struction will simulate the given 1d-PQCA with deterministic head position, i.e. if this QTM is observed at any time during its computation, the proba

Transcription

1 On One-Dimensional Quantum Cellular Automata John Watrous Computer Sciences Department University of Wisconsin Madison, Wisconsin Abstract Since Richard Feynman introduced the notion of quantum computation in 1982, various models of \quantum computers" have been proposed. These models include quantum Turing machines and quantum circuits. In this paper we dene another quantum computational model, one-dimensional quantum cellular automata, and demonstrate that any quantum Turing machine can be eciently simulated by a onedimensional quantum cellular automaton with constant slowdown. This can be accomplished by consideration of a restricted class of one-dimensional quantum cellular automata called one-dimensional partitioned quantum cellular automata. We also show that any one-dimensional partitioned quantum cellular automaton can be simulated by a quantum Turing machine with linear slowdown, but the problem of eciently simulating an arbitrary one-dimensional quantum cellular automaton with a quantum Turing machine is left open. From this discussion, some interesting facts concerning these models are easily deduced. 1 Introduction The idea that certain principles of quantum mechanics might be powerful computational tools has led to the study of various theoretical models of quantum computers. The archetypal quantum computer, as introduced by Richard Feynman in [1], is a computer which can simulate quantum physical processes, and which operates in accordance with quantum physical laws. Feynman noted that the problem of simulating quantum physics with a computer based on classical physics appears to be intractable, thereby suggesting that quantum computers may be inherently more powerful than classical computers. David Deutsch later formalized the notion of quantum computation by dening what has become known as the Supported in part by NSF Grant CCR quantum Turing machine (QTM) in [2], and in [3] Ethan Bernstein and Umesh Vazirani, expanding on Deutsch's work, showed that there exists a universal QTM which can simulate any QTM to any required accuracy with at most polynomial slowdown. Another quantum computational model, the quantum circuit, was introduced by Andrew ao in [4] and shown to be equivalent to the quantum Turing machine. It is not known whether or not these models are more powerful than their classical analogues. However, there is evidence to suggest that this is the case; most notably Peter Shor has shown in [5] that the integer factoring and discrete log problems can be solved in polynomial time using a QTM. (These problems are believed not to be solvable in polynomial time using a probabilistic Turing machine.) It is natural to extend the idea of quantum computation to other computational models; in this paper we discuss one-dimensional quantum cellular automata. Given any well-formed QTM, we give a construction of a one-dimensional quantum cellular automaton (1d- QCA) which will eciently simulate this QTM. This is accomplished by dening a restricted class of 1d- QCA called one-dimensional partitioned quantum cellular automata (1d-PQCA). A 1d-PQCA is a 1d-QCA in which each cell is partitioned into three subcells (left, middle, and right), and where the next states of any given cell depend only on the contents of the right subcell of its left neighbor, the middle subcell of itself, and the left subcell of its right neighbor. This is the quantum analogue of the partitioned cellular automaton discussed by Kenichi Morita and Masateru Harao in [6]. The advantage of the 1d-PQCA class is that it is a simple matter to determine whether or not a given 1d-PQCA is well-formed, while this is not a trivial matter for an arbitrary 1d-QCA. It is not clear that an arbitrary 1d-QCA can be eciently simulated by a QTM. However, it is shown that any given 1d-PQCA can be simulated by a QTM with linear slowdown. It is interesting to note that given any 1d-PQCA, the QTM which results from this con-

