 1 8 Computational Time Physical Time Figure 1: The average computational time for a cyclic Feynman computer with gates. In general, the acceleration does not vanish if t =. Incontrast to innite size computers, the speed is not constant. This can also be seen from gure 1. Here, the average computational time h (t) j N j (t)i is shown with (t) being the solution to the Schrodinger equation if we start from the single computational state j i =[1 ::: ]. It increases quickly to N= = 1, and then oscillates in an irregular way below N=. This behavior can be explained easily. Itmust be expected that all the probabilities to measure one of the computational states j i ::: j N;1i become independent for larger t. After a while, we expect a more or less random superposition of all possible computational states. The computational time n belongs to a computational state j ni. Therefore, the average computational time around N= belongs to a random superposition of states. Alternative denition for computational speed So far we have dened quantum speed over the average computational velocity based on the concept of average computational time. There is another possibility to approach the problem of quantum speed for nite size clock computers which comes with the special solution of the Schrodinger equation for this case. As outlined in [Gr95a], the probability to nd a nal result is given by N=(t), visualized in gure and gure 3 for a Feynman computer with and 1 gates. As an approximation for (5) the analysis in [Gr95a] yields for small t and jnj N n(t) ( Jjnj (t) for even n ;i J jnj (t) for odd n () 5

7 Time Figure : The probability to get a result upon measurement for a Feynman computer with gates and small times. J k (x) are the Bessel functions of order k. From () we see that for small t the probability to nd the result is approximately D N= j (t)e = N= J N= (t) : () For dierent N, the approximate probabilities () are depicted in gure. The probability to measure the result increases to a rst maximum which is absolute. There may be higher maxima at later times if we use the exact results (see gures and 3). However, they are not essentially higher than the rst one. Also, the larger N, the later and more abruptly the rst maximum arises. Therefore it makes some sense to say that the time T at which the rst maximum occurs in () is correlated to the speed of the computer. T can be calculated analytically. The derivative of () is d dt J N= (t) =8J N=(t) J N=;1 (t) ; J N=+1 (t) : There is a minimum if J N= (t) = and a maximum if J N=;1 (t) ; J N=+1 (t) = (8) as can be easily checked by calculating the sign of the second derivative. The solution for (8) with the smallest t = T is approximately for large N (from [SO8]): T =:5N +:39N 1 3 ; :5N ; 1 3 :

8 Time Figure 3: The probability to get a result upon measurement for a Feynman computer with 1 gates. Therefore, we have to wait an amount of time which is roughly proportional to the number of gates of our computer until we have achance to measure a result which diers essentially from zero (see gure 5). In gures the probability to measure the nal result, i.e. the height of the rst maximum is shown. 5 Summary In contrast to innite size clock computers, computers with nite clocks do not compute at a constant rate. As outlined in [Gr95b], the speed of a Margolus cellular automaton with a nite clock is in general also not constant. The computational velocity based on a denition in [Ma8] has been analyzed in detail. Furthermore, an alternative denition has been proposed. It is based on the fact that the probability for the Feynman computer to obtain a nal result is very small until it abruptly reaches a rst maximum. As one would expect for a serial computer, the time to reach this maximum is approximately proportional to the number of gates of the Feynman computer.

9 N=. N= N=1..1 N= N= Time N= Time Figure : Squared Bessel functions of order N= as approximations for the probabilities to measure a result on a Feynman computer with N gates. 8

10 1 1 8 Time Gates Figure 5: The time until the rst maximum of the probability to measure a result in dependence on the number of gates of the Feynman computer Gates Figure : The height of the rst maximum of the probability to measure a result in dependence on the number of gates of the Feynman computer. Only a computer with two or four gates is able to yield a result with certainty. 9

11 References [Bi93] M.Biafori (1993): \Few body cellular automata", Thesis, MIT/LCS/TR-59. [Fe85] R.Feynman (1985): \Quantum Mechanical Computers", Optics News 11, pp. 11-, also in Foundations of Physics 1, No., 198, pp [Gr9] T.Gramss (199): \On the speed of quantum computation", Santa Fe Institute Working Paper Series [Gr95a] T.Gramss (1995): \Solving the Schrodinger equation for the Feynman quantum computer", Santa Fe Institute Working Paper Series 95-9-???. [Gr95b] T.Gramss (1995). \Quantum Computation with Local Hamiltonians" Habilitationsschrift, submitted to the Mathematisch-Naturwissenschaftliche Fakultat der Christian-Albrechts-Universitat zu Kiel (University of Kiel). [Ma8] N.Margolus (198): \Quantum Computation" New Techniques in Quantum Measurement Theory, Annals of the New York Academy of Sciences 8, pp [Ma9] N.Margolus (199): \Parallel Quantum Computation" in \Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity", VIII, Ed. W.H.Zurek, Addison-Wesley, pp [Pe85] A.Peres (1985): \Reversible Logic and quantum computation", Physical Review A 3, No., pp [SO8] J.Spanier, K.B.Oldham (198): \An atlas of functions", Hemisphere Publishing Corporation, p. 5. 1