Energy needed to send a bit

Transcription

1 Energy needed to send a bit By Rolf Landauer IBM Thomas J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA Earlier work demonstrated that quantum-mechanical machinery can be used to transmit classical bits without a minimal unavoidable energy cost per bit. It is shown that a very minor variation of the earlier proposal also works for qubits. In addition, some casual remarks about error control in quantum parallelism are provided. Keywords: quantum communication; energy requirement; laws of physics; error immunity; bistable wells; reversible communication 1. Error control Benioff pointed out that reversible computation could be a totally coherent quantum process, if we allow ourselves to be optimistic and assume that systems can be untouched by the environment. Deutsch (1985) then pointed to the possibility of quantum parallelism, in which a computer can handle a coherent quantum superposition of different computational trajectories. This resulted in an active field which is celebrated by the companion papers in this issue. See Spiller (1996), Ekert & Jozsa (1996) and Barenco (1996) for reviews. This field was slow to face the need for error control, despite some early awareness of the need to do so (Peres 1985; Zurek 1984; and Landauer 1986). Eventually, however, Shor (1995) and Steane (1996) showed that, contrary to this author s published expectations (Landauer 1998), error-handling techniques are feasible. There is now a thriving industry of error correction proposals; we cite only two recent items (Steane 1997; Knill et al. 1997) and refer the reader to these, as well as to other papers in this issue, for further citations. Despite the unexpected and impressive progress in developing quantum mechanical redundancy and error correction techniques, difficulties remain. First of all, the actual physical techniques which have been suggested for quantum gates approximate the needed unitary transformations, they do not provide them exactly. They approximate them even if there are no errors in manufacturing, nor in the control of externally supplied pulses. Thus all gates, including those introduced to code and decode, deviate systematically from those that are needed. The physical computational machinery does not know about the unitary transform we would like to see executed. The machinery can only execute that which is embodied in its own physical structure. Perhaps the relationship of the redundancy codes, chosen with regard to the function of the gates, can provide a degree of stabilization against errors. But it would be appropriate for the literature to acknowledge more explicitly the problem we have emphasized. After all, the progress in error correction that has occurred came only after the need for that was acknowledged. Preskill (this volume) has made an effective start toward these additional error-control problems. Another difficulty, not yet clearly admitted in the existing literature, relates to 454, c 1998 The Royal Society Printed in Great Britain 305 TEX Paper

2 306 R. Landauer cross-talk, i.e. to unintended coupling between degrees of freedom not intentionally connected to each other in a gate structure. In the proposal by Cirac & Zoller (1995), for example, all ions are coupled to some degree. The actual physical structure, therefore, does not consist of a clean set of gates, each with one, two, or three inputs and the same number of outputs. Any theory that does not acknowledge this cannot be a totally convincing demonstration that quantum computation is feasible. Finally, I point out that error correction techniques which require that a gate executes its function with a probability of ending in an incorrect state, far less than 1%, are extremely optimistic. A computer structure requires a great many gates, each handling a somewhat complex analogue evolution. The chances that the error probability per gate can be contained to, say, less than 10 4, seem remote. An additional difficulty was recently pointed out by Myers (1997). If quantum parallelism is used for general purpose computation, then the well-known halting problem creates a difficulty. We will not know when all of the simultaneous and interfering computational histories are completed, and, therefore, will not know when we can make a measurement dependent on all the interfering histories. Undoubtedly, many special applications can avoid this problem. Communication, in contrast to computation, requires very limited handling of each bit, and is, therefore, far less demanding. Quantum cryptography (Muller et al. 1996) works, though its range of practical application is still unclear. As a result, in the remainder of this note, we emphasize the communications channel. 2. Energy required for bit transmission It has, historically, been assumed that it takes at least kt ln 2, where T is the temperature of the transmission channel and k is Boltzmann s constant, to send a bit. It has also been widely presumed that it takes more energy than that in a quantized channel if the photon energy, hν, associated with the signal frequency, is larger than kt. Landauer (1987, 1989) demonstrated that the classical channel, in principle, does not need to dissipate kt per transmitted bit. As emphasized, (Landauer 1996a) that conclusion was already implicit in reversible computation. But, despite its subsequent elaboration, the point is still not widely recognized by those who are primarily concerned with the theory of communication channels. The quantum case, however, is more complex. The older theories of quantum computation included only the information bearing degrees of freedom, without reference to the communication links. Furthermore, they specified Hamiltonians, not apparatus. The proposals in recent years which have specified specific apparatus have not attempted to minimize energy expenditure. They have typically invoked repeated external energy supply through radiative pulses or particle beams. Therefore, the energy requirement question has to be addressed more explicitly in the quantum case. A widely accepted result for a quantum channel used for classical bits, guaranteed tobeina0or1state, is (Caves & Drummond 1994) C = π 2P/3h, (2.1) ln 2 C is the bit transmission rate and P is the power. P/C is the energy per bit, and according to equation (2.1), is proportional to the bit rate. The energy per bit is,

