Unitary dilation models of Turing machines in quantum mechanics

PHYSICAL REVIEW' A

VOLUME 51, NUMBER 5

MAY 1995

Unitary dilation models of Turing machines in quantum mechanics Paul Benioff' Enoironmenta/

Assessment Diuision, Building 900, Argonne Rational Laboratory, Argonne, Illinois 60439 (Received 1 July 1994; revised manuscript received 6 January 1995)

A goal of quantum-mechanical models of the computation process is the description of operators that model changes in the information-bearing degrees of freedom. Iteration of the operators should correspond to steps in the computation, and the final state of halting computations should be stable under iteration. The problem is that operators constructed directly from the process description do not have these properties. In general these operators annihilate the halted state. If information-erasing steps are present, there are additional problems. These problems are illustrated in this paper by consideration of operators for two simple one-step processes and two simple Turing machines. In general the operators are not unitary and, if erasing steps are present, they are not even contraction operators. Various methods of extension or dilation to unitary operators are discussed. Here unitary power dilations are considered as a solution to these problems. It is seen that these dilations automatically provide a good solution to the initial- and final-state problems. For processes with erasing steps, recording steps must be included prior to the dilation, but only for the steps that erase information. Hamiltonians for these iHA) U, where U is a unitary processes are also discussed. It is noted that H, described by exp( — step operator for the process and b a time interval, has complexity problems. These problems and those noted above are avoided here by the use of the Feynman approach to constructing Hamiltonians directly from the unitary power dilations of the model operators. It is seen that the Hamiltonians so constructed have some interesting properties.

=

PACS number(s): 03.65.Bz

I. INTRODUCTION Background

There has been much work on developing models of computation in quantum. mechanics. The work is based on studies of Landauer [1—3] on reversible and irreversible computation, energy dissipation, and the physical nature of information and the computation process [4]. The work of Bennett [5], which showed that every Turing machine could be perfectly simulated by a reversible Turing machine, was also very important in stimulating work on quantum computation. Benioff [6—10] developed different types of quantummechanical Hamiltonian models of Turing machines. Some of these models were described on two dimensional lattices of spin- —, particles. Additional scattering systems were present in some of the models [6,7]. Feynman [11] initiated work on quantum-mechanical computational networks and proposed a ballistic Hamiltonian model of computation. Deutsch [12,13] provided the first description of quantum Turing machines (QTMs), which are the analog of classical nondeterministic or probabilistic Turing machines. He also described a universal quantum simulator and extended the work on quantum computational networks. Margolus [14,15] discussed serial and parallel quantum computation based on cellular automa-

ta. Peres [16] and Zurek [17] have worked on reversible logic and error correction in quantum computation. More recent work has included studies on the efficiency of quantum computers [18,19], on quantum complexity [20,21], and on unsolvability aspects of the spectral properties of model step operators [22,23]. Investigations of the computation power of QTMs [24 —26] suggest that for some cases these machines may be more efficient than classical machines. For example, Shor [24] has described a quantum algorithm that solves the discrete logarithm and integer factoring problem in polynomial time. More details can be found in the original papers and in a recent review of quantum computation [27].

Present mailing address: Physics Division, Argonne National Ar gonne, IL 60439. Electronic address: Laboratory, pbenio8@ANL. GOV

This paper is concerned with some aspects of model models of the step operators for quantum-mechanical computation process and Hamiltonians that describe the time evolution of these models. One method of model construction, which is widely used, is to use simple projection operators and unitary change operators to construct model step operators that model only the information-bearing degrees of freedom of the process. One goal of this construction is to provide a model operator description of the process where iteration of the operator corresponds to carrying out successive steps of the process. If the process halts, then further iterations of the model operator should not change the halt status or the result obtained. This is a necessary condition to enable measurements, to determine the halt status, and the Snal result if the system has halted, to be performed at leisure with minimal chance of disturbing the system [8,9]. As an operator on a space spanned by a set of orthogonal quantum basis states, which model process

1050-2947/95/51(5)/3513(12)/$06. 00

3513

51

1995

The American Physical Society

PAUL BENIOFF

states, the goal is equivalent to requiring that the model operator be unitary with stable initial and halting states. There are some problems with this method. One is that very often operators constructed directly from the description of the process do not have the desired properties. The process operator can be only partially defined so that iteration will annihilate the halted state. Another problem is that some steps are irreversible in that they are information-erasing steps. Iteration of these operators can also destroy the normalization of computation basis states. There are many solutions to these problems in the literature. These include addition of initial and final ballast states to provide for overall system motion with no change in the halting status at least for many iterations [9]. Implicit [8,9] or explicit [25] extensions of partially defined operators to cover the whole space have also been used. Irreversibility due to information erasure can be treated by expansion of the process to include a history recording of each step. This was described by Bennett [5] for Turing machines and applied to quantum-mechanical models by Benioff [7]. Deutsch's use [12] of unitary-step operators at the outset assumes a solution of these problerns, although no specific discussion is provided. In this paper use of minimal unitary power dilations of contraction operators (operators with norm llTll ~ 1) is investigated as a solution to these problems. The dilations will be combined with Feynman's construction of HamilHamiltonians to construct quantum-mechanical tonian models of Turing machines. In Secs. II and III a discussion of models of simple examples of one-step processes that illustrate the problems will be followed by a brief discussion of unitary dilations of the model process operators. Next comes a brief description of the theory of minimal unitary power dilations of contraction operators (Sec. IV A). Application to the simple examples discussed is done in Sec. IV B. Two simple examples of multistep processes as Turing computations, which illustrate the problems, are discussed in Sec. V with additional details in the Appendix. It is seen that unitary power dilation provides a very satisfactory solution to the initial- and the final-state problems. However, history recording for informationerasing steps is needed prior to dilation. Application of these results to Turing machines in general, including universal machines, is summarized. Hamiltonian models are described in Sec. VI. It is seen that there are problems of complexity in associating a finite-time interval 6 with each model step. Feynman's method [11] of Hamiltonian construction, applied to the unitary power dilation models, is shown to avoid this problem and any problems of initial- and final-state stability. Section VII concludes the paper with a discussion of of the unitary power dilation physical interpretations models. Physical models of the dilation are provided for which the dilation Hamiltonian is spatially local. The use of unitary power dilations in quantum mechanics is not new as it has been used in the study of master equations and in the dilation of semigroups of

51

contractions to unitary groups [28 —31]. Here the contraction T and the corresponding unitary power dilation V generate a discrete semigroup (T" for n =0, 1, . . . ) and discrete group [( V")" for n . . , —1, 0, 1, . . . ], respectively. This paper follows many of the previous papers in the field by neglecting the problem of physical constructibility. This problem, which has been emphasized by Landauer [4,32], needs remedying. Work in this area, which is in very preliminary stages, includes that of Lloyd [33] and Reck et al. [34].