2 struction will simulate the given 1d-PQCA with deterministic head position, i.e. if this QTM is observed at any time during its computation, the probability that the tape head will be observed in any given location will be either 0 or 1. This allows for the construction of a QTM which will simulate, with deterministic head position, any given QTM with linear slowdown. Thus, for example, the position of the tape head of any such QTM could be observed at every time step without aecting its computation. (This is generally not the case for an arbitrary QTM.) The remainder of this paper will be organized as follows. In section 2, the one-dimensional quantum cellular automata model is dened. In section 3, the class of one-dimensional partitioned quantum cellular automata is dened, and necessary and sucient conditions for the well-formedness of a 1d-PQCA are discussed. In section 4, the quantum Turing machine model is reviewed and in section 5, the equivalence of the quantum Turing machine model and the partitioned quantum cellular automata model is demonstrated. Finally, in section 6, some facts resulting from this discussion are mentioned. 2 One-Dimensional Quantum Cellular Automata A one-dimensional quantum cellular automaton M is a quadruple (Q; ; k; A) where Q is a nite set of states (including a distinguished quiescent state denoted by ), is a local transition function (described below), k is an integer denoting the acceptance cell, and A Q is a set of accepting states. M is assumed to have a two-way innite sequence of cells indexed by the integers (hereafter denoted by Z.) The neighborhood of each cell is dened to be that cell itself along with its closest neighbor on each side. A conguration of a 1d-QCA M is a map a : Z! Q where, for each integer n, a(n) denotes the state of the cell indexed by n. For any conguration a, it is assumed that there are only nitely many values of n for which a(n) is a non-quiescent state. Denote by C C(M) the set of all congurations of M, so that C is countable for any given 1d-QCA M. For any a 2 C, dene jaj (the length of a) to be the maximum number of consecutive cells such that the rst and last cells are non-quiescent. Any conguration of M in which the cell indexed by k contains an element of A is said to be an accepting conguration and all other congurations are non-accepting congurations. The local transition function is a map with : Q 4?! C 1 if q (; ; ; q) 0 if q 6 (1) which describes the evolution of M. (Here C denotes the eld of complex numbers.) Suppose that at some given time t, three consecutive cells of M are in states q 1 ; q 2 and q 3 respectively. Then for every state q 2 Q, the cell which contained q 2 at time t will, at time t+1, update to the state q with amplitude (q 1 ; q 2 ; q 3 ; q). Every cell updates simultaneously in this manner, so that globally, if M is in some conguration a 2 C at time t, then the amplitude with which M transforms to any conguration b 2 C at time t + 1 is dened to be the product of the amplitudes with which each cell of a transforms to the corresponding cell of b, i.e. n2zz (a(n? 1); a(n); a(n + 1); b(n)): (This product is guaranteed to exist by (1).) Thus, any conguration of M will transform into multiple \next congurations", where the transformation to each individual conguration has an associated amplitude. The evolution behaves as if all of these transformations occur simultaneously, so that after some number of steps, M will have evolved along many computation paths simultaneously. Each path has an associated amplitude which is dened to be the product of the amplitudes of the transformations along that path. Multiple paths with diering amplitudes may lead from one given conguration to another. This eect is known as interference. If M is assumed to be in some conguration a and is allowed to evolve for, say l steps, then M will be in a linear superposition of congurations. For each conguration b in such a superposition, we associate with it an amplitude which is the sum of the amplitudes of all paths of length l from a to b. It may therefore be the case, for example, that a conguration will have amplitude zero in some superposition, despite the fact that multiple paths of nonzero amplitude lead to it. If a 1d-QCA M is observed while in some superposition of congurations, the observer will not see this superposition. Rather, the act of observation forces the machine to choose one of the congurations in the given superposition randomly, so that exactly one con- guration will be observed. The probability that any

