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The Fundamental Physical Limits of Computation The Fundamental Physical Limits of Computation What constraints govern the physical process of computing? Is a minimum amount of energy required, for example, per logic step? There seems to be no minimum, but some other questions are open by Charles H. Bennett and Rolf Landauer computation, whether it is per- ber of other workers at other institu- In fact, computation as it is current- formed by electronic machin- tions have entered this field. ly carried out depends on many opera- A ery, on an abacus or in a biolog- In our analysis of the physical lim- tions that destroy information. The so- ical system such as the brain, is a physi- its of computation we use the term "in- called and gate is a device with two cal process. It is subject to the same formation" in the technical sense of input lines, each of which may be set at questions that apply to other physical information theory. In this sense infor- 1 or 0, and one output, whose value processes: How much energy must be mation is destroyed whenever two pre- depends on the value of the inputs. If expended to perform a particular com- viously distinct situations become in- both inputs are 1, the output will be 1. putation? How long must it take? How distinguishable. In physical systems If one of the inputs is 0 or if both are 0, large must the computing device be? In without friction, information can nev- the output will also be 0. Any time the other words, what are the physical lim- er be destroyed; whenever information gate's output is a 0 we lose informa- its of the process of computation? is destroyed, some amount of ener- tion, because we do not know which So far it has been easier to ask these gy must be dissipated (converted into of three possible states the input lines questions than to answer them. To the heat). As an example, imagine two eas- were in (0 and 1, 1 and 0, or 0 and 0). extent that we have found limits, they ily distinguishable physical situations, In fact, any logic gate that has more in- are terribly far away from the real lim- such as a rubber ball held either one put than output lines inevitably dis- its of modern technology. We cannot meter or two meters off the ground. If cards information, because we cannot profess, therefore, to be guiding the the ball is dropped, it will bounce. If deduce the input from the output. technologist or the engineer. What we there is no friction and the ball is per- Whenever we use such a "logically ir- are doing is really more fundamental. fectly elastic, an observer will always reversible" gate, we dissipate energy We are looking for general laws that be able to tell what state the ball start- into the environment. Erasing a bit of must govern all information process- ed out in (that is, what its initial height memory, another operation that is fre- ing, no matter how it is accomplished. was) because a ball dropped from two quently used in computing, is also fun- Any limits we find must be based sole- meters will bounce higher than a ball damentally dissipative; when we erase ly on fundamental physical principles, dropped from one meter. a bit, we lose all information about not on whatever technology we may If there is friction, however, the ball that bit's previous state. currently be using. will dissipate a small amount of ener- Are irreversible logic gates and era- There are precedents for this kind gy with each bounce, until it eventual- sures essential to computation? If they of fundamental examination. In the ly stops bouncing and comes to rest are, any computation we perform has 1940's Claude E. Shannon of the Bell on the ground. It will then be impos- to dissipate some minimum amount Telephone Laboratories found there sible to determine what the ball's ini- of energy. are limits on the amount of informa- tial state was; a ball dropped from As one of us (Bennett) showed in tion that can be transmitted through a two meters will be identical with a ball 1973, however, they are not essential. noisy channel; these limits apply no dropped from one meter. Information This conclusion has since been demon- matter how the message is encoded will have been lost as a result of ener- strated in several models; the easiest of into a signal. Shannon's work repre- gy dissipation. these to describe are based on so-called sents the birth of modern information reversible logic elements such as the science. Earlier, in the mid- and late ere is another example of informa- Fredkin gate, named for Edward Fred- 19th century, physicists attempting to H tion destruction: the expression kin of the Massachusetts Institute determine the fundamental limits on 2 + 2 contains more information than of Technology. The Fredkin gate has the efficiency of steam engines had cre- the expression =4. If all we know is three input lines and three outputs. ated the science of thermodynamics. that we have added two numbers to The input on one line, which is called In about 1960 one of us (Landauer) yield 4, then we do not know whether the control channel, is fed unchanged and John Swanson at IBM began at- we have added 1 + 3, 2 + 2, 0 + 4 or through the gate. If the control channel tempting to apply the same type of some other pair of numbers. Since the is set at 0, the input on the other two analysis to the process of computing. output is implicit in the input, no com- lines also passes through unchanged. If Since the mid-1970's a growing num- putation ever generates information. the control line is a 1, however, the CONVENTIONAL COMPUTING DEVICES, the abacus and the dissipative because of friction between its beads and rods. It could logic chip, both dissipate energy when they are operated. The 'Vog- not be built of frictionless components: if there were no static fric- ic gates"central to the design of a chip expend energy because they tion, the beadsypositions would change under the influence of ran- discard information. A chip consumes energy for a less fundamen- dom thermal motion. Static friction exerts a certain minimum force tal reason as well: it employs circuits that draw power even when no matter what the beads' velocity, and so there is some minium en- they merely hold information and do not process it. The abacus is ergy that the abacus requires no matter how slowly it is operated. outputs of the other two lines are if we erase them, it will cost us all it to its original state--without dissi- switched: the input of one line be- the energy dissipation we have been pating any energy. comes the output of the other and vice trying to avoid. versa. The Fredkin gate does not dis- Actually these bits have a most im- o far we have discussed a set of log- card any information; the input can al- portant use. Once we have copied S ic operations, not a physical device. ways be deduced from the output. down the result of our computation, It is not hard, however, to imagine Fredkin has shown that any logic de- which will reside in the normal output a physical device that operates as a vice required in a computer can be im- bits, we simply run the computer in Fredkin gate. In this device the infor- plemented by an appropriate arrange- reverse. That is, we enter the "gar- mation channels are represented by ment of Fredkin gates. To make the bage bits" and output bits that were pipes. A bit of information is repre- computation work, certain input lines produced by the computer's normal sented by the presence or absence of a of some of the Fredkin gates must be operation as "input" into the "back ball in a particular section of pipe; the preset at particular values [see lower il- end" of the computer. This is possible presence of a ball signifies a 1 and the lustration below]. because each of the logic gates in the absence of a ball signifies a 0. Fredkin gates have more output computer is itself reversible. Running The control line is represented by a lines than the gates they are made to the computer in reverse discards no in- narrow segment of pipe that is split simulate. In the process of computing, formation, and so it need not dissipate lengthwise down the middle. When a what seem to be "garbage bits," bits of any energy. Eventually the computer ball enters the split segment of pipe, it information that have no apparent use, will be left exactly as it was before the pushes the two halves of the pipe are therefore generated. These bits computation began. Hence it is possi- apart, actuating a switching device. must somehow be cleared out of the ble to complete a "computing cyclew- The switching device channels any in- computer if we are to use it again, but to run a computer and then to return put balls that may be in the other two pipes: when a ball is present in the con- trol line, any ball that enters an input AND GATE OR GAT pipe is automatically redirected to the INPUT OUTPUT INPUT OUTPUT other pipe. To ensure that the switch is A A closed when no control ball is present, there are springs that hold the two B halves of the split pipe together.
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