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II.

EXAMPI ES OF SIMPI.E ONE-STEP PROCESSES

Consider the establishment of correlation between two binary systems 5 and R. Let S be in the initial state and R in the initial state lo). The inp=alo)+pl teraction, which is much studied in the quantum theory of measurement and is the essential part of the recording step in models of Turing machines [6—10], is given by

1)

alo&lo&+pll

ylo&

&

11

&

.

(2. 1)

An operator for this as a one-step process is given by

pSpR

+ pS

RpR

(2.2)

„

where P, and Po are projection operators and O. is the Pauli spin operator. Both T and its adjoints are partial isornetries, i.e., T T and TT are projection operators. Note that T T=POWTT =POPO +PfP, . T is also a contraction operator as ll Tll ~ 1. From the viewpoint of this paper T is not a satisfactory model of the correlation process (as a free-standing onestep process). The reason is that T annihilates any state of the form Pl 1 ) (S in the state and R in the state 1 ) ). As a result, iteration of T destroys part of the final state. That is, &"1(lo) =alo) lo) for n &2. For another example consider the erasure of information from a single bit, i.e., the transformation + lo) 1 )— + lo). This is represented by an operator and lo) —

f

l

l

T= po+o „pi .

(2.3)

Here T is not even a contraction operator as llTll =2. T is clearly not suitable as a model operator for a one-step (free-standing) erasure process as iteration of T destroys the normalization of states that are linear combinations of lo) and 1 ). The operator W= T!&2 is a contraction operator. 8' and 8' are both partial isometrics as W W=PO and WW =(1+o„)/2 are both projection operators. However, this case is di6'erent from the first example in that 8'8 ~ is a projection operator in a rotated basis. For this reason iteration of 8'on a general state alo)+pl 1 ) destroys the state normalization by asymptotically annihilating the state. This case has been much discussed in the literature, especially by Landauer [1,2] with respect to reversibility of computations and information erasure. It is well known that erasure can be made reversible by creating and saving a record. For example, let R and S be two binary systems. The creation of a record [Eq. (2.2)] followed by use of the record to carry out an erasure is l

UNITARY DILATION MODELS OF TURING MACHINES

51

defined by the operator

T pSpR

+

SpS RpR

where, for convenience, the two steps are combined into one operator. As defined by Eq. (2.4), T is a contraction operator, but it has the same problem as the recording example; that is, T gives 0 when applied to S in any state and R in 1). T is a partial isometry in that both T~T and TT are projection operators on different two-dimensional subspaces of the four-dimensional Hilbert space of all configurations. It is to be emphasized that these examples are provided for purposes of illustration. In most computations these single-step operations are not free standing but are part of a more complex computation process. However, as shown in Sec. V and the Appendix, the same problems hold for quantum models of Turing machines. None of the problems discussed above appear in Deutsch's definition [12,13] of quantum Turing machines as the step operator is assumed to be unitary at the outset. His definition assumes that the problems described here are already solved at the elementary step level, e.g. , recording steps are provided and any partially defined operators are completely defined over a suitable set of basis states. I

III.

UNITARY DILATIONS

OF MODEL PROCESS OPERATORS A goal of this paper is to investigate the use of unitary dilations of model process operators as appropriate quanturn models of the process. That is, if T is a model contraction operator on a Hilbert space %, which is a subspace of a larger space H, a unitary dilation U of T must be equal to T on & in the sense that PU P = TP, where P projects & out of H [35]. In addition, it is required that iteractions of U model iterations of T in the sense that for n = 2, 3, . . . , P ( U ) "P = T". A unitary dilation that satisfies this condition is called a unitary power dilation

of T [35]. rhere are many extensions of contraction operators that are not unitary power dilations. Consider, for example, the correlation example of Eq. (2.2). T is defined on the three-dimensional subspace & spanned by I00), I10), and I 11), of the four-dimensional space H, which also includes I01 ). One unitary extension is given by UT

This

is

T+pSpR +pS

RpR

not

dilation

a unitary

3515

renormalizing by dividing by &2 is not necessary and erasure of the S bit is accomplished. However, the extension is not unitary. It is similar to the correlation operator in that it is defined on a three-dimensional subspace of

H. As was the case for the correlation, T has a fourdimensional unitary dilation. However, the extension is not a power dilation. It is also possible to give a direct unitary dilation of W=T/&2 to an operator on a fourdimensional space such as H using a canonical dilation procedure [35]. However, the dilation is also not a power dilation. The same holds for a direct unitary dilation of T to a unitary operator of an eight-dimensional space H+H. However, as before, the dilation is not a unitary power dilation.

IV. MINIMAL UNITARY POWER DILATIONS OF CONTRACTION OPERATORS A. Theory

Here a brief discussion of the theory of minimal unitary power dilations of contraction operators is given. The material is based on the work of Nagy and Foias [28], Halmos [35], and Fillmore [36]. For additional details, the literature, especially the work of Nagy and Foias [28], should be consulted. Let % be a Hilbert space of states of a system Q and T a contraction operator on &. Define operators Dz and

D tby

Dr =(1 —TtT)' D

t=(l —TT

)'

(4. 1)

(4.2)

where positive square roots are meant. Note that D p and DT are self-adjoint. They are also well defined as II

TII

—1

~

These operators are called defect operators on &. The reason for this can be seen from the fact that these operators satisfy

(3.1) as

(PU 11) = I10), but Tl 1 1 ) =0). The U T T+ ~SpS~RpR + Sx pSpR x 0 x 1

IN. . .

1

1

TAPU

on extension given by

(3.2)

is a unitary dilation of T. It is not a unitary power dilation of T as P( U ) AT on &. The erasure operator, given by Eq. (2.2), can also be extended in several ways. One such extension to a fourdimensional space is given by Eq. (2.4). As noted, history recording and erasure are combined into one step. The extended operator is a partial isometry (both TtT and TT are projection operators). It has the advantage that

for any g in &. Note that for any P for which IIT@ll=ll@ll Dr@=0. A»o if T@=0 the»T+ Similar relations hold for D y and T . Commutation relations for DT, T, D p, and T are given by

TDT=D

yT

(4.3)

and

T D g=DTT

(4.4)

Define the Hilbert space H by

H =Me 1'(Z ),

(4.5)

3516

PAUL BENIOFF

where l (Z) is the Hilbert space of square summable seA general state in l (Z) has the quences from —00 to form P=gc Im ). A general state in H has the form where I' =g Im) with g denoting a

~.