3 given conguration is observed is the absolute square of the amplitude associated with that conguration, i.e. the absolute square of the sum of the amplitudes of all paths which lead to that conguration. The act of observation has the eect of altering the machine, so that immediately after the observation occurs the machine will be in a single conguration, and no longer a superposition of congurations. (More general types of observations are possible, but will be ignored for the purposes of this paper.) For any 1d-QCA M and input conguration a, we are interested in whether or not an accepting conguration will be observed after some number of steps. The probability that M accepts a after l steps is simply the probability of observing any accepting conguration after l steps, assuming that M is initially placed in conguration a and is not observed before l steps have passed. Since each conguration in a given superposition is observed with a certain probability, it is necessary that the sum of the probabilities be exactly one. Thus, allowed local transition functions must be restricted to those that guarantee this condition for any superposition resulting from any given input. A machine with such a local transition function is said to be wellformed. This will be formalized presently. Given any 1d-QCA M, let `2(C) denote the space of all complex valued functions with domain C C(M) and bounded `2-norm, i.e. `2(C) 8 < : x : C! C 12 9 x(a)x(a)! < 1 ; : Then `2(C) is a Hilbert space with respect to the inner product h; i : `2(C) `2(C)! C dened by hx 1 ; x 2 i x 1 (a)x 2 (a): Any superposition of M can now be identied with an element x 2 `2(C), where x(a) 2 C denotes the amplitude associated with conguration a in this superposition. Thus, the probability of observing a from this superposition is jx(a)j 2 for any a 2 C. We must therefore have jx(a)j 2 1 (2) for any superposition x. The sum in (2) is exactly kxk 2 by denition, so we dene S fx 2 `2(C) j kxk 1g to be the set of all possible superpositions of M. Each superposition of a given 1d-QCA M will, in one time-step, evolve into a new superposition according to the local transition function of M. We can associate with any M a function E which will map any given superposition to this next superposition; if x is some superposition of M at time t, then M will be in the superposition Ex at time t + 1. In general, after l steps, M will be in the superposition E l x. E is the time-evolution operator of M, and can be explicitly dened as follows. For all a; b 2 C let (a; b) denote the amplitude with which conguration a transforms into conguration b, i.e. (a; b) (a(n? 1); a(n); a(n + 1); b(n)); n2zz and for each x 2 `2(C) and b 2 C dene Ex(b) (a; b)x(a): Thus, we say that M (Q; ; k; A) is a well formed one-dimensional quantum cellular automaton (write M 2 1d-QCA) if and only if the corresponding timeevolution operator E preserves `2-norm, i.e. x 2 S () Ex 2 S: 3 Partitioned Quantum Cellular Automata In general, given an arbitrary M (Q; ; k; A), it is not a trivial matter to determine whether or not M is well formed. For this reason we now dene a restricted class of 1d-QCA called partitioned quantum cellular automata for which this can easily be determined. This is a generalization of (deterministic) partitioned cellular automata discussed by Morita and Harao in [6]. A one-dimensional partitioned quantum cellular automaton is a 1d-QCA in which each cell is partitioned into three subcells: a left subcell, a middle subcell and a right subcell. (The set of states Q is decomposed accordingly.) The next state(s) of any cell may now only depend upon the states of the left subcell of the right neighbor, the middle subcell of the cell itself, and the right subcell of the left neighbor. More formally, let M (Q; ; k; A) as in the 1d- QCA case, where Q and are restricted as follows. Let Q Q l Q m Q r

4 for nite sets Q l, Q m and Q r, (and again let denote the distinguished quiescent element of Q.) Let be an jqj jqj matrix over C having the form 0 (q 1 ; q 1 ) (q 1 ; q 2 ) : : : (q 1 ; q jqj ) (q 2 ; q 1 ) (q 2 ; q 2 ) : : : (q 2 ; q jqj ) (q jqj ; q 1 ) (q jqj ; q 2 ) : : : (q jqj ; q jqj ) where : Q Q?! C must satisfy 1 if q (; q) (q; ) 0 if q 6 : For any state q (q l ; q m ; q r ) dene and dene as l(q) q l ; m(q) q m ; r(q) q r ; (q 1 ; q 2 ; q 3 ; q) ( (l(q 3 ); m(q 2 ); r(q 1 )); q) 1 C A (3) for all q 1 ; q 2 ; q 3 ; q 2 Q. Since a given matrix completely determines in this manner, we may write M (Q; ; k; A) rather than M (Q; ; k; A) for a 1d-PQCA M whenever convenient. The behavior of a partitioned quantum cellular automaton M is, in general, simpler than that of an arbitrary 1d-QCA. Suppose that M is in conguration a. Then after one time step, for any given n 2 Z, the state of the cell indexed by n will be transformed to each state q with amplitude (a(n? 1); a(n); a(n + 1); q) ( (l(a(n + 1)); m(a(n)); r(a(n? 1))); q): (4) In order to simplify this, we will dene a permutation : C?! C as a(n) (l(a(n + 1)); m(a(n)); r(a(n? 1))) for all n 2 Z. The action of the permutation on any conguration a can be illustrated as follows: a(n? 1) a(n) a(n + 1) a(n? 1) a(n) a(n + 1) Now, (4) is equivalent to (a(n? 1); a(n); a(n + 1); q) (a(n); q); so that applying to each neighborhood of some con- guration a is equivalent to rst applying to a and then applying to each individual cell of