3 Energy needed to send a bit 307 in its order of magnitude, equal to the quantum associated with the frequency corresponding to the bit rate. Equation (2.1) is applicable to a linear boson channel, and assumes that the power in the signal needs to be consumed. Communication, of course, need not be done via electromagnetic or acoustic waves. We can transport information engraved on stone tablets. Transmission of information via particle beams has been discussed on a few occasions, (see, for example, Pendry 1983), but typically without concern about our ability to recycle either the particles or their kinetic energy. We can easily improve on the limit, given in equation (2.1), as pointed out long ago (Landauer & Woo 1973). If we divide the available power among n channels, then the total channel capacity becomes C = π ln 2 n 2P/n 3h = π 2P n ln 2 3h. (2.2) Lowering the power in each channel reduces the frequency range invoked in each channel, and therefore the typical photon energy. Ultimately, for large enough n, ω <kt, and we face classical channels which, as discussed, clearly put no lower limit on the energy required per bit. Therefore, in principle, there is no real need to face the quantum channel. But there will be those whose taste leads them to ask what about a single undivided channel with a high bit rate? As a result, we continue with a more explicit discussion of that case. We will, in that analysis, make the same optimistic assumption made in other quantum information analyses, that coherent quantum machinery is available. Error control will, however, be relatively easy in our case. 3. Transport of bistable wells The general nature of the quantum communications scheme, for classical bits, proposed by Landauer (1996a) is shown in figure 1. Information is loaded into bistable wells moved along a carrier somewhat like a ski lift. The information-bearing particle sits either in the left-hand pocket or the right-hand pocket of a symmetrical bistable well. The particle is not in the ground state; in the ground state both pockets are occupied. But if we choose a sufficiently high barrier between the two pockets, then the energy elevation above the ground state can be made as small as desired. Furthermore, we also choose the barrier high enough to make the tunnelling, during information transport, unlikely. Note that I have said unlikely, not impossible. The handling of rare errors causes no difficulty, and will be discussed subsequently. If the two separate pockets of the bistable well are made sufficiently deep and narrow, then they will have very little polarizability. This minimizes elevation to higher lying intra-pocket states during acceleration or deceleration of the wells. Alternatively, we can make these velocity changes adiabatic. In the unloading step in figure 1, the information that has been carried in the bistable well is copied for subsequent use (the exact meaning of copy will become clearer later) and the particle in the well is reset to a standardized state, say 0. The resetting is done in interaction with the new copy; it is not simple erasure. The latter would be a dissipative step. Note that the time-dependent potentials we will invoke, including the moving bistable potential well, do not require the motion of machinery over long distances,

4 308 R. Landauer Figure 1. Schematic characterization of the communications link. Information is sent along the bottom link. Standardized bits (say, all 0) are returned in the top link. At each end, the wells slow down (or even stop) for loading and unloading. Figure 2. Controlling A well is perpendicular to controlled B well. The position of A controls the barrier of B. The A barrier can be increased temporarily to lock the A particle firmly in its position. except for the information-bearing particle. If we consider this to be a charged particle, then we can bring other charges toward or away from its path to generate any needed time-dependent force field. In mesoscopic physics time-dependent potentials, created via gate electrodes, are common. Charge-coupled devices (CCD), used in video cameras, transport electrons through time modulated potentials which cause an effective pocket to move in space. Figure 2 shows the B well which has just been returned and is known to be in the 0 state. The A well contains the qubit which is to be loaded, and which will control tunnelling in the B well. Let us assume attractive interaction between the two particles; the discussion to follow can easily be modified to accommodate the case of repulsive interaction. The barrier in the A well can, temporarily, be raised to insure that the particle in the A well will remain in its state, though that is not really necessary. The two wells, A and B, are perpendicular to each other, so that the position of the A particle controls the barrier height seen by the B particle. If A is in 0, and further from B, no tunnelling occurs in B. If A is in 1, and closer to the B barrier, the two particles are allowed to interact just long enough for the B particle to tunnel from 0 to 1. Let p, q denote the initial state, where both p and q can be 0or1andpdenotes the state of the A particle and q the state of the B particle. q is actually known to be 0 for the B particle which has just come back on the return link. If the initial state is 0, 0, no tunnelling occurs and after the time allowed for tunnelling, we end up in the same 0, 0 state. If the initial state is 1, 0, then the B particle tunnels and we end up in e iθ 1, 1. The factor e iθ represents the fact that if the A and B particles are closer, their interaction energy causes a different rate of phase advance than if A is in the 0 state, initially. Thus, if we are initially in a coherent superposition of the 0 and 1 state in the A well, we find: 0,0 +β 1,0 α 0,0 +βe iθ 1,1. (3.1) Now in the second part of the cycle we want to reset the A well to the 0 state, if it initially was in the 1 state, for further reuse. B now must control the tunnelling in