4,

4=gd

state of Q. A unitary power dilation' V on H of T on & is given by (Nagy and Foias [28], Halmos [35], Fillmore [36])

VT=

g

J= 00 j@0,—1

(UP )+(T+UDT)PO+(UD t

U— T )P (4.6)

Here P is

j

th. e projection operator for state ) in l (Z) and U is the bilateral shift, that is, U ) = + 1 ) . U satisfies UP =PJ+1U. DT and D p are as defined by Eqs. (4. 1) and (4.2). The unitarity of V follows from the definitions and commutation relations for Dz and D y. The adjoint ( V ) of V, obtained from Eq. (4.6) as I

(V

)

=

g

J= oo jAO, —1

j

I

I

j

(P)U )+Po(T +DTUt)

+P, (D

tU

TUt )— ,

is also a unitary power dilation

(4.7)

of T [35].

Physically, unitary power dilation can be thought of as adding a degree- of freedom, such as a particle moving along a one-dimensional lattice from —~ to Do, to the original system. In this case, states P represent wave packets of the particle position along the lattice. General states in H of the form 4 = gd 4 m ) represent overall system correlated state wave packets for finding the system Q in state g and the particle at lattice position Im ). Other physical interpretations are discussed in Sec.

51

Here 2) t and 2)z. &0 are Hilbert spaces spanned by all vectors of the form gd~ P m ) with g in Xl t and or P in 2)z and m 0, respectively. &o contains m vectors of the form QIO) with P in &. 2)z and 2) t are the defect subspaces of % defined as the completion of 2)z& and 2) t%, respectively. For our purposes, it is more convenient to work with H instead of E even though the interactions of interest are those on states in E. This causes no problem because both K and the orthogonal complement HBK of K in H are invariant under V . The action of iterations of V on states 4' in HBE is very simple. It is sufficient to consider a state PIm ) with —1 and D„tg=o. Iterations of V moves the partim cle stepwise down to position —1 with no change in the state g of Q. The next iteration moves the particle to lattice position +1 and operates with —T on g. Additional iterations move the particle stepwise along the lattice with no further changes in the state g. Since these states are not of interest here they will not be discussed further. The action of iterations of V on states can also be expressed directly. For the purposes of this paper it is sufficient to consider a state 0'=ttjI —1) with the particle at position —1 and Q in state g. One has from Eq. (4.6)

)

(0

I

(

n

—1

(v')"+=T" 'D, &BIO)-+ y D, T"

&

'D, ,yIJ

)-(4.9)

I

VII.

The

will be used for the particle interpretation remainder of this paper. Based on this, V refers to the system Q fixed at lattice positions —1 and 0 and interacting with a particle moving along the lattice from left to right (from negative to positive lattice positions). The factor multiplying P 1 in the definition for V splits a general state into "go" (UD t) or "no-go" (U Tt) components. The factor multiplying Po splits the "go" component further into a history component ( UDT) and a T active compc, nent. This will become clear from the study of examples. As defined, V on H is not minimal. A minimal unitary power dilation is obtained by restricting V to the subspace E of H defined by

K=2)rt oS&p&2)z)o .

(4.8)

8. Examples For the correlation example, the unitary power dilation is given by V, Eq. (4.6) with T given by Eq. (2.2). Here DT=P, and D =P P +P P". Let O'= IO, O) Im ) with m —1 an initial state of the dilated model system with S and R both in state IO) and the extra particle p at lattice position m to the left of the origin. Iteration of V on this state moves p along the lattice to position —1 with no changes in the state of S or R. At the next iteration of V the switch term UD t UTt is ac— tive Since . D tIO, O) =0 and T IO, O) =Io, o), the effect of the next iteration is to model

(

transform IO, O)I —1) to —Io, o)I1) with no change in the state of S or R. Additional iterations just move p along the lattice with no further changes in S or R. For the initial state Il, o) —1) with S in state I 1), —1), R in state Io), and p in state VrI1, 0)I —1) =Il, o)IO). For this first iteration, the switch term acts difFerently because D tIO, O) = IO, O) and I

I

T'Io, o) =o.

The presentation given here is isomorphic to that in the literature. There H is given by k&k, where the k sum ranges have the form from —~ to ~ and &k=&. States in k, . . . , $0, . . . , fk, . . . ] with each component state in ~k. no~ is give»y II pll'= Xk Ilfk II' A»o, in the literature, V~ is given in matrix form only.

0

4

»e

2It is sufficient to consider initial states with p at position —1 —1 just move p to the left since iteractions with p at positions with no changes in the other systems.

(

IN. . .

UNITARY DILATION MODELS OF TURING MACHINES

51

For the next iteration the term T+UDT is active. Since Dr 1, 0) =0, one gets (V ) ~1, 0) —1) = ~1, 1) lo) as T is active. For the third iteration T is inactive and DT is active to give the state 1, 1 ) 1 ). Additional iterations move p to the right with no further changes in the state of S or R. This example illustrates the desirable features of the unitary power dilation models. Both initial and final ballistic states are automatically provided, much like in Feynman's [11] and Benioff's [10] models. For example, one can start with an initial wave-packet state c ~1, 0) ~m ) with c %0 for m After n iterations of V, where n+m &1 for all m for which c %0, the final state is a wave-packet state c 1, 1 ) m + n —1 ). Iterations move the wavepacket state to the right with no further changes in the state of S or R. Another good feature is that T is active for at most one iteration. This is what is desired in a free-standing model for this example (where recording is not part of a larger process). In this way the undesirable e6'ects from multiple iterations of T are avoided. Another property of this example needs discussion. g'~ 0 ) ~ —1 ), where initial state an Consider which is a linear superposition of states g=a~o) +P~ of S. After n & 1 iterations of V the overall system state is given by a~o, o) ~n )+P~ 1, 1 ) ~n —1). The coherence between the different components of S and R is destroyed as the state of p is also correlated with the outcomes of the S-R correlation establishment. This is a consequence of the fact that 0 iterations of T are needed to generate the output if S is in the state ~0) and 1 iteration is needed if S is in the state ~1). This loss of coherence would be undesirable in computations with recording steps, su. ch as those that occur in the Turing machine models [9, 10], which consist of repetitive sequences of record computations, and head shift steps. However, the problem does not occur in the models because of the presence of switch systems, which have the effect of requiring one iteration of T for the recording step irrespective of the state of S. For the single bit information erasure, the unitary power dilation model operator is given by Eq. (4.6) with T replaced by W= T/&2 and T given by Eq. (2.3). Here D s = ( I —o „)/2 and D t = P i are both projection operators, but in different bases. For the initial state ~0) —1) with S in state ~0) and p in state —1), the first iteration of V generates the rotated state — g+ 1 ) . Here g+ = 1/&2( ~0 ) + 1 ) ). Note that D~t~0) =0. Additional iterations move p along the lattice with no further changes in the state of S. For the initial state 1) —1) the first iteration of V generates the state ~1) ~0). The next iteration gives the state I/&2~0) ~0)+ I/V2f ~1). This is the first case encountered where both terms of a switching plus action factor in Eq. (2.6) give nonzero contributions. The third iteration gives the state ~

~

((0.