a, i.e. a(n) is transformed into each state q with amplitude (a(n); q) for every n 2 Z: If, for such an M, we have M 2 1d-QCA we will say that M is a well formed one-dimensional partitioned quantum cellular automaton (write M 2 1d-PQCA:) The following theorem allows us to characterize onedimensional partitioned cellular automata in terms of the matrix. Theorem 3.1 Let M (Q; ; k; A) with as above. Then M 2 1d-PQCA if and only if is unitary. The remainder of this section will be devoted to proving this theorem. For each a; b 2 C dene and u b (a) (a; b); v b (a) (b; a): Lemma 3.1 Let M (Q; ; k; A), where satises (3). Then we have and for any b 2 C. # fa 2 C j u b (a) 6 0g < 1 (5) # fa 2 C j v b (a) 6 0g < 1 (6) Proof. Given b 2 C, assume that u b (a) 6 0. By denition u b (a) (a(n); b(n)): n2zz Since b 2 C and (q; ) 0 for q 6, a must be such that a(n) b(n) for all but nitely many n. There are nitely many such a 2 C: (6) follows similarly. and By Lemma 3.1, the sums u b (a)u b (a) v b (a)v b (a) contain only nitely many nonzero terms and therefore converge, so we have that u b and v b are elements of `2(C) for each b 2 C:

5 Lemma 3.2 unitary ) fu b g b2c and fv b g b2c are orthonormal sequences in `2(C). Proof. Given a; b 2 C we will show 1 if a b hu a ; u b i 0 if a 6 b: For each m 2 Z, q 2 Q, dene S q m : C?! C as S q m(c)(n) 8 < : c(n) q c(n + 1) if n > m if n m if n < m for all n 2 Z, and also dene Rm q : C?! C as c(n) if Rm(c)(n) q n 6 m q if n m for all n 2 Z: Since is one-to-one and onto, we have hu a ; u b i c2c u a (?1 c)u b (?1 c) (7) (c(n); a(n))(c(n); b(n)): (8) c2c n2zz By Lemma 3.1 there are only nitely many values of c for which the summand is nonzero, so we are free to change the order of summation in (8) as follows where hu a ; u b i c2c q2q P (q; a(m))(q; b(m)) P (Sm(c)(n); q a(n))(sm(c)(n); q b(n)) n6m and m is any integer. P is independent of q, so we have hu a ; u b i c2c P q2q (q; a(m))(q; b(m)): (9) Suppose that a 6 b: Then there must exist m 2 Z such that a(m) 6 b(m): is unitary, so that q2q (q; a(m))(q; b(m)) 0 and therefore by (9) we have hu a ; u b i 0: Now consider hu a ; u a i. From (9) hu a ; u a i c2c P q2q j(q; a(m))j 2 for P j(sm(c)(n); q a(n))j 2 : n6m For any p 2 Q we have P q2q j(p; q)j2 1, since is unitary. Thus hu a ; u a i c2c c2c n6m n6m j(s q m(c)(n); a(n))j 2 j(s m(c)(n); a(n))j 2 j(c(n); Rm(a)(n))j 2 c2c n2zz hu a1 ; u a1 i (10) for a 1 R m(a). Now a 2 C, and therefore there exists a nite set fm 1 ; : : : ; m k g for which n 62 fm 1 ; : : : ; m k g ) a(n) : If we repeat the process resulting in (10) for each m m 1 ; m 2 ; : : : ; m k, we get hu a ; u a i hu ak ; u ak i for a k R m 1 (R m 2 ( (R m k (a)) )). Since a k (n) for all n 2 Z we have hu a ; u a i hu ak ; u ak i j(c(n); )j 2 1; c2c n2zz and so we have shown (7). A slight modication of the above argument shows that fv b g b2c is also an orthonormal sequence. Proof of Theorem 3.1. Assume that we are given M (Q; ; k; A) with unitary. We will show that E, the time-evolution operator of M, is unitary. By denition, we have so that Ex(b) x(a)u b (a): Ex(b) hx; u b i for every x 2 `2(C) and b 2 C. Also dene F as so that F x(b) x(a)v b (a); F x(b) hx; v b i for every x 2 `2(C) and b 2 C.

6 By Lemma 3.1, fu b g b2c is an orthonormal sequence. So, by Bessel's inequality kexk 2 b2c Ex(b)Ex(b) b2c kxk 2 ; jhx; u b ij 2 for any x 2 `2(C). Thus E is a bounded linear operator, and hence we have that hex; yi b2c x(a)u b (a)y(b) (11) is a bounded bilinear form. This allows us (see [7], for example) to change the order of summation in (11) to get hex; yi b2c x(a)u b (a)y(b) x(a) y(b)v a (b) b2c hx; F yi for any x; y 2 `2(C), so that E F. Now, for any x 2 `2(C) and b 2 C we have, by applying Lemma 3.1 and Lemma 3.2, E Ex(b) Ex(a)v b (a) c2c c2c x(c)u a (c)v b (a) x(c)v c (a)v b (a) c2c x(c)hv c ; v b i x(b) so that E E I. Similarly EE x(b) c2c x(c)hu c ; u b i x(b) for any x 2 `2(C); b 2 C, so that EE I. Thus, E is unitary and hence preserves `2-norm. We therefore have that M 2 1d-PQCA. The converse is straightforward. Corollary 3.1 For any M 2 1d-PQCA, the associated time-evolution operator E is unitary. 