5 Energy needed to send a bit 309 A. To do this we turn both of the wells through 90 in space. If we start in the 0, 0 state, no tunnelling occurs in the A state, and we end up in the same 0, 0 state. If we are initially in the 1, 1 state, then tunnelling occurs and we end up in e iθ 0, 1, with the same reason for occurrence of the e iθ factor as in the first part of the cycle. Thus, the result given in equation (3.1) now changes as specified α 0, 0 + βe iθ 1, 1 0,0 +βe 2iθ 0, 1. (3.2) At the end the A particle is known to be in the 0 state. We are no longer in an entangled state, as was the case after the first part of the cycle. Therefore, α 0, 0 + βe 2iθ 0, 1 = 0 ( 0 +βe 2iθ 1 ). (3.3) In equation (3.3) on the right-hand side, the initial 0 denotes the state of the A particle. The final term on the right-hand side, including the α and β coefficients, represents the state of the B particle. Thus, the initial A state, at the beginning of the whole sequence, is now reflected in the B state, except for an undesired e 2iθ factor. We can easily eliminate such an undesired phase. We apply a temporary bias force to the B well, giving the 0 and 1 state different energies, and therefore differing rates of phase evolution. After that, the B state will be 0 +β 1, except for an irrelevant overall additional phase factor. Thus, the initial state in A has been loaded into B, through what is essentially a set of controlled-not or exclusive-or operations. Unloading and resetting at the right-hand end of figure 1 can be done in the same way. As already indicated, this machinery will have some error rate. Tunnelling may occur, when it is not intended. The time allowed for tunnelling may not be exactly correct. Acceleration will induce transitions to higher lying states. There will be some unintended interaction between far apart bits. This is not a true two-level system like an electron spin; when the controlled tunnelling is supposedly complete, there will be a little residual wave function left in the unintended well. We believe, in fact, that all proposals for quantum computation have defects of this sort. That is why, in 1, we stressed the deviation of the actual unitary time evolution, in each gate, from that needed to do the logic. In our communications link rare errors in a long chain of successive bits can easily be corrected through a little redundancy, using a small percentage of additional bits and encoding techniques. For classical bits this follows from Shannon s (1948) theory. But it is also true for qubits (Bennett et al. 1996). Not only do we need to correct errors in the transmitted bits, we also need to prevent the long-term accumulation of errors in the return link of figure 1. This can be handled by the technique discussed by Landauer (1996a), adapted from Lloyd (1993). During a portion of the return trip, a bias force favouring the desired 0 state is applied, possibly with a simultaneous barrier lowering. Inelastic tunnelling will cause the occasional 1 to become a 0. This is a dissipative process, but the energy dissipation will be proportional to the error rate, and can be made small by the use of accurate machinery. To make the radiative decay, invoked here for error correction, likely, in an environment where interaction with the environment has hopefully been minimized, extra precautions may have to be taken. The particle, for example, can be temporarily coupled to a damped cavity or oscillator tuned to the energy difference in the inelastic transition. An alternative to the long-range motion of wells, symbolized in figure 1, would invoke a shift register. Here the techniques discussed in connection with figure 2 would be used to move a bit from one bistable well to the next. In that case, however,