~

~

1),

~

~

~

~

~

~

I/2(IO&

IO&

Iteration of V

—@ n

~

Il &)+I/&2Q

+ 1 times

j=0

~

gives the state

2

~n) .

(4. 10)

This state is complex, yet it is exactly 'what is to be expected. The sum represents a sum over history components with one history component generated for each iteration of 8'. The powers of 2 factors, which appear automatically, ensure the correct normalization of the state. Generation of history components is clearly necessary for a unitary operator since each iteration of 8 is an operation. Note that this occurs information-erasing only for the initial state 1 ) —1 ), which is the state undergoing erasure. History components are noi generated for the initial state ~0) —1). Examination of this and the previous example shows another satisfying property of unitary power dilations. Not only are initial and final ballast states automatically generated, but history component states are generated when needed. The main problem with the iterated state [Eq. (4. 10)] is that the leading (left-hand) term is exponentially small. This is the main term of interest resulting from the n + 1 iterations (in iterations of W) and is not a history component. Almost all the normalization is taken up by history generation. This problem can be avoided by adding history recording to the model T operator prior to unitary power dilation. An example of this is given by Eq. (2.4). It is left to the reader to show that iteration of the unitary power dilation of T defined by Eq. (2.4) does not generate history components and does not have the normalization probabove. The same holds if history lems encountered recording and information erasure are separated into two steps. This problem will be revisited later on.

j

~

~

~

V. UNITARY POWER DILATION OF TURING MACHINE MODELS

In this section unitary power dilations of quantum models of multistep processes exemplified by deterministic Turing machines will be considered. Two simple examples that illustrate the problems encountered in the previous sections will be considered: adding 1(mod2) to a binary string and adding 1 to the end of a string of 1s. For the purposes of this paper it is sufhcient to consider Turing machines with one two-way infinite tape C of binary cells and one head h, which can assume any one of a finite number of states. The head in state m at position and expression s on the tape is modeled by a computation basis state ms ) . At any given time, at most a finite number of tape cells contain a 1. All computations are assumed to begin in a standard initial state 0 of h, at position 0, and Os in all cells of C to the left of position 0. Halting computations end with h in state f, at position 0, and Os in all C cells to the left of position 0.

j

~

j

3518

PAUL BENIOFF

lated by T . Thus T annihilates any standard initial state. Based on this, T does not represent a satisfactory model of the addition of 1(mod2).

A. Adding 1(mod2) to a binary string mec-hanical model

Quantum

For this example the head moves to the right from position 0 past C cells containing a 1 to find the first cell containing a 0. The 0 is replaced with a 1, followed by the head retracting its steps, replacing 1s with Os as it moves back. This back motion stops when the head arrives at the first cell (at position —1) containing a 0. The head then moves one step to the origin and halts. The model operator for this process is given by

T

y

J=

[phg

h+ Uhph

g + Uhphg

ht

oo

+ph~ g

uh

+ Uhphg

uhy

(5. 1)

The adjoint T is given by

Tt=

g

J=

p [uhtg

ph+g

phUht+

hg phUh't

The unitary power dilation model of the addition of 1(mod2) is given by Eq. (4.6) with T given by Eq. (5. 1). Recall that U moves particle p one step along a lattice and pk is the projection operator for finding p at position k. Let iIIo, ok = ~OsO) ~k ) describe a standard initial state of the Turing machine with h in state 0 at position 0, expression s on C, and p at position k with k &0. The actions of iterations of V on this state are a generalization of the actions described for the one-step correlation example. Initially, p is moved along the lattice with no changes in the Turing rnachine state. When p arrives at position —1, the switch factor UD y —U T~ becomes active. Since D t~OsO) =~OsO) and T ~OsO) =0 [Eqs. (2.2) and (5.2)], the switch transforms the state ~OsO) —1) to ~

oo

+uhg ~ ph+uhtg

phUht)

(5 2)

In these expressions I';" and p are the respective projection operators for finding the head in state i and at C and Qk, is the projection operator for finding position symbol s (0 or 1) at C position k. U" and u" are bilateral shift operators for changing the head states and the positions. They satisfy U"P;"=P;"+iU (mod4) and u hp =p. +, u h . The unitary operator for exchanging 0 and 1 at cell of C is given by o. . From left to right the terms of T in Eq. (5. 1) correspond to shifting the head to the right past 1s, changing the first 0 encountered to a 1, moving back one step, moving back and replacing 1s with 0s, and moving right one step to the origin. The operators T T and TT~ can be determined from the expressions for T and T to give

j

j

TtT=Ph+P" +P"

g

(5.3)

Q ip

and

TT

2. Unitary power dilation

=P2+ Q lPog, ip, +i+Pig, ip, +P3gjop,

j

ioso) io). Iterations of V on Vo, oo= ~OsO) ~0) have the effect of iterations of T on the Turing machine state without changing the position of p at 0. This occurs because the switch factor UDT is inactive for all n for which (and DrT"IOsO) =0). The strucf/T"JOsO) f/=///OLO)// ture of T given in Eq. (5. 1), particularly the lack of overlap between the terms, guarantees that norm preservation occurs for all steps of the model calculation. Iterations of V on the model state continue to generate states representing successive computation steps until the standard mode1 halting state 43, +&QQ is reached. For this state T~OsO) =0 and the switch factor UDT becomes active and stops the computation by giving V T4 3 + ]QQ 0 3 + ]Q] Continued iterations of V T continue to shift p down the lattice with no further changes in the Turing machine state. The discussion shows that the unitary dilation automatically generates predecessor states for %Q, QQ and successor states for %3 +$QQ There are an infinite number of these "ballast" states and they are pairwise orthogonal. As was noted for the correlation example, the action of V on wave-packet states follows immediately. The action of V on a wave-packet state % Q Q defined by

(5.4)

j

where Qjo+Qji=l for each and gP pj=l have been used. These equations show that both T and T~ are partial isometrics as both T T and TT are projection operators. T is clearly a contraction operator. T represents a multistep process with the number of steps depending on the number of successive 1s to the right of the initial head position and the initial head state. For the standard initial state ~OsO) with n successive 1s to the right of position 0 in s after 2n + 3 iterations of T the final state 3, s + 1, 0) is obtained. One more iteration of T annihilates this state as T~3, s, ) =0 for any s and Similarly, any state with h in the initial state 0 and at any position next to a cell at containing a 0 is annihi-

j

~

j+1

j

j.