4 Quantum Turing Machines In this section we will review the quantum Turing machine model as dened in [3]. A quantum Turing machine M is a quintuple (K; ; ; k; A) where K is a nite set of states, is a nite tape alphabet (including a distinguished blank symbol, denoted by b), is a local transition function (described below), k is an integer denoting the distinguished acceptance tape square, and A is a set of accepting symbols. M is assumed to have a single read-write tape head and a single two-way innite tape with tape squares indexed by Z. A conguration of a QTM M is a triple (s; h; c), where s 2 K denotes the current state of M, h 2 Z denotes the index of the tape square over which the tape head is currently located, and c is a map from Z to which describes the contents of the tape. For any conguration of M it is assumed that the number of tape squares containing non-blank symbols is nite. Any conguration of M in which the tape cell indexed by k contains an element of A is said to be an accepting conguration and all other congurations are non-accepting congurations. The local transition function is a map : K K fl; Rg?! C which describes the evolution of M. Suppose that, at some particular time, the current state of M is s and the tape head of M is located above a tape square which contains the symbol. Then for every triple ( 0 ; s 0 ; d) 2 K fl; Rg, M will write the symbol 0 in the currently scanned tape square, change internal state to s 0 and move the tape head in direction d with amplitude (s; ; 0 ; s 0 ; d). As in the 1d-QCA case, the computation behaves as if all of these transitions occur simultaneously, so that after some number of steps M will have traversed many dierent computation paths simultaneously, each with an associated amplitude. The amplitude associated with each path is dened to be the product of the amplitudes of the transitions along that path. After l steps, say, M will be in a linear superposition of congurations, and the amplitude associated with each conguration will be the sum of the amplitudes along all paths of length l to that conguration. If observed while in a superposition of congurations, M will randomly choose a single conguration which will be seen; again the probability that a given conguration will be observed is the absolute square of the amplitude associated with that conguration. Allowed local transition functions must be restricted

7 to those that guarantee that the sum of these probabilities will be one for any given superposition. Such a QTM is said to be well-formed. (Write M 2 QTM whenever M is a well-formed quantum Turing machine.) We state, without proof, the following theorem, due to Bernstein and Vazirani, which allows us to characterize well-formed quantum Turing machines in terms of their local transition functions: Theorem 4.1 (Bernstein & Vazirani) Given any M (K; ; ; k; A) as above, M 2 QT M if and only if (i) 8 (s 1 ; 1 ); (s 2 ; 2 ) 2 K : ;s;d (s 1 ; 1 ; ; s; d)(s 2 ; 2 ; ; s; d) 1 if (s1 ; 1 ) (s 2 ; 2 ) 0 if (s 1 ; 1 ) 6 (s 2 ; 2 ) and (ii) 8 (s 1 ; 1 ; 1 ); (s 2 ; 2 ; 2 ) 2 K : s (s 1 ; 1 ; 1 ; s; L)(s 2 ; 2 ; 2 ; s; R) 0: (P1) M 0 enters each state while moving in exactly one direction, i.e. if 0 (s 1 ; 1 ; 1 ; s 0 1 ; d 1) and 0 (s 2 ; 2 ; 2 ; s 0 2 ; d 2) are both nonzero, then d 1 d 2 : Thus, given M tm (K; ; ; k tm ; A tm ) satisfying (P 1), K can be partitioned into two sets: K l and K r, such that M enters states in K l only when moving left, and enters states in K r only when moving right. We now dene a 1d-PQCA M ca (Q; ; k ca ; A ca ) which will simulate M tm. Dene Q Q l Q m Q r with Q l K l [ f#g; Q m ; Q r K r [ f#g: The quiescent element of Q is dened to be (#; b; #), where b is the blank symbol of M tm. Dene as follows: (i) for each (s 1 ; 1 ); (s 2 ; 2 ) 2 K l let ((s 1 ; 1 ; #); (s 2 ; 2 ; #)) (s 1 ; 1 ; 2 ; s 2 ; L); (ii) for each (s 1 ; 1 ) 2 K l, (s 2 ; 2 ) 2 K r let 5 Equivalence of QTM and 1d-PQCA Models We now show that for any M tm 2 QTM, there is an M ca 2 1d-PQCA which simulates M tm with constant slowdown, and similarly for any M ca 2 1d-PQCA there is an M tm 2 QTM which simulates M ca with linear slowdown d-PQCA Simulation of a QTM Let M tm (K; ; ; k tm ; A tm ) 2 QTM and let M ca (Q; ; k ca ; A ca ) 2 1d-PQCA. We will say that M ca simulates M tm if and only if there exists a lineartime computable function T : C(M tm )! C(M ca ) and a function f : Z + Z +! Z + such that for any given conguration a 2 C(M tm ), the probability that M tm accepts a after t steps is equal to the probability that M ca accepts T (a) after f(t; jt (a)j) steps. Without loss of generality, we can assume that the QTM which is to be simulated has the property that any given state is only entered while the tape head is moving in one direction, due to the following lemma of Bernstein and Vazirani. Lemma 5.1 (Bernstein & Vazirani) Given any M 2 QTM, there exists an M 0 2 QTM which simulates M with constant slowdown, and which satises the following property: ((s 1 ; 1 ; #); (#; 2 ; s 2 )) (s 1 ; 1 ; 2 ; s 2 ; R); ((#; 2 ; s 2 ); (s 1 ; 1 ; #)) (s 2 ; 2 ; 1 ; s 1 ; L); (iii) for each (s 1 ; 1 ); (s 2 ; 2 ) 2 K r let ((#; 1 ; s 1 ); (#; 2 ; s 2 )) (s 1 ; 1 ; 2 ; s 2 ; R); (iv) for any q 1 ; q 2 2 Q for which (q 1 ; q 2 ) has not already been dened in (i)? (iii), let (q 1 ; q 2 ) 1 if q1 q 2 0 otherwise. Let k ca k tm and dene A ca f(q l ; q m ; q r ) 2 Q j q m 2 A tm g: A straightforward application of Theorem 3.1 and Theorem 4.1 yields the following Lemma 5.2 If M tm 2 QTM then M ca 2 1d-PQCA. For (s; h; c) 2 C(M tm ); dene T ((s; h; c)) 2 C(M ca ) as follows. For each n 2 Z let T ((s; h; c))(n) 8 >< >: (#; c(n); #) if s 2 K l and n 6 h + 1 (s; c(n); #) if s 2 K l and n h + 1 (#; c(n); #) if s 2 K r and n 6 h? 1 (#; c(n); s) if s 2 K r and n h? 1:

