Information, Physics and Computation 1 Introduction

Information, Physics and Computation Subhash C. Kak



Received April 26, 1995 Abstract

This paper presents several observations on the connections between information, physics and computation. In particular, the computing power of quantum computers is examined. Quantum theory is characterized by superimposed states and non-local interactions. It is argued that recently studied quantum computers, which are based on local interactions, cannot simulate quantum physics.

1 Introduction We ask the question: What is the relationship between information and computation, and what is the physical basis of this relationship? This will be done by de ning the context for the kind of physics that can be simulated by a computer. The other side of this question is: How do classical and quantum computers compare? Information has a central role in quantum theory. Although Schrodinger's equation is linear, the reduction of the wave packet, upon observation, is a nonlinear phenomenon. Observation is thus tantamount to making a choice [20]. From another perspective, there is an asymmetry in the preparation of a state in a quantum system and that of its measurement, because the measurement can only be done in terms of its observables. Furthermore, the uncertainty in the measurement of complementary variables de nes the fundamental ground entropy associated with such a system [9, 10]. In a classical system also there is intervention at the beginning and the end of the physical process that de nes the computation. The question of simulating physics was raised by Feynman in 1982 [5]. Simulation can be an ambiguous term. Feynman de ned it as the computer doing \exactly the same as nature... [And] everything that happens in a nite volume of space and time would have to be exactly analyzable with a nite number  Department of Electrical & Computer Engineering, Louisiana State University, Baton Rouge, LA 70803-5901

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of logical operations." (page 468) Elsewhere (page 467), Feynman prefers that such a machine be locally interconnected and he states clearly that he rules out a computer where the computation is related in an exponential relationship to the size of the problem. If we were to cast this de nition in terms of computational complexity classes, such simulation excludes the NP and the PSPACE classes. The fact that the many body problem of mechanics is unsolvable indicates that a simulation of an arbitrary physical system in the sense of Feynman should not be possible in general. In practice, conventional computers might be perfectly adequate because we deal with systems that are isolated so that nonlinearities that lead to chaotic behavior can be ignored. In his paper [5], Feynman implicitly relaxed his own declared requirements for simulation and considered the computation of problems for which proper theory exists. In other words, he was now considering the computation of the models of tractable physical systems. A simulation is a computation on the projection of the complete model on the space associated with the variable of interest. It is in this limited sense that we will use the term simulation in the remainder of this paper. Feynman considered quantum mechanical systems and argued that classical computers cannot simulate certain aspects of quantum mechanics. This was one of the motivations for the development of the idea of quantum computers. It was assumed that quantum computers will take advantage of the superpositions that are inherent in a quantum description and thereby do more than what classical computers can achieve. For accounts of research in the quantum computing area see [2, 6, 3, 15, 17, 16, 19]. The development of the idea of a quantum mechanical computer has paralleled, more or less, the structure of a classical computer. Whether such a structure, based on local interconnections, exhausts the capabilities of quantum computing, we do not know yet. Is it possible that current quantum computing models, which are visualized to operate on lattice atoms, do not capture the entire complexity of the processing in a quantum physical system? In other words, are quantum computers too constrained to be able to simulate a quantum physical process? This article examines these issues.

2 Energy and computing speed A much discussed perspective on the computation process is that of energy requirement. One may begin here by considering computation as a physical process; a computer may be viewed in terms analogous to a heat engine. Now the equations of classical as well as quantum mechanics are time-reversible. Since one can, in principle, use reversible models of computation [13, 14, 15, 2, 7]

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there will be no energy requirement in such situations. But if computation is carried out with reversible gates the speed of computation is reduced greatly [15]. Furthermore, loading and erasing of information comes with its energy costs. Clearing the circuits expends energy. A computer must dissipate at least kTln2 (about 3  10?21 J at room temperature) per bit of irreversible information. The minimum, with respect to the T0 = 2:7 K background microwave radiation, is 2:583  10?23 J. The characteristic time for information processing in the presence of this background noise is: t0 = kT hln2  = 4:08  10?12 sec: 0 This indicates that the characteristic data rate is 245  109 bits=sec. Other characteristic units, with regard to information, are that of mass and length. Mass is obtained by:  0:287  10?36 grams: m0 = kTc02ln2 = On the other hand, length may be measured by: 2 l0 = kTe ln2  = 8:056  10?4 cms: 0 These units of mass and length are a bit smaller than the Planck units.