+Oso

g

Ck +Osok

(5.5)

gk~ck~ =1 and eh%0 only if k(0, moves the packet along the lattice without changing the machine state. Continued iterations of V generate a linear superposition of states with the particle at position 0, but each component representing a different stage of the computation. If the wave-packet width is much less than the number of steps in the computation, then, for a range of iterations, essentially all components will be at position 0, but in different stages of the model computation. That is

where

( V')"'Po. o=

ck T"+"+o,m . X k

(5.6)

IN. . .

UNITARY DILATION MODELS OF TURING MACHINES

)

The value of n is such that n +k 0 for all k ( (0) for which eh%0. As the computation is completed the wave packet will emerge from position 0 and p will move along the lattice with no changes in the final halted Turing machine state ~3, s+1,0). This description of the action of V also applies to states that are linear superpositions of standard initial states each with a different tape expression. That is, iterations of V on the state 4'oo=g, d, 'po, o with po, o given for each s by Eq. (5.5) (where the ch can also depend on gs carry out the model computation independently on each different s component wave packet. As noted by Deutsch [12], computations can be carried out simultaneously on different inputs in this manner. As noted in Sec. IV, the operator V defined on H is a The unitary power dilation, but it is not minimal. minimal unitary power dilation is obtained by restricting V to the subspace IC defined by Eq. (4.8). It is of interest to briefly discuss which model states are in K and which are in H but not in K. The standard initial state 'IIp, pk for k 0 is in K; but this state for k & 0 is not in K. The reader can verify this from the definitions of T, E, Dz, and D y. Similarly the standard final state 3 + ipse is not in E for k & 0. More generally, any model state k in the domain of T T is not in K for k &0. Similarly any model state is not in E for k if it is in the domain of TT . The action of V on states in H but not in E are given in Sec. II. As noted, both X and its orHBE are invariant subspaces thogonal component for V .

4',

(0

B. Addition of 1 to the

end of a string of 1s

The Turing machine for addition of 1 to the end of a string of 1s is modeled here as a multistep process with an information-erasing step. As such it would be expected to show properties in common with the single information-erasing step discussed in Secs. III and IV. The model operator T is defined by

[P "Q,uh+

J=

UhPho

ht Q u

oo

+phg uh'f+ Uhphg

uh]

(5.7)

From left to right the terms shift h to the right past 1s until a 0 is encountered. The head moves back one cell after changing the 0 to 1. Then h moves back past the 1s until a 0 is found. The computation halts with h moving one cell to the right. T is not a contraction operator because of the overlap of the second and the third right-hand terms. In particular ~~T~~ =2. To see this, note that for a state g of the form 1

—( ~os 2

j ) + j ) ), ~

Lt

where the tape expressions s and t are related by s(j ) =0, t(j)=1, and s=t elsewhere, Tf=&2~1tj —1). Based on this and the consideration of other states T can be converted into a contraction operator W by dividing the second and the third terms by 1/&2, that is,

J=

Pogjiu

+

3519

(U"Pocr Q.ou "t+Phig,

u ht)

oo

+ UhPhg

uh

(5.8)

It is tedious but straightforward

to show that W and isometrics in that both W W and WW~ are projection operators. In particular, W~ are partial

WW='po+P, + '

—,

g

[

—Ptg o —P"Q +POU"

Q~oo

+P", U Q, crj]p)

(5.9)

and

WW

=

g

J=

[Pog~jp, +i+P2Q&OPJ+i+P", QJ@~, ] . oo

(5. 10) Details of the iteration of the unitary power dilation of Won the state %'p, p i are given in the Appendix. There it is seen that the iterations generate history components only for those steps for which the second and the third terms of W are active. No history components appear for steps in which the first and the fourth terms of W are active. The main problem with the unitary dilation of W is that the component of the state giving the result of the is multiplied completed computation by a factor of is the number of steps in which the where second or the third term of W is active. As a result, for long model calculations, there is an exponentially small probability of observing the result of the calculation; almost all of the probability is taken up by the history components in the state. For this reason direct dilation of the partially renormalized W will not be used further. Instead the model will be changed by adding history recording, but only for the second and the third terms in T. The unitary power dilation of the changed model is satisfactory in that no additional history components are generated. Details are given in the Appendix. V

2,

I

C. Turing machines in general The above can be applied to construct unitary power dilation models for deterministic Turing machines in general, including universal machines. A brief description of the procedure follows. To each Turing machine is associated an operator T that can be constructed directly from the machine tuples. If all the terms are pairwise orthogonal (no overlap), then T and T are partial isometrics and a unitary power dilation constructed from T will have no history components, as was the case for the first example, the addition of 1(mod2). If overlapping terms are present, signifying iniormation erasure, then T must be changed to include history recording terms that are switched on whenever the over-

PAUL BENIOFF

3520

lapping terms are active. The resulting mode1 is quite complex because switching, etc. , must be arranged so that all terms in the changed model are pairwise orthogonal in both the changed operator and its adjoint. Unitary power dilation of the changed model is satisfactory in that no additional history is generated by the dilated model and the norm of the successive computation states is preserved. Details on one method for constructing a changed model and its dilation are given elsewhere [37]. Qne feature of the general case is worth discussing here, namely, that the computation may not halt for some values of the input. This is the case if T models a universal Turing machine with recording added where necessary. Let %p, p;pI, be a standard initial dilated state with particle p at position k &0. The subscript Os0 refers to the initial state and the position of the head and the expression on the computation tape. i0 denotes the initial expression on the record tape(s) and a switch Z in position 0. If s is such that the machine never halts, iteration of V moves the particle p to k =0, where iterations of V correspond to iterations of T. Since the computation never halts, p never leaves k =0 to move to positive values of k. In a similar fashion, a standard initial state wave packet 0'o, o;0 = pi, cq +o,o;OI„ localized in the k & 0 region, moves to position k =0 under the iteration of V . There it remains until the computation halts and the packet exits position 0 and the particle moves along the k & 0 region. If the computation does not halt the packet never exits, i.e., it "disappears into a hole" at position 0. During operations at position 0, the initial spread of the wave packet is manifested as a spread among iteration states

of T.