8 Lemma 5.3 For any n 2 Z +, the probability that M tm accepts (s; h; c) after n steps is equal to the probability that M ca accepts T ((s; h; c)) after n steps. Proof. [Sketch] For any conguration a of M tm, the symbol contained in the tape square indexed by n is equal to the contents of the middle subcell of the cell indexed by n in T (a). Recall that the evolution of a 1d-PQCA may be decomposed into two steps: applying the permutation to the current conguration, then transforming each cell according to. Now, for each n, T ((s; h; c))(n) will have have a non-# state in its left or right subcell exactly when h n, thereby simulating the presence of the tape head at this location. By inspection of the denition of, it is clear that M ca will evolve in accordance with M tm. Together, Lemma 5.1, Lemma 5.2 and Lemma 5.3 give us Theorem 5.1 Given any M tm 2 QTM there is an M ca 2 1d-PQCA which simulates M tm with constant slowdown. 5.2 QTM Simulation of a 1d-PQCA Again, let M tm (K; ; ; k tm ; A tm ) 2 QTM and let M ca (Q; ; k ca ; A ca ) 2 1d-PQCA. Then M tm simulates M ca if and only if there exists a linear-time computable function T : C(M ca )! C(M tm ) and a function f : Z + Z +! Z + such that for any given conguration a 2 C(M ca ), the probability that M ca accepts a after t steps is equal to the probability that M tm accepts T (a) after f(t; jt (a)j) steps. Given M ca (Q; ; k ca ; A ca ) 2 1d-PQCA we will now dene M tm (K; ; ; k tm ; A tm ) 2 QTM which will simulate M ca. Each tape square of M tm will represent the cell of M ca with the same index. Dene (Q l Q m Q r ) [ (Q l Q m Q r )? where (Q l Q m Q r )? is a copy of (Q l Q m Q r ), so that (p l ; p m ; p r )? 2 (Q l Q m Q r )? whenever (p l ; p m ; p r ) 2 (Q l Q m Q r ) : (M tm will need to simulate only a nite portion of M ca, since we are only concerned with nite congurations of M ca, so tape squares containing a symbol in (Q l Q m Q r )? will be used to mark the leftmost and rightmost cells of M ca being simulated at that time.) Let be the blank symbol of M tm. Dene K Q l fs 0 ; s 1 ; s? 1 ; s 2; s? 2 ; s 3g Q r : Each state of M tm will consist of one element from Q l and one element from Q r (in order to \move" states from one tape square to another when performing the permutation, as described in section 3) and one element from fs 0 ; s 1 ; s? 1 ; s 2; s? 2 ; s 3g. The simulation of each step of M ca takes place in three stages. The rst stage reversibly performs the permutation on the conguration of M ca currently represented by the contents of the tape. In order to do this, let take the following values: for each (l; r) 2 Q l Q r ; (p l ; p m ; p r ); (q l ; q m ; q r ) 2 Q l Q m Q r, dene ((l; s 0; r); (pl; pm; pr)? ; (pl; pm; r)? ; (l; s 1; pr); R) 1 ((l; s 1; r); (pl; pm; pr); (pl; pm; r); (l; s 1; pr); R) 1 ((l; s 1; r); (pl; pm; pr)? ; (pl; pm; r); (l; s? 1; pr); R) 1 ((l; s? 1; r); (pl; pm; pr)? ; (pl; pm; r)? ; (l; s? 1; pr); R) 1 ((l; s? 1; r); (pl; pm; pr); (l; pm; r)? ; (pl; s 2; pr); L) 1 ((l; s 2; r); (pl; pm; pr); (l; pm; pr); (pl; s 2; r); L) 1 ((l; s 2; r); (pl; pm; pr)? ; (l; pm; pr); (pl; s? 2; r); L) 1 ((l; s? 2; r); (pl; pm; pr)? ; (l; pm; pr)? ; (pl; s? 2; r); L) 1 (Note that not all of these transitions will be used given proper input, but are added to guarantee wellformedness.) In the second stage of the simulation, the state contained in each tape square of M tm is transformed in accordance with, so for each (l; r) 2 Q l Q r ; (p l ; p m ; p r ); (q l ; q m ; q r ) 2 Q l Q m Q r, dene ((l; s? 2; r); (pl; pm; pr); (ql; qm; qr)? ; (pl; s 3; r); R) ((l; pm; pr); (ql; qm; qr)) ((l; s 3; r); (pl; pm; pr); (ql; qm; qr); (l; s 3; r); R) ((pl; pm; pr); (ql; qm; qr)) ((l; s 3; r); (pl; pm; pr)? ; (ql; qm; qr)? ; (l; s 0; r); L) ((pl; pm; pr); (ql; qm; qr)) The third stage of the simulation simply returns the tape head to the tape square representing the (new) leftmost cell of M ca being simulated. For each (l; r) 2 Q l Q r ; (p l ; p m ; p r ); (q l ; q m ; q r ) 2 Q l Q m Q r, dene ((l; s 0; r); (pl; pm; pr); (pl; pm; pr); (l; s 0; r); L) 1 Finally, let take the value 0 everywhere not dened above. Let k tm k ca and dene A tm fq j q 2 A ca g [ fq? j q 2 A ca g: The following lemma follows from Theorem 3.1 and Theorem 4.1. Lemma 5.4 If M ca 2 1d-PQCA then M tm 2 QTM. Given a 2 C(M ca ) let n l and n r denote the indices of the leftmost and rightmost non-quiescent cells of a respectively. (If n l n r then set n r n l + 1. If all