3 On computable numbers Physical theories are based on the continuum so what are the issues related to computing these models? According to the Church-Turing thesis, a function is computable if it can be computed by a Turing machine. Being a constructive procedure, the physical realizability of a Turing machine|within the constraints of a large enough memory|is guaranteed. The number of algorithms is countable because they can be put in a one-to-one correspondence with the integers. Therefore, predictions by algorithms cannot form a continuum. Thus limits of sequences of computable numbers are generally non-computable. A computable number is one whose digits can be computed p by a nite algorithm. A nite algorithm can generate in nite strings, and so 2, e, and  are computable numbers. Classical physics is represented in terms of dierential equations which are in turn often represented by the corresponding dierence equations. The representation of the continuum by discrete points may be objected to, since most irrational numbers are non-computable.

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The universe of classical objects appears to have non-computable numbers, although the numbers do not appear in any computation. Consider the problem: Find a; for which the transformation x ! ax(1 ? x) gives an in nite period sequence for x 2 (0; 1): This point is at the end of the sequence of period doubling bifurcations that are in the sequence 3, 3.449490,..., and so on, where we witness the rst onset of chaos [18]. This point is inferred as being the limit point of another sequence of super-attractors 2.0, 3.23606.., 3.49856.., 3.55464... and so on. This point appears to be not-computable since this value can only be de ned as a limit point. Nevertheless, we do acknowledge that a transition to such an a occurs in nature during chaotic behavior. If we consider the logistic map x ! 4x(1 ? x) for x 2 (0; 1); it can be easily shown that a chaotic sequence is obtained if we start with an x that is irrational [18]. If it is accepted that chaotic sequences in nature are generated by this or similar maps, then Turing computability further requires that the initial conditions will always be Turing computable irrational numbers. When space and time are viewed as a discrete lattice, the diculty with the non-computable numbers goes away. But the problem with this view is that it leads to anisotropies in physics, as well as preferred directions aligned to the lattice, that have never been observed. It is sometimes argued that non-computable numbers have nothing to do with the physical world and that a classical physical system can be represented by a Turing machine. In such a view, the chaotic sequences obtained in nature are either long-period sequences or computable in nite period sequences. Thus consider the physical version of the Church-Turing hypothesis [3]: Every nitely realizable physical system can be perfectly simulated by a universal model computing machine operating by nite means. Although the context does not

make it clear, it may be assumed that this statement implies the measurements generated by a physical system, rather than the mathematical models of it. In other words, physical models may not be computable but physics is.

3.1 Computability in quantum theory

Notwithstanding the physical Church-Turing hypothesis [3], the simulation of quantum interactions by a computer is problematic. Feynman [5] argued that \it is impossible to represent the results of quantum mechanics with a classical universal device" (page 476). In other words, a probabilistic classical computer cannot represent the equations of quantum mechanics. This is owing to the fact that no satisfactory hidden-variable models of quantum mechanics, that oer a deterministic substratum in terms of unknown variables have been found. The probability amplitudes of quantum theory are like the \complex square roots" of probabilities. Since the evolution of a system takes place in terms of these amplitudes we nd interference between quantum mechanical alternatives.

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Quantum theory has a structure which, if expressed in classical terms, can amount to the use of negative probabilities [5]. Such negative probabilities cannot be simulated on a classical computer. Furthermore, we have quantum non-locality, so that a computer that operates solely on local correlations cannot capture this behavior. Feynman argued that a quantum computer should be able to simulate such probabilities and, therefore, it should be able to simulate any physics. But Deutsch [3] claims that, although a quantum computer may solve a speci c problem much faster, a universal quantum computer and a classical Turing machine are equivalent in the capability to solve problems. So the question arises: Why should a quantum computer be able to simulate quantum

physics if a classical computer cannot?