machines. There are two problems with this method. One is that for models that do not have ballast state transitions built into U, to stabilize the desired final state, time evolution under the Hamiltonian can destroy the final state. A very simple example of this is the time evolution of the one-step correlation process given by a Hamiltonian that satisfies e U, where U is given by either Eq. (3. 1) or (3.2). After time b, the action of H on the initial state 10) gives the desired final state 11 But this state is not stable as it is transformed back to the initial state after time 2h or 3b, for U defined by Eq. (3. 1) or (3.2), respectively. Other examples can be constructed from unitary extensions or dilations of the Turing machine model operators. This problem does not exist for Hamiltonians describing unitary dilations that are power dilations. These are constructed so that the final process state, such as that for halting computations, is stable under additional iterations of V . Examples are the unitary power dilations of the correlation process (Sec. IV 8) and of Turing machine models with history recording for overlapping steps

™=

~

~

}.

(Sec. V). A more fundamental

problem with this method of constructing Hamiltonians concerns the complexity of the Hamiltonian. In essence, it can be shown that for times t that are not integral multiples of the step time 6, e ' ' is a linear superposition over all basis states reached arbitrarily far into both the future and the past. This is an immediate consequence of the spectral decomposition of e and of H, where H is defined by e ' =U . This holds irrespective of whether U is or is not a power dila-

™

tion.

Iterations of V on wave packets of the form Too, o=g, d, +0,0,.0 can result in a sum over components for nonhalting computations with particle p at position 0 and halted components with p at positions k &0. Based on Lloyd's work [22,23], one concludes that it is impossible to efFectively decide whether a given initial component % p p p will or will not exit from position 0.

VI. MADEL HAMILTQNIANS There are several methods of constructing Hamiltonians for the model quantum systems considered here. Qne method is to consider a model process operator T to describe the possible steps of the process that occur in a finite time A. This requires that the Harniltonian H satis' =T(%=1 is assumed). For conservative isolated fy e systems H is self-adjoint, which requires that T be uni-

tary. Because of this requirement this method cannot be applied to the model operators T that are not unitary. This includes both one-step examples discussed in Sec. II and the Turing machine models described in Sec. V. The method can be used, however, for any extension or dilation U of T that is unitary. That is, define H by e ' = U . This includes all the unitary extensions or dilations discussed in this paper and includes the methods used by Benioff [6 —9) and Deutsch [12, 13] in describing quantum-mechanical Hamiltonian models of Turing

For models of Turing machines, this property of H, which is called "time global" [9], has the consequence that H appears to have the complexity of all computation paths. This is very undesirable because it means that construction of the Harniltonian implies knowledge of all computation paths prior to carrying out the model computation. For universal Turing machines this implies that the halting problem must be solved to construct H. Qne method of solving this problem is to construct a time-dependent Hamiltonian model of the process and add additional driver systems to construct a timeFor these models of Turing model. independent machines H has the complexity of the machine program, which is a desirable result [6,7]. Another solution is to follow Feynman [11]by use of T and its adjoint, or any extension thereof, as infinitesimal generators of the time translation. For example, H can be defined by H =K(X+Xt) or H =iK(X X'), where — X is T or any unitary extension of T. K is a suitable constant.

It is clear from the above and the definition of X that for Turing machines, H so defined has the complexity of the machine program. However, for models X that do not have suitable ballast states, computation halting states are not stable under time evolution. Examples include X=T with T given by Eqs. (2.2), (2 4), (3. 1), (3.2), or (5. 1). Feynman's method [11] can also be used to construct

UNITARY DILATION MODELS OF TURING MACHINES Harniltonians from the unitary power dilation models of Turing machines. Both problems discussed above are thereby avoided. To this end define H' for each Turing machine ~ by

H

=K[2 —v' —(v')'],

(6. 1)

V and its adjoint are defined by Eqs. (4.6) and (4.7). T is a Turing machine model step operator that includes recording or history steps where necessary [for example, Eq. (5. 1) or (A4} in the Appendix]. The superscript ~ on H refers to the fact that the Hamiltonian depends on ~. The motion of the particle p along the lattice given by H is that of a free particle, except for an interaction potential between it and the Turing machine at positions —1 and 0. To see this, H can be written as a sum of a lattice kinetic-energy term given by where

EKE=K(2 —U —U and a potential-energy

EpE=

term given by

+ U(DT K[[T— +[U(D

(6.2)

)

1)]Pa+— Po[T +(D

1)U

]—

T ]P 1) —U —

t

+P i[(D

t

i

1)U

Ut T—

. ]] —

(6.3)

These expressions are obtained from Eqs. (4.6) and (4.7) adding and subtracting UPo, Po U, UP „and P &U from the kinetic- and the potential-energy terms . is the bilateral shift operator and using gP~ =1. Here U and P the projection operator for position on the lattice for the particle. Note that the expression for the kinetic energy is essentially the lattice expression of the second derivative in position. Determination of the motion of the particle and time development of the Turing machine (TM) model and the eigenstates of H appears difficult as H' represents a particle moving along a lattice with a complex interaction with the TM at positions —1 and 0. However, this is not the case. This follows from the fact that, over the Hilbert space of states of the particle plus TM model, which are of interest, V is a bilateral shift. Thus the Hamiltonian, defined by Eq. (4. 1), describes motion of a free complex system (particle plus TM) not over lattice position space but over orbit position space. The basis states of each orbit can be generated from a state 0'O, o;k representing the TM in a standard initial state (the subscript i denotes the standard initial state of the record system and switch if present) and the particle at a position k &0. The states 0'„, =( V )"'Po, o;k for all integral n (note that (V ) "=[(V ) ]"}, which are pairwise orthonormal because V is a bilateral shift, are the orbit basis states for the orbit representing particle motion and computation by the TM ~ on input s. There is a separate orbit for each input s. In this representation, the motion of the TM plus particle, as a free system on each orbit, is the same as that of a free system on a lattice. This is clear from the form of the Hamiltonian given by Eq. (6.3), which is that of a kinetic-energy operator on a one-dimensional lattice, and by

j

—

IN. . .