9 cells of a are quiescent, then choose n l arbitrarily and set n r n l + 1.) Dene (s; h; c) T (a) as follows: let s (; s 0 ; ), let h n l, and let c : Z! be dened as Dene c(n) a(n) if n 6 nl and n 6 n r a(n)? if n n l or n n r : f(t; jcj) 4t 2 + 4jcjt? t: This is the number of steps required for M tm to simulate t steps of M ca on input c.. Lemma 5.5 For any t 2 Z + and c 2 C(M ca ), the probability that M ca accepts c after t steps is equal to the probability that M tm accepts T (c) after f(t; jt (c)j) steps. Proof. [Sketch] Assume that we begin the simulation of each time step of M ca with the tape head located over the leftmost cell of the nite section of M ca being simulated, and with the current internal state of M tm equal to (; s 0 ; ): First, the permutation is performed reversibly in two passes of the tape contents, rst moving to the far right then returning to the left. The nite section of M ca being simulated will grow by one cell on each end during this stage, since the right and left starred symbols are shifted one cell to the right and left respectively. Next, in a single pass to the right, the state of each tape square is \quantumly" transformed in accordance with. When this has been completed, the tape head is returned to the tape square representing the (new) leftmost cell of M ca being simulated. Since the tape squares outside of the region between the starred symbols are assumed to contain blanks (representing quiescent cells), the current internal state of M tm will again be (; s 0 ; ). Thus, after four passes of the current tape contents, M tm will be in a linear superposition of congurations which corresponds to the superposition of M ca after one time step, i.e. each conguration of M tm will have the same amplitude as the conguration in the superposition of M ca which it represents. The long-term evolution of M tm will thus behave exactly as that of M ca. By Lemma 5.4 and Lemma 5.5 we have Theorem 5.2 Given any M ca 2 1d-PQCA, there exists M tm 2 QTM which simulates M ca with linear slowdown. 6 Some Resulting Facts From the preceeding discussion, we are immediately led to two straightforward results. The rst result is an observation concerning the construction in section 5.2. The QTM which results from this construction has the interesting property that the motion of its tape head is deterministic, formalized as follows. Let M 2 QTM and let a be an arbitrary conguration of M. If M is assumed to be in conguration a and is run (unobserved) for l steps and then observed, then for any integer h there is some probability that the tape head of M will be located over the tape square indexed by h. We say that M has deterministic head position if, for all a 2 C(M), l 2 Z + and h 2 Z, we have that this probability is either zero or one. By Theorem 5.1 and Theorem 5.2 we have Corollary 6.1 For any QTM M, there exists a QTM M 0 such that M 0 simulates M with linear slowdown, and such that M 0 has deterministic head position. In addition to being deterministic, the head position of the QTM resulting from the previously mentioned construction is also oblivious; for any input of a given length, the position of the tape head at a given time will depend only upon this length. The second result regards the notion of acceptance and rejection by a 1d-PQCA. Consider M (Q; ; k; A) 2 1d-PQCA: Recall that for any a 2 C and l 2 Z +, M accepts a after l steps with probability equal to that of observing an element of A in the cell indexed by k after l steps. Of course, this probability may dier greatly with any change in l, so that if we are interested in whether or not M recognizes some language L C(M), we must pair some l 2 Z + with each input a 2 C and observe M after exactly l steps have passed. This may seem unsatisfactory since it implies that the observer must carefully clock the machine and observe it at precisely the right moment, or else the computation may be invalid. However, we see that this is a reasonable definition of acceptance/rejection, due to the following Theorem 6.1 For any M 2 1d-PQCA there exists an M 0 2 1d-PQCA so that for any a 2 C(M) and l 2 Z + there exists an a 0 2 C(M 0 ) (computable in O(l + jaj) steps) such that the probability that M accepts a after l steps is equal to the probability that M 0 accepts a 0 after l 0 steps for any l 0 l.