6 310 R. Landauer the number of operations carried out on a transmitted bit increases with the length of the link. This brings with it all the error handling problems of quantum computation, emphasized in Long-range interactions While we need the interaction between bits as invoked in figure 2, the unintentional interaction between other bits must be minimized. The route to this will be discussed, but not as definitively as might be desired. At the simplest level we can assume short-range forces, without specifying their origin. Alternatively, we can use a nearby metallic screening electrode. The image charges in that electrode will move slowly if the bit motion is slow and dissipation due to currents in the screening electrode can be kept as small as desired. In principle, the screening electrode can be superconducting, though that will not completely avoid dissipation. Note that slow motion does not necessarily result in a low bit rate. The bits can follow each other as closely as desired. The main problem with a metallic screening electrode comes not from the energy dissipation but from the associated decoherence. This is not a problem in the earlier scheme (Landauer 1996a) intended to transmit classical information, but is a difficulty for qubit transmission. We can, of course, invoke all the coding and redundancy apparatus invoked in quantum computation, to protect against decoherence. But that is unfair, after all we have emphasized that communication is simpler than computation. There are additional ways to diminish long-range interactions. One possibility is to represent bits by complimentary charges. Each positive charge is accompanied by a nearby negative charge, with both charges put through essentially identical operations. Also, instead of using charges in a well, we can use the bistable quadrupole moments invoked in quantum cellular automata proposals (Lent et al. 1993). Or, we can go further and use pairs of complimentary quadrupole patterns. A more quantitative analysis of the various alternatives we have listed seems unwarranted in this conceptual discussion. 5. Overview As stressed in the earlier discussion of quantum machinery used for classical bit transmission (Landauer 1996a), this discussion provides an existence theorem, not a serious technological proposal. The latter would require an invention which eliminates the need for active machinery all along the link. Perhaps a way can be found to do this with particle beams, without requiring unreasonable precision and without facing chaotic motion. Or, perhaps, photons can be used. But the loading and unloading steps may be hard to provide for photons. After all, charged particles have a natural Coulomb handle for interaction, whereas nonlinear optics typically require high energies which are likely to be accompanied by dissipation. In this author s view, the real motivation for understanding the handling of qubits in computation and communication does not come from the practical possibilities. Instead, it arises from a concern about the ultimate nature of the laws of physics. As discussed elsewhere (Landauer 1996b), these are essentially algorithms for information handling and must be executable in our real physical universe. Therefore, the computational power available to us in principle, in our actual universe, becomes relevant.

7 Energy needed to send a bit 311 References Barenco, A Quantum physics and computers. Contemp. Phys. 37, Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K Mixed-state entanglement and quantum error correction. Phys Rev. A 54, Caves, C. M. & Drummond, P. D Quantum limits on bosonic communication rates. Rev. Mod. Phys. 66, Cirac, J. I. & Zoller, P Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, Deutsch, D Quantum theory, the Church Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, Ekert, A. & Jozsa, R Quantum computation and Shor s factoring algorithm. Rev. Mod. Phys. 68, Knill, E., Laflamme, R. & Zurek, W. H Resilient quantum computation: error models and thresholds. Online preprint, quant-ph/ , 26 February Landauer, R Computation and physics: Wheeler s meaning circuit. Found. Phys. 16, Landauer, R Energy requirements in communication. Appl. Phys. Lett. 51, Landauer, R Computation, measurement, communication and energy dissipation. In Selected topics in signal processing (ed. S. Haykin), pp Englewood Cliffs, NJ: Prentice- Hall. (The printed version has some figures oriented incorrectly.) Landauer, R. 1996a Minimal energy requirements in communication. Science 272, Landauer, R. 1996b The physical nature of information. Phys. Lett. A 217, Landauer, R Is quantum mechanically coherent computation useful? In Proc. Drexel-4 Symp. on Quantum Nonintegrability Quantum Classical Correspondence, Philadelphia, PA, 8 September 1994 (ed. D. H. Feng & B.-L. Hu). Boston, MA: International. Landauer, R. & Woo, J. F Cooperative phenomena in data processing. In Synergetics (ed. H. Haken), pp Stuttgart: Teubner. Lent, C. S., Tougaw, P. D., Porod, W. & Bernstein, G. H Quantum cellular automata. Nanotechnology 4, Lloyd, S A potentially realizable quantum computer. Science 261, Lloyd, S A potentially realizable quantum computer. Science 263, 695. Muller, A., Zbinden, H. & Gisin, N Quantum cryptography over 23 km in installed underlake telecom fibre. Europhys. Lett. 33, Myers, J. M Can a universal quantum computer be fully quantum? Phys. Rev. Lett. 78, Pendry, J. B Quantum limits to the flow of information and entropy. J. Phys. A 16, Peres, A Reversible logic and quantum computers. Phys. Rev. A 32, Shannon, C. E A mathematical theory of communication. Bell Sys. Tech. J. 27, (parts I and II); (part III). Shor, P. W Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493 R2496. Spiller, T. P Quantum information processing: cryptography, computation, and teleportation. Proc. IEEE 84, Steane, A. M Error correcting codes in quantum theory. Phys. Rev. Lett. 77, Steane, A. M Active stabilization, quantum computation, and quantum state synthesis. Phys. Rev. Lett. 78, Zurek, W. H Reversibility and stability of information processing systems. Phys. Rev. Lett. 53,

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