3521

Eq. (6. 1), which is the kinetic-energy operator on "orbit space" spanned by the states O'„, . The spectrum of H ranges from 0 to 4 K on each orbit Hilbert space and all eigenfunctions have the same form. The spectrum is degenerate as it is the same for each orbit space. The spectral decomposition of H' can be written as H= of r, where the orbit co~d(E& } independent kinetic energy co = 2K [1 — cos(p ) ]. For each orbit momentum p, the momentum "eigenstate" ~p is given for each input expression s by

f

},

(6.4) Here ~OsOi 1) r—epresents the standard initial state of the computation with the particle p at position —1. Wave-packet states move along the orbit with no change in kinetic energy or momentum. The internal part of the state changes as the computation progresses, but this does not affect orbit motion. The orbit computation speed is given by U =2iK( V —V As the wave packet moves, it spreads out. Minimization of the effects of spreading, especially for a long computation, can be achieved by making the computation, including the recording steps, as efficient as possible. Additional details on the rate of wave-packet spreading, the eigenfunctions, and the spectrum of H' are given elsewhere [10,27]. When viewed as motion not on orbit states but as positions on a one-dimensional space lattice, the motion is quite different. For initial wave-packet states of the form %0 o gk ek Vo, o k with the TM in a standard initial state and p localized in the k &0 region of the lattice, p moves towards the origin with no interaction occurring. At positions —1 and 0, p interacts with the TM, which is equivalent to an external potential at these positions. The particle motion stops at lattice position 0 and the wave-packet dispersion shows up as a dispersion through steps of the computation given by iterations of T and T~. If and when the computation halts, the particle moves away from the k =0 position off to the region of k 0 with no further interaction occurring and the TM in a fixed final state. Dispersion shows up as wave-packet spread over positions of p on the lattice. This description also clearly applies to more general standard initial wave-packet states of the form 0 00 = gk ck % p p k which are superpositions of standard initial states with different input strings. As Deutsch [12] has observed, one quantum computation with such an input state is equivalent to carrying out many computations in parallel, each with a different input string. Of course, to build up statistics, such a computation would have to be repeated many times. The models developed here and in earlier work [10] are a generalization of the work of Deutsch [12,13] in that, as defined here, e ' ' is not local even if H is local. Deutsch [12] requires that a similar unitary operator U(t), which operates on a tape plus head system over a finite time interval t, be local. There does not seem to be any physical reason to require that U(t) be local in that exponentiation of a local operator gives a nonlocal operator.

}.

)

PAUL BENIOFF

3522

VII. PHYSICAL INTERPRETATION OF DILATION The physical interpretation of the dilated model used in this paper is that of a particle moving along a lattice and the TM fixed at positions —1 and 0 of the lattice. Other interpretations for the extra degree of freedom are also possible. Examples include motion of the TM along a lattice or motion of the head h along a lattice. Motion of the TM is not reasonable as it is hard to conceive of motion of an infinitely large system. Motion of the head along a lattice in the z direction with no extra particle present is possible since all head motion arising by iterations of T is in the x direction on the lattice. Other interpretations of the extra degree of freedom are possible. The locality of V depends on the physical interpretation of the extra degree of freedom. The main problem is with the term TPo in the expression for V as this expresses interactions between "the extra degree of freedom at 0" and the head h at arbitrary x positions on the lattice. If motion of an extra spinless particle along the lattice in the z direction is used, then there is no locality problem for V because I'z is the projection operator for finding the particle at z=0. Physically the particle can be incident along the negative z (k &0) lattice, join the head for computation motion in the x lattice direction, and exit in the positive z direction if and when the computation halts. V is also local if the extra degree of freedom consists of positions of h along the z lattice. This work was not funded through Argonne National

Laboratory. APPENDIX Unitary dilation of W [Eq. (5.8)] is given by Eq. (4.6) with 8'replacing T. Iteration of V on the initial state i=~OsO) —1) generates history components, but only when the second and the third terms of 8' are active. This can best be seen from Eq. (4.9), which is repeated here for convenience with 8'replacing T ~

(

yw)n@

—Wn —1D n

Note that W

—1

~OsO)

'D~tfl~&

=0, so D

t~OsO)

W"pin&.

(A— il

//

~~

o,

//

~~

j

j

active. The Turing machine model is expanded by adding a record tape and a two-position switch z. States of z are considered as internal states of h. Details are shown in Fig. 1. Computation basis states are expanded to include states for the recording tape and switch. That is, ~m~s') ~tk ) describes the expanded system with h in state m at position j, expressions s and t on the computation and record tapes, and switch z in state k (=0 or I). Each computing step is now divided into two steps, record or pass and act, where the act step follows the record or pass step. The record or pass operator R is given by

g

J=

[POQ&ip~+PohQ

Op

o

"] .

(A2)

oo

In this expression, P", Q.„and p are the projection operators already defined and o. is the Pauli spin opera-

= ~OsO). For those = W" fOsO) /f,

WW" /OsO)

values of n for which Dii W"~OLO) =0, so no history components are generated. This is the case when the first and the fourth terms of 8' are active. For all values of n such that the second or the third W" ~OsO) and term of Wis active, [[ WW" ~OsO) =2 ' Dii, W" ~OsO) =2 '~ W" ~OsO), so by Eq. (Al) history components are generated. Note that if the second term is first active on the mth iteration, i.e., m =n — 2 in Eq. (Al), then after n m iterations, there are n —m comwith components multiplied by ponents in ( V )"Vo,

)

j

8 =P", +

y~O)

+ X D~W" '

increasing powers of 2 ' as the sequence position decreases. The component at position 0 is multiplied by — — 2 (n m)/2 To aid in understanding the above, note that in Eq. (Al) the left-hand component is the component that may or may not generate history as n is increased. If the comsum plus the ponent is nonzero, then all terms in the right-hand term are history components. If and when the computation halts, the left-hand term is equal to 0 and the components in the sum express particle motion to values with no further changes in the machine positive state. Additional details are left to the reader. Because of the loss of normalization into the history components noted above, unitary power dilation of 8' is not useful. However, the results suggest that adding history recording steps to the second and the third terms of T only and then dilating the result will be satisfactory. This is based on the fact that the changed T is like the first example, the addition of 1(mod2), in that there are no overlapping terms. To see that this is the case, T [Eq. (5.7)] will first be expanded by adding recording for the second and the third terms only. The fact that the record cells initially all contain Os (are blank) is used to reduce the amount of history recording needed. For the example considered here, Os can be used as a record for all steps for which the third term is active. History recording is needed for just the one computation step for which the second term of T is

/f

R

0 0

0000000000000 0 0 C

1

1

1

1

1

1

1

0

1

b

h

~~

FIG. 1. Graphic representation of Turing machine with simple record system. The recording tape with one record cell for each cell of C is shown by R. The initial state of R with all cells containing 0 is shown. C is the computation tape. The switch z is added to h so internal states of h are (m, z) with z =0 or 1.