Benio [2] and Feynman [6] de ned quantum computation in terms of a suitable Hamiltonian. It was suggested that a quantum computer could be visualized on a lattice where the data could be represented by the spins of the atoms. But this suggestion implies discretization of space. Through externally applied forces, time could likewise be discretized. In [6], the computation was visualized as being carried out through quantum analogs of reversible switches. Deutsch [3] examined certain characteristics of a quantum universal computer visualized in terms analogous to a classical universal computer with twostate observables. Such a machine is a generalization of a classical Turing machine, and it can simulate any nitely realizable physical system; but the nite assumption excludes many quantum systems. Deutsch claimed that one property of such a quantum computer, that is not reproducible by a classical computer, is that of `quantum parallelism'. But quantum parallelism only implies a speed-up and not the capability to solve problems that are unsolvable otherwise. Quantum computers, in the style of Deutsch, are eectively no better than a Turing machine. The claims of Deutsch are clearly at variance with the assertion of Feynman that classical and quantum computations are not equivalent. In view of this, Feynman's assertion [5] that a lattice system can simulate any quantum system needs to be questioned (page 475). Furthermore, Deutsch's view needs to be questioned. Certainly, if a classical computer cannot simulate a quantum system, classical and quantum computers should not be equivalent computationally. Is it that the assumption at the basis of the current quantum computing models, namely that of discretization of space and time, renders quantum computation much more limited than it can be otherwise? To look at the comparative capabilities of classical and quantum models, consider the billiard ball computer of Fredkin and Tooli [7]. In such a model, the shape of the container represents the hardware, and the initial conditions of the balls de nes the software for a speci c computation. The collisions are elastic so the computation proceeds without any energy loss. This model does suer

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from the problem that there would be some uncertainties in the initial speeds and directions, and the errors would quickly build up. But one might assume that some kind of an error-correction system is employed at each collision. Now consider the quantum analog of this model. In the Feynman path integral formulation of quantum mechanics [4], one needs to compute the sum of an in nite number of paths at each step. To summarize: 1. The probability P(a; b) of a particle moving from point a to point b is the square of the absolute value of a complex number, the transition function K(a; b): P(a; b) = jK(a; b)j2: 2. The transition function is given by the sum of a certain phase factor, which is a function of the action S, taken over all possible paths from a to b: K(a; b) =

X kei S=h ; 2

paths

where the constant k can be xed by K(a; c) =

X K(a; b)K(b; c);

paths

and the intermediate sum is taken over paths that go through all possible intermediate points b. This second principle says that a particle \snis" out all possible paths from a to b, no matter how complicated these paths might be. In quantum theory a future state, within the bounds of time and space uncertainty, can also in uence the present. In brief, even if the workings of such a billiard-ball model are approximated by consideration of a nite number of paths, it requires the tracking of an exponentially exploding number of entangled histories. This suggests that the determination of the hardware (shape of the container) for this problem, while being de nable in terms of a Hamiltonian, will not be computationally feasible. Classical computation is possible due to localization of balls, and a true quantum system does not permit localization. Furthermore, correction of errors at the collision sites, subsequent to some observation, will collapse the state into speci c ones, and the computer will become no dierent from its classical analog. If no corrections are made, the balls would spread inside in an unpredictable fashion and no useful computations would be possible.

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4 Computation and observation Time-reversible equations of physics cannot, in themselves, explain communication of information. Creation of information requires reduction in entropy. Owing to the fact that these models must carry the input along, reversible models can become extremely slow, so as to become unable to solve the problem in any reasonable period of time. Since measurement in a quantum system is a time-asymmetric process, one can speak of information transfer in a quantum observation. Let v; with eigenvalue equation vj n i = vn j ni; be the dynamical variable of the system S being measured. Let the measurement apparatus A be characterized, correspondingly, by the eigenvalue equation: M jAni = Mn jAn i: Let the system and the apparatus be in the states P i and A0i, respectively at the beginning of the measurement, where i = m am j m i. The state of the system and the apparatus, S + A, will be iA0 i. The Schrodinger equation ih dtd j i = H j i will now de ne the evolution of this state. If one uses the reduction postulate, the state j ijA0i collapses to j m ijAm i with the probability jam j2. But if one uses the Schrodinger equation then the initial state evolves into the unique state:

Xa m

m j m ijAM i

If one were to postulate another apparatus measuring the system plus the original apparatus the reduction problem is not solved. In fact one can visualize such a process in the so-called von Neumann chain. According to Wigner [21], the reduction should be ascribed to the observer's mind or consciousness. According to the orthodox view, namely the Copenhagen interpretation, the workings of the measuring instruments must be accounted for in purely classical terms. This means that a quantum mechanical representation of the measuring apparatus in not correct. In other words, a measurement is contingent on localization. In such a view the slogan that `computation is physics' loses its generality. It is not surprising that certain types of computations are taken to indicate the participation of conscious agents. Thus if we were to receive the digits of  in transmissions from outer space, we will take that as indication of life in some remote planet. The same will be true of other arithmetic computations. In brief, signals or computations that simulate the world using models much less complex than the real world indicate intelligence.