10 Proof. Suppose that a 1d-PQCA M (Q; ; k; A), with quiescent state, is given. First, dene M t (Q t ; t ; k; A t ) as follows. Let Q t f0; 1g f0; 1g f0g (with t (0; 0; 0) as the quiescent state), let t be the identity matrix and let A t f(1; 1; 0)g. The action of M t on any conguration is identical to the permutation. For any given l 2 Z +, if we dene a t 2 C(M t ) as a t (n) 8 < : (0; 1; 0) if n k (1; 0; 0) if n k + l (0; 0; 0) otherwise, then on input a t, the cell of M t indexed by k will be in the state (1; 1; 0) after exactly l steps have passed, and at no other time will any cell contain this state. Now dene M 1 (Q 1 ; 1 ; k; A 1 ) as follows. Let Q 1 Q Q t (with quiescent state 1 (; t ), let A 1 A A t and let 1 be as dened by 1 ((p; p t ); (q; q t )) (p; q) t (p t ; q t ) for (p; p t ); (q; q t ) 2 Q Q t. Clearly M 1 2 1d-PQCA. M 1 can be viewed as M and M t \stacked" on top of one another, evolving independently. For any given a 2 C(M) and l 2 Z +, if we dene a 1 2 C(M 1 ) as a 1 (n) (a(n); a t (n)); where a t is as dened above, then the probability that M 1 accepts a 1 after l steps is the same as the probability that M accepts a after l steps. Furthermore, the cell of M 1 indexed by k will not contain an accepting state at any other time, and at no time will any other cell contain an accepting state. Finally, dene Q a f0g f0; 1g f0g, and dene M 0 (Q 0 ; 0 ; k; A 0 ) as follows. Let Q 0 Q 1 Q a, let A 0 Q 1 f(0; 1; 0)g and dene 0 as 0 ((p 1 ; p a ); (q 1 ; q a )) 8 < : 1 (p 1 ; q 1 ) 0 otherwise if q 1 62 A 1 and p a q a or q 1 2 A 1 and p a 6 q a for every (p 1 ; p a ); (q 1 ; q a ) 2 Q 1 Q a. Since 1 is unitary, we must have that 0 is unitary, so that M 0 2 1d-PQCA (with quiescent state ( 1 ; (0; 0; 0)).) Now, for given a 2 C(M) and l 2 Z +, dene a 0 2 C(M 0 ) as a 0 (n) (a 1 (n); (0; 0; 0)); where a 1 is as dened above. Consider the evolution of M 0 on input a 0. By the denition of 0, we see that any state (p 1 ; p a ) will be transformed (with nonzero amplitude) into state (q 1 ; q a ) with q a 6 p a (toggling acceptance/rejection) exactly when q 1 2 A 1. But if any cell of M 0 ever contains a state in A 1 Q a, then we know that this must be the cell indexed by k and exactly l steps must have passed (by the property of M 1 on input a 1 mentioned above.) Thus, such a transformation will occur at most one time on any given computation path, and only on those paths which correspond to paths of M 1 which accept a 1 after l steps. Acknowledgements Many thanks to Eric Bach for his valuable insights and advice, and for introducing me to quantum computing. Thanks also to Anne Condon and Walter Ludwig for their helpful comments and suggestions, and to Christoph Durr for pointing out an error in an earlier version of this paper. References [1] R. Feynman, Simulating Physics with Computers, International Journal of Theoretical Physics, Vol. 21, nos. 6/7 (1982) [2] D. Deutsch, Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer, Proc. R. Soc. Lond., Vol. A400 (1985) [3] E. Bernstein and U. Vazirani, Quantum Complexity Theory, Proc. 25th Ann. ACM Symp. on Theory of Computing (1993) [4] A. ao, Quantum Circuit Complexity, Proc. 34th Ann. Symp. Foundations of Computer Science (1993) [5] P. Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, Proc. 35th Ann. Symp. Foundations of Computer Science (1994) [6] K. Morita and M. Harao, Computation Universality of One-Dimensional Reversible (Injective) Cellular Automata, Transactions of the IEICE, Vol. E 72 (1989) [7] R. Cooke, Innite Matrices and Sequence Spaces, Macmillan and Co., London, [8] N. oung, An Introduction to Hilbert Space, Cambridge University Press, Cambridge, 1988.