UNITARY DILATION MODELS OF TURING MACHINES

51

tor for exchanging 0 and 1 in the jth cell of the record tape. This operator shows that no history recording occurs except when the machine is in state 0 and h scans a 0 in C. For all other cases, it functions as a pass operator with no changes made.

The operator that carries out the compute step is given

IN. . .

3523

third terms of A. The cell and the head change operators u", and U" are unitary. The expanded model operator Y is constructed by using the switch z to turn on R and A in succession. Define

0,

Fby

F=o'qoR +0'q& A,

by where qo and q are projection operators for the switch z in states 0 and 1 and o. exchanges these states. It is straightforward to verify that iterations of F on model initial states carry out the model computation with history recording at only one step. An appropriate initial state is given by +o,ooo, which shows h in state 0 at position 0, expression s (containing a string of ls starting at position 1) on C, all record cells blank (indicated by 0), and z in position 0. Note that F annihilates the final state of the computation (which always halts) when h arrives in &

J=

[Pogltu p, oo

+U

&to., gjou "tp, ii, ,

+P~)g, u "tp II 0+ U"P", Q

ou

"p) j . (A3}

o and HJ& are projection operators for finding 0 and 1 in the jth record cell and o. changes the contents of the jth computation tape cell. As is the case for the Q projection operators, H o and Hk& are orthogonal only if k=j. U", u", andy. are as defined for T. The first to the fourth terms in A correspond, respectively, to the corresponding terms of T. Note that in A the first and the fourth terms carry out actions independent of the record. The second and the third terms depend on what is in the record cell at the position of h. Proper functioning of A requires that, as part of preparation of the initial state, all record cells are blank (contain 0) as this is used as a record for the third

Here H

term. Operators R and A have been constructed so that all nondiagonal terms in the four products R ~R, RR ~, A ~A, and AA~ are equal to 0. This is the function of the record cell projection operators in the second and the

[1] R. Landauer, IBM J. Res. Dev. 5, 183 (1961). [2] R. Landauer, Ber. Bunsenges. Phys. Chem. 80, 1048 (1976). [3] R. Landauer, Int. J. Theor. Phys. 21, 3/4, 283 (1982}. [4] R. Landauer, Phys. Today 44 (5), 23 (1991). [5] C. Bennett, IBM J. Res. Dev. 17, 525 (1973). [6] P. Benioff, J. Stat. Phys. 22, 563 (1980). [7] P. Benioff, Int. J. Theor. Phys. 21, 177 (1982). [8] P. Benioff, Phys. Rev. Lett. 48, 1581 (1982). [9] P. Benioff, J. Stat. Phys. 29, 515 (1982). [10] P. Benioff, Ann. N. Y. Acad. Sci. 480, 475 (1986). [11]R. Feynman, Opt. News, 11, 11 (1985) [reprinted in Found. Phys. 16, 507 (1986)]. [12] D. Deutsch, Proc. R. Soc. London Ser. A 400, 97 (1985). [13] D. Deutsch, Proc. R. Soc. London Ser. A 425, 73 (1989). [14] N. Margolus, Ann. N. Y. Acad. Sci. 480, 487 (1986). [15] N. Margolus, in Complexity, Entropy, and the Physics of Information, Santa Fe Institute Studies in the Sciences of Complexity, edited by W. Zurek (Addison Wesley, Redwood City, CA, 1990), Vol. VIII, pp. 273 —287. [16] A. Peres, Phys. Rev. A 32, 3266 (1985). [17] W. Zurek, Phys. Rev. Lett. 53, 391 (1984). [18] R. Jozsa, Proc. R. Soc. London Ser. A 435, 563 (1991). [19] D. Deutsch and R. Jozsa, Proc. R. Soc. London Ser. A 439, 553 (1992).

state 2.

It is easy to verify that the unitary dilation V of F has exactly the same properties as shown for the addition of 1(mod2}. No additional history recording occurs as all nondiagonal terms in Y F and FF are 0. This is the reason for the presence of the switch z. Iterations of V on % p oook with k & 0 move the particle to k =0. There it remains while the iterations of F carry the machine plus record system state through the model pass or record, and compute steps in succession. When h arrives in state 2, the computation is finished and further iterations of V move the particle along the lattice to positive k values with no changes in the machine plus record state. Details are left to the reader. The discussion in Sec. VA about wave-packet states also applies here without change.

[20] A. Yao, in Proceedings of the 34th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Los Alamitos, CA, 1993), pp. 352 —361. [21] E. Bernstein and U. Vazirani, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing (ACM, New York, 1993), pp. 11 —20. [22] S. Lloyd, Phys. Rev. Lett. 71, 943 (1993). [23] S. Lloyd (unpublished). [24] P. Shor, in Proceedings of the 35th Annual Symposium on of Computer Foundations Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), pp. 124-134. [25] D. Simon, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (Ref. [24]), pp.

116-122. [26] V. Cerny, Phys. Rev. A 48, 116 (1994). [27] P. Benioff, Trends in Statistical Physics (Council of Scientific Information, Trivandrum, India, in press). [28] B. Sz. Nagy and C. Foias, Harmonic Analysis of Operators in Hilbert Space {North-Holland, New York, 1970). [29] L. Horwitz, J. LaVita, and J. Marchand, J. Math. Phys. 12, 2537 {1971). [30] D. Evans and J. Lewis, Commun. Math. Phys. 50, 219 (1976). [31] A. Frigerio and C. Arateri, in Quantum Probability and

3524

PAUL BENIOFF

Applications IV, edited by L. Accardi and W. Von WalVol. 1396 denfels, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1987), pp. 89-98. [32] R. Landauer, Proc. R. Soc. London Ser. A (to be published).

[33] S. Lloyd, Science 261, 1569 (1993); 263, 695 (1994). [34) M. Reck, A. Zeilinger, H. Bernstein, and P. Bertani, Phys.

51

Rev. Lett. 73, 58 (1994). [35] P. Haltnos, A Hilbert Space Problem Book, 2nd ed. (Springer-Verlag, New York, 1982). [36] P. Fillmore, Notes on Operator Theory (Van Nostrand Reinhold, New York, 1970). [37] P. BeniofF (unpublished).