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5 Quantum computer proposals In these proposals Hamiltonians|Hermitian operators|are speci ed that cause an interacting set of up and down spins (like 0s and 1s) to evolve in time, analogous to a time-reversible computer. The Hamiltonians could be time dependent, as in the proposal of Benio [2], or one could visualize the motion of a wave packet along a periodic lattice, as was suggested by Feynman [6]. In Feynman's system, reversible logic gates, each made up of several interacting spins, are laid next to a chain of \clock" spins. Only one gate is active at a time step and the clock spins carry the computation forward. In this form, the system is a serial model. Parallel models have also been proposed. Lloyd [17] suggested that a quantum computer could be assembled out of organometallic polymers, where laser pulses could send the superposed states down the polymer chain analogous to electrons owing down a wire. Landauer [15, 16] has subjected these proposals for quantum computing to a penetrating scrutiny. He wonders: \Physical Hamiltonians are Hermitian; that does not mean that Hermitian operators are physically realizable. Why the speci cation of Hamiltonians has become accepted in this eld as equivalent to a computer design is a remarkable mystery."[16] He points to two major problems with these implementations:  Localization. If one visualized transfer of state using alternating Hamiltonians H1, H2. For example, H1 executing a certain computational step, such as changing a bit from 0 to 1, whereas H2 executes the reverse of it, which is the restoration of the 0 bit. If the energy dierence between the two states is given by
E, then the transfer of the state should be done exactly
t = h=
E seconds later. One sees that a slight error in time can lead to parts of the state being left behind. If the states are pushed in one direction by the erection of impenetrable barriers, then the states can get re ected in the wrong direction. Localization, as arising out of such unintended re ections, can cause the system to lose coherence.  Errors in the Hamiltonian. If the actual Hamiltonian is H = Hc + He where He is the error component, then this will also cause the computational trajectory to go o on incorrect tracks. Correction of errors will destroy quantum coherence.

6 Biological computing Animals are much more ecient than traditional computers at solving many problems. Examples of this are vision, speech understanding, and a host of other pattern recognition problems [12].

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The fundamental dierence between algorithmic and biological systems at the level of memories is that the latter are associative. Although algorithmic systems that are associative can be easily designed, such systems lack the capacity to reorganize themselves based on the nature of the problem. This reorganization of the biological system de nes the right context. The reorganization is out of a very large number of possibilities. Such a reorganization process may be compared to the sequence of operations in a Turing machine. It may be claimed that the associative memory paradigm is a generalization of the table look-up. We conclude then that a starting point which is generally not considered a legitimate computation leads, in association with the reorganization of the biological processing system, to a paradigm that has not been matched by traditional computers [11, 12]. The following question may be asked: Does the reorganization process de ne operations that cannot be simulated by a Turing machine?

7 Concluding remarks This paper has presented a short overview of the problem of computing models of physics. The laws of classical as well as quantum physics are time-reversible so the question of what kind of computations can be performed by classical or quantum devices becomes relevant. Computation need not be irreversible, but it does not follow that any computation can be performed by a speci c model of reversible computation. For example, aspects of classical as well as quantum physics cannot be simulated by classical computers. The study of quantum computers has been motivated by the search for computing structures that would be better than classical computers at simulating quantum physics. It is known that quantum mechanical versions of Turing machines are equivalent to the Turing machine in their ability to solve problems, although such machines might solve some problems faster. Our examination of this problem of information and computation points to two conclusions:  Currently analyzed quantum computers, which mimic traditional computers, are too limited to be able to simulate quantum physics.

 The diculties regarding the implementation of quantum Hamiltonians

suggest that such computers may not be realizable due to the localization problem. Reminding ourselves that physics is computable, because each physical system may be viewed as a computing machine, the question as to why certain models of physics remain non-computable may be asked. Does one conclude that we have not exhausted the range of all possible computing models? Whether

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new approaches to quantum computing, dierent from the current straightforward generalization of Turing machines, can be developed remains an open and intriguing problem. On the other hand, since localized states in a polymer, or quantum dots in a semiconductor, or spins in a lattice do support a quantum process, there clearly exist quantum computing structures in nature that could, in principle, be harnessed. The idea that protein polymers, to be found in the microtubules of neuronal cells, could support quantum processes[8] raises the intriguing possibility that a hybrid quantum + classical system may oer something unique. We also suggest that there exist such unresolved issues in understanding the nature of biological computing that the claim that the universe is a computer is as simplistic as the one from the 18th century that spoke of a clockwork universe.

Acknowledgements

The author wishes to thank R. Landauer, J. Ramanujam, and J.P. Sutton for comments on earlier versions of this paper.

References

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